Finiteness of for some Elliptic Curves of Analytic Rank >1>1

1 Introduction

For an elliptic curve $\mathcal{E}$ over a global field, it is conjectured that the Tate-Shafarevich group is finite. This is known in many cases, and in particular for function fields, when the analytic rank of $\mathcal{E}$ is at most $1$ ([Ulmer, Qiu, Kolyvagin]). However, there are very few examples when the conjecture is known when the analytic rank is $>1$ . For function fields the only previously known examples correspond to elliptic surfaces which are rational, K3 ([ASD]), or dominated by products of curves (e.g., [UlmerLargeRank]).

One of the purposes of this paper is to prove this finiteness for a new class of examples in the function field case. Let $k$ be a finite field of characteristic $p.$ Then we show

Theorem A.

Assume $p\geq 5$ . Let $\pi\colon X\to C$ be a minimal elliptic surface (with a zero section) over a smooth proper $k$ -curve $C$ of genus $1$ and $\mathcal{E}=X_{\eta}$ be the fiber over the generic point $\eta\in C$ . Suppose that

(i)

$\mathrm{ht}(\mathcal{E})=1$ ;
(ii)

all geometric fibers of $\pi$ are irreducible.

Then $\Sha(\mathcal{E})$ is finite.

Here $\mathrm{ht}(\mathcal{E})$ denotes the height of $\mathcal{E}$ , i.e., the degree of the fundamental line bundle $L\coloneqq(R^{1}\pi_{*}\mathcal{O}_{X})^{\vee}$ . We remark that by the Shioda-Tate formula, condition (ii) is equivalent to $\mathrm{rank\,}\mathrm{NS}(X)=\mathrm{rank\,}\mathrm{MW}(\mathcal{E})+2$ , where $\mathrm{NS}(X)$ is the Néron-Severi group of $X$ and $\mathrm{MW}(\mathcal{E})$ is the Mordell-Weil group of $\mathcal{E}$ . Moreover, on the coarse moduli space of $\pi\colon X\to C$ satisfying all conditions except possibly (ii) in the theorem, (ii) is generically satisfied, and there are infinitely many $9$ -dimensional families of $\mathcal{E}$ such that $\mathrm{rank\,}\mathrm{MW}(\mathcal{E})\geq 2$ up to replacing $k$ by a finite extension (see (7.4.5)).

For function fields, it is known that the finiteness of is equivalent to the BSD conjecture ([KatoTrihan, MilneAT]), which is further equivalent to the Tate conjecture for the corresponding elliptic surfaces. In fact, Thm A is obtained by refining a special case of a more general theorem on the Tate conjecture. To explain this, we consider arithmetic families of the following type: Suppose that $K$ is a field of characteristic $0$ and $S$ is a smooth connected $K$ -variety. We say that a smooth projective morphism $f:\mathcal{X}\to S$ is a $\heartsuit$ -family if for every geometric point $s\to S$ , the fiber $\mathcal{X}_{s}$ is connected, $\dim\mathrm{H}^{0}(\Omega^{2}_{\mathcal{X}_{s}})=1$ (i.e., $h^{2,0}(\mathcal{X}_{s})=1$ ), and the Kodaira-Spencer map

\nabla\colon T_{S/K}\to\underline{\mathrm{Hom}}(R^{1}f_{*}\Omega^{1}_{\mathcal{X}/S},R^{2}f_{*}\mathcal{O}_{\mathcal{X}})

(1)

is nontrivial. Our general theorem states:

Theorem B.

Let $\mathsf{M}$ be a connected and separated scheme over $\mathrm{Spec}(\mathbb{Z})$ which is smooth and of finite type and let $f\colon\mathcal{X}\to\mathsf{M}$ be a smooth projective morphism with geometrically connected fibers. If the restriction of $\mathcal{X}$ to $\mathsf{M}_{\mathbb{Q}}$ is a $\heartsuit$ -family, then for $p\gg 0$ , every fiber of $\mathcal{X}$ over a point $s\in\mathsf{M}_{\mathbb{F}_{p}}$ satisfies the Tate conjecture in codimension $1$ , i.e., for every prime $\ell\neq p$ , the Chern class map

c_{1}:\mathrm{Pic}\,(\mathcal{X}_{s}){\otimes}\mathbb{Q}_{\ell}\to\mathrm{H}^{2}_{\mathrm{{\acute{e}}t}}(\mathcal{X}_{\bar{s}},\mathbb{Q}_{\ell}(1))^{\mathrm{Gal}(\bar{s}/s)}

is surjective, where $\bar{s}$ is a geometric point over $s$ .¹¹1Over a finite field there is also a crystalline version of the Tate conjecture, which in codimension $1$ is known to be equivalent to the $\ell$ -adic version (see [Morrow, Prop. 4.1]).

The above theorem is roughly saying that under a mild assumption, the Tate conjecture in codimension $1$ is “generically true” in an arithmetic family of varieties with $h^{2,0}=1$ . Clearly, it is both a positive characteristic analogue of Moonen’s main result in [Moonen], and a generalization of the Tate conjecture for K3 surfaces, for which people made great progress in the past decade (e.g., [MPTate, Maulik, Charles]). The base scheme $\mathsf{M}$ above should be thought of as the moduli of varieties of a certain type. Note that we do not require in the theorem above that $s$ is a closed point—it is allowed to have positive transcendence degree over $\mathbb{F}_{p}$ . Also, although the above theorem is non-effective in $p$ , for concretely given families we often can make it effective, as in the case of Thm A. To illustrate this, we also analyze a class of surfaces of general type:

Theorem C.

Assume that $k$ is a field finitely generated over $\mathbb{F}_{p}$ for $p\geq 5$ . Let $X$ be a minimal smooth projective geometrically connected surface over $k$ . Let $K_{X}$ denote its canonical divisor, $p_{g}$ the geometric genus $h^{0}(K_{X})$ , and $q$ the irregularity $h^{1}(\mathcal{O}_{X})$ . If $p_{g}=K_{X}^{2}=1$ , $q=0$ and $K_{X}$ is ample, then $X$ satisfies the Tate conjecture.

We remark that condition (ii) in Thm A is an analogue of the condition “ $K_{X}$ is ample” above. Over $\mathbb{C}$ , surfaces with the above invariants were classified by Todorov and Catanese ([Todorov, Catanese0]). They are simply connected, have a coarse moduli space of dimension $18$ and were among the first examples of $p_{g}=1$ surfaces for which both the local and the global Torelli theorems fail ([Catanese]).

Sketch of Proofs

We first explain how to prove Theorem B. We build on the overall strategy of Madapusi-Pera [MPTate], which has two main steps. The first is to construct an integral period morphism $\rho:\mathsf{M}\to\mathscr{S}$ , where $\mathscr{S}$ is the canonical integral model of a Shimura variety $\mathrm{Sh}(G)$ defined by a suitable special orthogonal group $G$ .²²2For the exposition of ideas, we temporarily suppress Hermitian symmetric domains and level structures from the notation of Shimura varieties. Up to passing to its spinor cover, $\mathscr{S}$ is equipped with a family of abelian schemes $\mathcal{A}$ . The second is to construct, for each geometric point $s\to\mathsf{M}$ and $t=\rho(s)$ , a morphism $\theta:\mathrm{L}(\mathcal{A}_{t})\to\mathrm{NS}(\mathcal{X}_{s})$ , where $\mathrm{L}(\mathcal{A}_{t})$ is a distinguished subspace of $\mathrm{End}(\mathcal{A}_{t})$ . Then the Tate conjecture for $\mathrm{NS}(\mathcal{X}_{s})$ follows from a variant of Tate’s theorem for $\mathrm{L}(\mathcal{A}_{t})$ . As a crucial step, Madapusi-Pera proved that $\rho$ is étale. This boils down to the geometric fact that a K3 surface $X$ has unobstructed deformation and $\mathrm{H}^{1}(\Omega_{X})\,{\cong}\,\mathrm{H}^{1}(T_{X})$ . Unfortunately, this is rarely true when $X$ is not a close relative of a hyperkähler variety.

The main contribution of our paper is a method to remove the dependence of the above strategy on any good local property of $\rho$ (not even flatness). Indeed, the condition on the Kodaira-Spencer map contained in the definition of a $\heartsuit$ -family just amounts to asking $\dim\mathrm{im}(\rho_{\mathbb{C}})>0$ , so even the situation $\dim\mathsf{M}<\dim\mathscr{S}$ is allowed in Thm B. Below we explain in more detail the difficulties in extending the two steps above and how to overcome them.

(1.0.1)

It is not hard to construct a period morphism $\rho_{\mathbb{C}}:\mathsf{M}_{\mathbb{C}}\to\mathrm{Sh}(G)_{\mathbb{C}}$ over $\mathbb{C}$ , as the target is a moduli of variations of Hodge structures with some additional data. As in [MPTate], the idea to construct $\rho$ is to descend $\rho_{\mathbb{C}}$ to a morphism $\rho_{\mathbb{Q}}$ over $\mathbb{Q}$ , and then appeal to the extension property of $\mathscr{S}$ . If for every $s\in\mathsf{M}(\mathbb{C})$ , the motive $\mathfrak{h}^{2}(\mathcal{X}_{s})$ defined by $\mathrm{H}^{2}(\mathcal{X}_{s},\mathbb{Q})$ is an abelian motive (e.g., in the K3 case), then one shows that the action of $\rho_{\mathbb{C}}$ on $\mathbb{C}$ -points are $\mathrm{Aut}(\mathbb{C})$ -equivariant, so that $\rho_{\mathbb{C}}$ descends to $\mathbb{Q}$ just as in [MPTate].

To treat the general case, we absorb inputs from Moonen’s work [Moonen]. Let $b\in\mathsf{M}(\mathbb{C})$ be a point lying over the generic point of $\mathsf{M}$ , and let $\mathrm{T}(\mathcal{X}_{b})$ be the orthogonal complement of $(0,0)$ -classes in $\mathrm{H}^{2}(\mathcal{X}_{b},\mathbb{Q}(1))$ . Let $E$ be the endomorphism algebra of the Hodge structure $\mathrm{T}(\mathcal{X}_{b})$ , which is known to be either a totally real or a CM field. In the latter case, one can still show that $\mathfrak{h}^{2}(\mathcal{X}_{s})$ is abelian for each $s$ (see (2.2.7)), so that the argument of [MPTate] still applies. In the former case, we need to consider an auxiliary Shimura subvariety $\mathrm{Sh}(\mathcal{G})_{\mathbb{C}}\subseteq\mathrm{Sh}(G)_{\mathbb{C}}$ , defined by the Weil restriction $\mathcal{G}$ of a special orthogonal group over $E$ .

Up to replacing $\mathsf{M}$ by a connected étale cover and $b$ by a lift, the restriction of $\rho_{\mathbb{C}}$ to $\mathsf{M}^{\circ}$ factors through a morphism $\varrho_{\mathbb{C}}:\mathsf{M}^{\circ}\to\mathrm{Sh}(\mathcal{G})_{\mathbb{C}}$ , where $\mathsf{M}^{\circ}$ is the connected component of $\mathsf{M}_{\mathbb{C}}$ containing $b$ . We will show that the field of definition $F$ of $\mathsf{M}^{\circ}$ always contains $E$ . Interestingly, to descend $\rho_{\mathbb{C}}$ to $\mathbb{Q}$ it suffices to descend $\varrho_{\mathbb{C}}$ to $F$ . The trick is to consider the left adjoint to the base change functor from $\mathbb{Q}$ -schemes to $F$ -schemes. To show that $\varrho_{\mathbb{C}}$ descends to $F$ , consider the submotive $\mathfrak{t}(\mathcal{X}_{b})$ of $\mathfrak{h}^{2}(\mathcal{X}_{b})$ defined by $\mathrm{T}(\mathcal{X}_{b})$ . When $\dim_{E}\mathrm{T}(\mathcal{X}_{b})$ is odd, although we do not know that $\mathfrak{t}(\mathcal{X}_{b})$ (or equivalently $\mathfrak{h}^{2}(\mathcal{X}_{b})$ ) is abelian, Moonen’s work [Moonen] tells us that (a slight variant of) “the $E/\mathbb{Q}$ -norm” $\mathrm{Nm}_{E/\mathbb{Q}}(\mathfrak{t}(\mathcal{X}_{b}))$ is abelian. This allows us to show that $\varrho_{\mathbb{C}}$ descends to $F$ because considering $\mathrm{Nm}_{E/\mathbb{Q}}(\mathfrak{t}(\mathcal{X}_{b}))$ amounts to considering a different faithful representation of $\mathcal{G}$ . As we document in a separate reference file [YangSystem], under some hypotheses the canonical models of Shimura varieties of abelian type over their reflex fields have a moduli interpretation attached to any faithful representation. When $\dim_{E}\mathrm{T}(\mathcal{X}_{b})$ is even, some adaptation is needed, which we omit here.

(1.0.2)

Next, we explain how to construct $\theta:\mathrm{L}(\mathcal{A}_{t})\to\mathrm{NS}(\mathcal{X}_{s})$ , where the main novelty of our method lies. As in [MPTate], the key to construct $\theta$ is to find, for each $\zeta\in\mathrm{L}(\mathcal{A}_{t})$ , a characteristic $0$ point $\widetilde{t}$ on $\mathscr{S}$ lifting $t$ , such that (i) $\zeta$ deforms to $\mathcal{A}_{\widetilde{t}}$ and (ii) $\widetilde{t}$ comes from a lifting $\widetilde{s}$ of $s$ via $\rho$ . The existence of $\widetilde{t}$ which satisfies (i) was already shown in [CSpin]. When $\rho$ is étale (or at least smooth), (ii) is then automatically satisfied. This is where [MPTate] crucially relies on the étaleness of $\rho$ .

The challenge to generalize this, especially when $\dim\mathsf{M}<\dim\mathscr{S}$ , is that there is no general way to characterize locally the image of $\rho$ in $\mathscr{S}$ , so for any given $\widetilde{t}$ one cannot decide directly whether it satisfies (ii) or not. Indeed, this is essentially a local Schottky problem, which is famously hard. To overcome this, we revisit Deligne’s insight for [Del02, Thm 1.6] but replace the local crystalline analysis by a global and topological argument.

Pretend for a moment that $\mathsf{M}_{\mathbb{C}}$ and $\mathsf{M}_{k}$ are both connected, where $k$ is the (separably closed) field defining $s$ , and set $p=\mathrm{char\,}k$ . Let $\bar{\eta}$ and $\bar{\eta}_{p}$ be geometric generic points over $\mathsf{M}_{\mathbb{C}}$ and $\mathsf{M}_{k}$ respectively. Pick a prime $\ell\neq p$ and restrict $f:\mathcal{X}\to\mathsf{M}$ to $\mathbb{Z}_{(p)}$ . Suppose for the sake of contradiction that for some $\zeta$ , there is no lifting $\widetilde{t}$ which satisfies (i) and (ii). Then using that the deformation of $\zeta$ is controlled by a single equation, we can show that $\zeta$ deforms along the formal completion of $\mathsf{M}_{k}$ at $s$ , and hence gives rise to an element of $\mathrm{L}(\mathcal{A}_{\rho(\bar{\eta}_{p})})$ . This further induces an element in $(R^{2}f_{*}\mathbb{Q}_{\ell})_{\bar{\eta}_{p}}$ which is stabilized by an open subgroup of $\pi_{1}^{\mathrm{{\acute{e}}t}}(\mathsf{M}_{k},\bar{\eta}_{p})$ . On the other hand, we show using the theorem of the fixed part that all elements of $(R^{2}f_{*}\mathbb{Q}_{\ell})_{\bar{\eta}}$ stabilized by an open subgroup of $\pi_{1}(\mathsf{M}_{\mathbb{C}},\bar{\eta})$ come from $\mathrm{L}(\mathcal{A}_{\bar{\eta}}){\otimes}\mathbb{Q}_{\ell}$ . Therefore, to derive a constradiction it suffices to show that $\mathsf{M}_{k}$ does not have more “ $\pi_{1}$ -invariants” than $\mathsf{M}_{\mathbb{C}}$ .

Using Hironaka’s resolution of singularities and a spreading out argument, we can find an open subscheme $U$ of $\mathrm{Spec\,}(\mathbb{Z})$ such that $\mathsf{M}_{U}$ admits a compactification whose boundary is a relative normal crossing divisor. Then we apply Grothendieck’s specialization theorems for tame fundamental group and Abhyankar’s lemma to show that $\mathsf{M}_{k}$ indeed cannot have more “ $\pi_{1}$ -invariants” than $\mathsf{M}_{\mathbb{C}}$ , when $(p)\in U$ .³³3We thank Aaron Landesman for pointing out to us the applicability of Abhyankar’s lemma, which simplifies our original argument. This proves Thm B.

(1.0.3)

To prove Theorem A and C, we need to avoid the spreading out argument above. Although compactifying moduli spaces is in general a hard geometric problem, one can find for $\mathsf{M}$ in question a partial compactification. That is, a morphism $\mathsf{M}\to B$ and a smooth proper $\mathscr{P}$ over $B$ such that $\mathsf{M}$ is an open subscheme of $\mathscr{P}$ . Then we can find lots of smooth proper curves in $\mathscr{P}$ . If the boundary $\mathfrak{D}=\mathscr{P}-\mathsf{M}$ is generically reduced modulo $p$ , then we can find a curve on $\mathsf{M}_{k}$ which deforms to characteristic $0$ such that by looking at the curve we can already prove that $\mathsf{M}_{k}$ does not have more “ $\pi_{1}$ -invariants” than $\mathsf{M}_{\mathbb{C}}$ . Of course, such curve needs to be chosen wisely, and we do this by repeatedly applying the Baire category theorem. We will also need $\rho$ to satisfy a stronger condition, namely $\dim\mathrm{im}(\rho_{\mathbb{C}})>\dim B_{\mathbb{C}}$ , as opposed to just $\dim\mathrm{im}(\rho_{\mathbb{C}})>0$ . This is known to hold for the surfaces in question.

The boundary $\mathfrak{D}$ is essentially a discriminant scheme, i.e., $\mathcal{X}$ extends to a family over $\mathscr{P}$ , and $\mathfrak{D}$ is precisely the locus where the extension fails to be smooth. In general, it is possible for a discriminant scheme over $\mathbb{Z}$ to be generically non-reduced modulo a certain prime (cf. [Saitodisc, Thm 4.2]), so the task is to determine an effective range of $p$ for which this does not happen. Drawing ideas from enumerative geometry, we show that this happens only when a general fiber over $\mathfrak{D}_{\bar{\mathbb{F}}_{p}}$ is “more singular” than that over $\mathfrak{D}_{\mathbb{C}}$ . To exclude this possibility when $p\geq 5$ for the surfaces in Thm C, it suffices to adapt Katz’ results on Lefschetz pencils. Doing this for Thm A is much more involved. In particular, we need to develop some nonlinear Bertini theorems (see §7.2) tailored to handle Weierstrass equations, the key input being Kodaira’s classification for the singular fibers in an elliptic fibration.

Remark (1.0.4).

Recently Fu and Moonen proved the Tate conjecture for Gushel-Mukai varieties in characteristic $p\geq 5$ . The middle cohomology of these varieties behaves like that of a K3 surface up to a Tate twist. An earlier version of the paper also discussed these varieties. For our method, these varieties can be treated in a similar way as the surfaces in Thm C. However, as Fu-Moonen [FuMoonen] gave much more thorough treatment of these varieties and proved that the relevant integral period morphism $\rho$ is indeed smooth, we removed the section from the current version. In general, it is hard to determine whether $\rho$ is smooth integrally even when $\rho_{\mathbb{C}}$ is smooth, as one can tell from [FuMoonen], but when this can be achieved, $\rho$ has many more potential applications than the divisorial Tate conjecture (e.g., CM lifting and the Tate conjecture for self-correspondences, as shown in [IIK]).

On the other hand, our main purpose is to deal with the situation when the smoothness of $\rho$ cannot be hoped for. In particular, Thm B implies that the characteristic $p$ counterparts of the surfaces in Moonen’s list [Moonen, Thm 9.4] satisfy the Tate conjecture for $p\gg 0$ , and we expect that our methods for the refinements Thm A and C can be adapted to make Thm B effective for most classes of these surfaces.

Organization of Paper

In section 2, we review and mildly extend Moonen’s results in [Moonen] on the motives of fibers of $\heartsuit$ -families. In particular, we recap the norm functors used in loc. cit. In section 3, we discuss the moduli interpretations of Shimura varieties of abelian type over their reflex fields, as documented in [YangSystem], and recap the integral models of orthogonal Shimura varieties from [CSpin]. In section 4, we construct the period morphisms for $\heartsuit$ -families in characteristic $0$ , using results from sections 2 and 3. In section 5, we prove Thm B after giving a more effective version (see (5.3.3)). In section 6, we set up some basic tools to analyze deformation of curves on parameter spaces, including applications of the Baire category theorem. Finally, in section 7 and 8, we use tools from section 6 as well as (5.3.3) to prove Thm A and C respectively. In particular, in section 7 we study the geometry of natural parameter spaces of elliptic surfaces, which we hope to be of independent interest.

(1.0.5)

Finally we introduce some notations and conventions.
(a) Let $f:X\to S$ be a morphism between schemes. If $T\to S$ is another morphism, denote by $X_{T}$ the base change $X\times_{S}T$ and by $X(T)$ the set of morphisms $T\to X$ as $S$ -schemes. By a geometric fiber of $X$ , we mean $X_{s}$ for some geometric point $s\to S$ . If $x$ is a point (resp. geometric point) on a scheme $X$ , we write $k(x)$ for its residue field (resp. field of definition). By a variety over a field $k$ we mean a scheme which is reduced, separated and of finite type over $k$ .
(b) The letters $p$ and $\ell$ will always denote some prime numbers and $\ell\neq p$ unless otherwise noted. We write $\widehat{\mathbb{Z}}$ for the profinite completion of $\mathbb{Z}$ and $\widehat{\mathbb{Z}}^{p}$ for its prime-to- $p$ part. Define $\mathbb{A}_{f}:=\widehat{\mathbb{Z}}{\otimes}\mathbb{Q}$ and $\mathbb{A}^{p}_{f}:=\widehat{\mathbb{Z}}^{p}{\otimes}\mathbb{Q}$ . If $k$ is a perfect field of characteristic $p$ , we write $W(k)$ for its ring of Witt vectors.
(c) For a field $k$ of characteristic $\neq 2$ , we consider a quadratic form $q$ on a finite dimensional $k$ -vector space $V$ simultaneously a symmetric bilinear pairing $\langle-,-\rangle$ such that $q(v)=\langle v,v\rangle$ for every $v\in V$ .
(d) For a finite free module $M$ over a ring $R$ , we write $M^{\otimes}$ the direct sum of all the $R$ -modules which can be formed from $M$ by taking duals, tensor products, symmetric powers and exterior powers. We also use this notation for sheaves of modules on some Grothendieck topology whenever it makes an obvious sense.
(e) We use the following abbreviations: VHS for “variations of Hodge structures”, LHS (resp. RHS) for “left (resp. right) hand side”, and ODP for “ordinary double point”. Unless otherwise noted, the local system in a VHS is a local system of $\mathbb{Q}$ -vector spaces. Moreover, we always assume that the VHS is pure, i.e., it is a direct sum of those pure of some weight (or the weight filtration is split).

2 Preliminaries

2.1 Motives and Norm Functors

(2.1.1)

Let $k$ be a field of characteristic $0$ . We denote by $\mathsf{Mot}_{\mathrm{AH}}(k)$ the neutral $\mathbb{Q}$ -linear Tannakian category of motives over $k$ with morphisms defined by absolute Hodge correspondences (cf. [Pan94, §2] where it is denoted by $\mathscr{M}_{k}$ ). We have the Tate objects $\mathbf{1}(n)$ for every $n\in\mathbb{Z}$ in this category. For any object $M\in\mathsf{Mot}_{\mathrm{AH}}(k)$ , we write $M(n)$ for $M{\otimes}\mathbf{1}(n)$ and by an absolute Hodge cycle on $M$ we mean a morphism $\mathbf{1}\to M$ .

Following [MPTate, §2], we denote by $\omega_{\ell}$ the $\ell$ -adic realization functor which sends $\mathsf{Mot}_{\mathrm{AH}}(k)$ to the category of finite dimensional $\mathbb{Q}_{\ell}$ -vector spaces with an action of $\mathrm{Gal}_{k}:=\mathrm{Gal}(\bar{k}/k)$ , where $\bar{k}$ is some chosen algebraic closure of $k$ . Putting $\omega_{\ell}$ ’s together we obtain $\omega_{\mathrm{{\acute{e}}t}}$ , which takes values in the category of finite free $\mathbb{A}_{f}$ -modules with a $\mathrm{Gal}_{k}$ -action. Let $\omega_{\mathrm{dR}}$ denote the de Rham realization functor, which takes values in the category of filtered $k$ -vector spaces. If $k=\mathbb{C}$ , we additionally consider the Betti realization $\omega_{B}$ (resp. the Hodge realization $\omega_{\mathrm{Hdg}}$ ) which takes values in the category of $\mathbb{Q}$ -vector spaces (resp. Hodge structures). For a smooth projective variety $X$ over $k$ , $\mathfrak{h}^{i}(X)$ denotes the object such that $\omega_{?}(\mathfrak{h}^{i}(X))$ is the $i$ th $?$ -cohomology of $X$ , for $?=B,\ell,\mathrm{dR}$ whenever applicable.

Let $\mathsf{Mot}_{\mathsf{Ab}}(k)\subseteq\mathsf{Mot}_{\mathrm{AH}}(k)$ be the full Tannakian sub-category generated by the Artin motives and the motives attached to abelian varieties. We will repeatedly make use of the following fact ([LNM900, Ch I], cf. [MPTate, Thm 2.3]):

Theorem (2.1.2).

The functor $\omega_{\mathrm{Hdg}}$ is fully faithful when restricted to $\mathsf{Mot}_{\mathsf{Ab}}(\mathbb{C})$ . In particular, for every $M\in\mathsf{Mot}_{\mathsf{Ab}}(\mathbb{C})$ , every element $s\in\omega_{B}(M)\cap\mathrm{Fil}^{0}\omega_{\mathrm{dR}}(M)$ is given by an absolute Hodge cycle.

We often refer to objects in $\mathsf{Mot}_{\mathsf{Ab}}(k)$ as abelian motives.

(2.1.3)

We will often consider the automorphism $-{\otimes}_{\sigma}\mathbb{C}$ on $\mathsf{Mot}_{\mathrm{AH}}(\mathbb{C})$ defined by an element $\sigma\in\mathrm{Aut}(\mathbb{C})$ (cf. [LNM900, §II 6.7], see also [MPTate, Prop. 2.2]). For $M\in\mathsf{Mot}_{\mathrm{AH}}(\mathbb{C})$ , we write $M^{\sigma}$ for $M{\otimes}_{\sigma}\mathbb{C}$ . Base change properties of étale (resp. de Rham) cohomology give us a $\mathbb{A}_{f}$ -linear (resp. $\sigma$ -linear) canonical isomorphism $\omega_{\mathrm{{\acute{e}}t}}(M)\,{\cong}\,_{\mathrm{bc}}\omega_{\mathrm{{\acute{e}}t}}(M^{\sigma})$ (resp. $\omega_{\mathrm{dR}}(M)\,{\cong}\,_{\mathrm{bc}}\omega_{\mathrm{dR}}(M^{\sigma})$ ). Here the subscript “bc” is short for “base change”. For an absolute Hodge class $s\in\omega_{B}(M)$ , we write $s^{\sigma}$ for the class in $\omega_{B}(M^{\sigma})$ which has the same étale and de Rham realizations as $s$ under the “ $\,{\cong}\,_{\mathrm{bc}}$ ” isomorphisms.

Finally, we remark that for $M\in\mathsf{Mot}_{\mathrm{AH}}(\mathbb{C})$ , it makes sense to say whether a tensor $s\in\omega_{B}(M)^{\otimes}$ (see (1.0.5)(d)) is absolute Hodge, because $s$ has to lie in $\omega_{B}$ of a finite direct sum of tensorial constructions on $M$ . And when $s$ is absolute Hodge, we may form $s^{\sigma}\in\omega_{B}(M^{\sigma})^{\otimes}$ for $\sigma\in\mathrm{Aut}(\mathbb{C})$ , extending the notation in the previous paragraph.

(2.1.4)

Next, we recall the basics of norm functors. The reader may refer to [Moonen, §3] for more details. Let $k$ be a field of characteristic $0$ and $E$ be a finite field extension of $k$ . Let $\mathscr{C}$ be any Tannakian $k$ -linear category and $\mathscr{C}_{(E)}$ be the category of $E$ -modules in $\mathscr{C}$ . For any object $M\in\mathscr{C}_{(E)}$ , we write $M_{(k)}$ for the underlying object in $\mathscr{C}$ when we forget the $E$ -linear structure. In [Ferrand], Ferrand gave a general construction of a norm functor $\mathrm{Nm}_{E/k}\colon\mathscr{C}_{(E)}\to\mathscr{C}$ , which was summarized in [Moonen, §3.6]. Note that we restricted to the case when $E$ is a field, as opposed to a general étale $k$ -algebra in loc. cit. This is good enough for our purposes.

We first consider the case when $\mathscr{C}$ is the category of $k$ -modules $\mathsf{Mod}_{k}$ . For any $M\in\mathsf{Mod}_{E}$ , there is a functorial polynomial map $\nu_{M}\colon M\to\mathrm{Nm}_{E/k}(M)$ such that $\nu_{M}(em)=\mathrm{Norm}_{E/k}(e)\nu_{M}(m)$ for any $e\in E$ and $m\in M$ . The norm functor $\mathrm{Nm}_{E/k}$ is a ${\otimes}$ -functor and is non-additive (unless $E=k$ ). However, for any $M_{1},M_{2}\in\mathsf{Mod}_{E}$ , there is an identification

\mathrm{Nm}_{E/k}(\mathrm{Hom}_{E}(M_{1},M_{2}))=\mathrm{Hom}_{k}(\mathrm{Nm}_{E/k}(M_{1}),\mathrm{Nm}_{E/k}(M_{2})).

The above identification gives us a structural map

\eta:\mathrm{Res}_{E/k}\mathrm{GL}(\mathcal{V})\to\mathrm{GL}(\mathrm{Nm}_{E/k}(\mathcal{V}))

for any $\mathcal{V}\in\mathsf{Mod}_{E}$ , which sends an $E$ -linear automorphism $f$ to $\mathrm{Nm}_{E/k}(f)$ . Let $T_{E}$ denote the torus $\mathrm{Res}_{E/k}\mathbb{G}_{m,E}$ and $T_{E}^{1}$ denote the kernel of the norm map $T_{E}\to T_{k}$ . Viewing $\mathbb{G}_{m,E}$ as the diagonal torus of $\mathrm{GL}(\mathcal{V})$ , so that $T_{E}^{1}$ is a subgroup of $\mathrm{Res}_{E/k}\mathrm{GL}(\mathcal{V})$ , we have $\ker(\eta)=T^{1}_{E}$ .

Notation (2.1.5).

Take $k=\mathbb{Q}$ and $E$ to be a totally real field. Let $\mathcal{V}$ be an $E$ -vector space equipped with a quadratic form $\widetilde{\phi}:\mathcal{V}\to E$ . We often drop $\widetilde{\phi}$ from the notation when it is assumed.

(a)

Write $\mathcal{V}_{(\mathbb{Q})}$ for the underlying $\mathbb{Q}$ -vector space of $\mathcal{V}$ , equipped with a quadratic form $\phi:\mathcal{V}_{(\mathbb{Q})}\to\mathbb{Q}$ given by $\mathrm{tr}_{E/\mathbb{Q}}\circ\widetilde{\phi}$ .
(b)

Write $\mathcal{G}(\mathcal{V})$ for $\mathrm{Res}_{E/\mathbb{Q}}\mathrm{SO}(\mathcal{V})$ , $Z^{1}_{E}(\mathcal{V})$ for $T^{1}_{E}\cap\mathcal{G}(\mathcal{V})$ , and $\mathcal{H}(\mathcal{V})$ for $\mathcal{G}(\mathcal{V})/Z^{1}_{E}(\mathcal{V})$ , or simply $\mathcal{G},Z^{1}_{E}$ and $\mathcal{H}$ when $\mathcal{V}$ is understood.
(c)

Denote by $\mathrm{Cl}^{+}(\mathcal{V})$ the even Clifford algebra of $\mathcal{V}$ over $E$ and by $\mathrm{Cl}^{+}_{E/\mathbb{Q}}(\mathcal{V})$ its norm $\mathrm{Nm}_{E/\mathbb{Q}}\mathrm{Cl}^{+}(\mathcal{V})$ .
(d)

Let $\mathcal{E}$ denote the $1$ -dimensional quadratic form over $E$ given by equiping $E$ with the form $\widetilde{\phi}(\alpha)=\alpha^{2}$ .

Recall our convention (1.0.5)(c). The association $\mathcal{V}\mapsto\mathcal{V}_{(\mathbb{Q})}$ in (a) above defines an equivalence of categories between quadratic forms over $E$ and quadratic forms over $\mathbb{Q}$ with an self-adjoint $E$ -action ([Knus, Ch 1, Thm 7.4.1]). We call $\mathcal{V}_{(\mathbb{Q})}$ the transfer of $\mathcal{V}$ , and $\mathcal{V}$ the $E$ -bilinear lift of $\mathcal{V}_{(\mathbb{Q})}$ .

Lemma (2.1.6).

Let $E$ be a totally real field, and let $\mathfrak{w}\in\mathsf{Mot}_{\mathrm{AH}}(\mathbb{C})_{(E)}$ be a motive equipped with an $E$ -action and a symmetric $E$ -bilinear form $\widetilde{\phi}:\mathfrak{w}{\otimes}_{E}\mathfrak{w}\to\mathbf{1}_{E}$ . If $\dim_{E}\omega_{B}(\mathfrak{w})$ is odd, then $\mathfrak{N}(\mathfrak{w}):=\mathrm{Nm}_{E/\mathbb{Q}}(\mathfrak{w}){\otimes}_{\mathbb{Q}}\det(\mathfrak{w}_{(\mathbb{Q})})$ is (noncanonically) isomorphic to a submotive of $\mathrm{Cl}^{+}_{E/\mathbb{Q}}(\mathfrak{w},\widetilde{\phi})$ .

The meaning of $\mathrm{Cl}^{+}_{E/\mathbb{Q}}(\mathfrak{w},\widetilde{\phi})$ is explained in the proof.

Proof.

This follows from the content in [Moonen, §5.4]. We give a sketch so that the reader can easily check the details from loc. cit. Set $W:=\omega_{B}(\mathfrak{m})$ and consider $\mathfrak{m}$ as a representation of its motivic Galois group $G_{\mathrm{mot}}(\mathfrak{w})$ , which respects $\widetilde{\phi}$ and the $E$ -action. The motive $\mathrm{Cl}^{+}_{E/\mathbb{Q}}(\mathfrak{w},\widetilde{\phi})$ then corresponds to the $G_{\mathrm{mot}}(\mathfrak{w})$ -representation $\mathrm{Cl}^{+}_{E/\mathbb{Q}}(W,\widetilde{\phi})$ .

Let $\Sigma$ be the set of embeddings $\sigma:E\hookrightarrow\mathbb{C}$ , $\mathfrak{S}$ be the symmetric group of $\Sigma$ , and $Q$ be the set of $\mathfrak{S}$ -orbits of $\mathbb{N}^{\Sigma}$ . Then under the assumption that $\dim_{E}W$ is odd, there is an ascending filtration $\mathrm{Fil}_{\bullet}$ on $\mathrm{Cl}^{+}_{E/\mathbb{Q}}(\mathfrak{w},\widetilde{\phi})$ indexed by $Q$ such that for some $q_{1},q_{2}\in Q$ , $\mathrm{Fil}_{q_{1}}/\mathrm{Fil}_{q_{2}}\,{\cong}\,\mathfrak{N}(\mathfrak{w})$ . Therefore, $\mathfrak{N}(\mathfrak{w})$ is a subquotient of $\mathrm{Cl}^{+}_{E/\mathbb{Q}}(\mathfrak{w},\widetilde{\phi})$ . As $\mathsf{Mot}_{\mathrm{AH}}(\mathbb{C})$ is semisimple, $\mathfrak{N}(\mathfrak{w})$ is in fact (non-canonically) a subobject. ∎

2.2 Motives of Varieties with $h^{2,0}=1$

Definition (2.2.1).

A polarized Hodge structure $V$ of weight $0$ is said to be of K3-type if $\dim V_{\mathbb{C}}^{(-1,1)}=\dim V_{\mathbb{C}}^{(1,-1)}=1$ , and $V_{\mathbb{C}}^{i,j}=0$ when $|i-j|>2$ . The transcendental part $T(V)$ of $V$ is the orthogonal complement of $V^{(0,0)}:=V\cap V_{\mathbb{C}}^{(0,0)}$ .

We recall the following fundamental result of Zarhin:

Theorem (2.2.2) ([Zarhin, §2]).

Let $V$ be a Hodge structure of K3-type such that $V^{(0,0)}=0$ and let $\phi$ be its polarization form. Then the endomorphism algebra $E\coloneqq\mathrm{End}_{\mathrm{Hdg}}V$ is either a totally real field or a CM field, and the adjoint map $e\mapsto\bar{e}$ defined by $\phi(ex,y)=\phi(x,\bar{e}y)$ is the identity map when $E$ is totally real and is complex conjugation when $E$ is CM.

To discuss motives in families, we first give a definition:

Definition (2.2.3).

Let $S$ be a connected smooth $\mathbb{C}$ -variety. For every $\mathbb{Q}$ -local system $\mathsf{V}_{B}$ over $S$ and $b\in S(\mathbb{C})$ , we write $\mathrm{Mon}(\mathsf{V}_{B},b)$ for the Zariski closure of the image of $\pi_{1}(S,b)$ in $\mathrm{GL}(\mathsf{V}_{B,b})$ , and $\mathrm{Mon}^{\circ}(\mathsf{V}_{B},b)$ for its identity component. When $\mathsf{V}=(\mathsf{V}_{B},\mathsf{V}_{\mathrm{dR}})$ is a polarizable VHS⁴⁴4This means that $\mathsf{V}_{B}$ is the $\mathbb{Q}$ -local system and $\mathsf{V}_{\mathrm{dR}}$ is the filtered flat vector bundle in the VHS. Similar conventions apply throughout the paper. over $S$ , we say that $\mathsf{V}$ has maximal monodromy if $\mathrm{Mon}^{\circ}(\mathsf{V}_{B},b)$ is equal to the derived group of the Mumford-Tate group $\mathrm{MT}(\mathsf{V}_{b})$ , where $b$ is any Hodge-generic point.

For the terminology “Hodge-generic points”, see for example [Moonen-Fom, 31]. Note that [AndreMT, §5] says that in the above notation, $\mathrm{Mon}^{\circ}(\mathsf{V}_{B},b)$ is always a normal subgroup of the derived group of $\mathrm{MT}(\mathsf{V}_{b})$ (see also [Peters, Thm 16]).

(2.2.4)

For the rest of (2.2), let $S$ be a connected smooth $\mathbb{C}$ -variety and $f:\mathcal{X}\to S$ be a $\heartsuit$ -family of relative dimension $d$ . Let $\mathsf{V}=(\mathsf{V}_{B},\mathsf{V}_{\mathrm{dR}})$ be the VHS on $S$ defined by $R^{2}f_{*}\mathbb{Q}(1)$ . Note that by the definition of a $\heartsuit$ -family, $\mathsf{V}$ satisfies condition (P) in [Moonen, Prop. 6.4]. Let $\boldsymbol{\xi}$ be a relatively ample line bundle on $\mathcal{X}/S$ , which defines a symmetric bilinear pairing $\langle-,-\rangle$ on $\mathsf{V}$ such that $\langle\alpha,\beta\rangle=\alpha\cup\beta\cup c_{1}(\boldsymbol{\xi})^{d-2}$ for local sections $\alpha,\beta$ . Choose a Hodge-generic base point $b\in S(\mathbb{C})$ .

Suppose for a moment that $\mathrm{Mon}(\mathsf{V}_{B},b)$ is connected. Then for $\rho:=\dim\mathrm{NS}(\mathcal{X}_{b})_{\mathbb{Q}}$ , $\mathsf{V}$ admits an orthogonal decomposition $\mathbf{1}_{S}^{\oplus\rho}\oplus\mathsf{P}$ , where $\mathsf{P}$ is a VHS polarized by the pairing induced by $\boldsymbol{\xi}$ and $\mathbf{1}_{S}$ is the unit VHS. Note that the Hodge structure $\mathsf{P}_{b}=(\mathsf{P}_{B,b},\mathsf{P}_{\mathrm{dR},b})$ , together with its polarization, is of K3-type and satisfies the hypothesis of (2.2.2). Let $E$ be the endomorphism field of $\mathsf{P}_{b}$ . As $\mathrm{Mon}(\mathsf{V}_{B},b)=\mathrm{Mon}^{\circ}(\mathsf{V}_{B},b)\subseteq\mathrm{MT}(\mathsf{V}_{b})=\mathrm{MT}(\mathsf{P}_{b})$ and $\mathrm{MT}(\mathsf{P}_{b})$ commutes with $E$ , the $E$ -action on $\mathsf{P}_{B,b}$ commutes with $\pi_{1}(S,b)$ and hence extends to an action on $\mathsf{P}$ (see e.g., [Peters, Cor. 12]). If $\mathsf{V}$ (or equivalently $\mathsf{P}$ ) has maximal monodromy, we say that $(\mathcal{X}/S,\boldsymbol{\xi})$ is in case (R+) if $E$ is totally real and (CM) if $E$ is CM; we further divide (R+) into case (R1) for $\dim_{E}\mathsf{P}_{B,b}$ odd and case (R2) for $\dim_{E}\mathsf{P}_{B,b}$ even. We say we are in case (R2’) if $\mathsf{V}$ has non-maximal monodromy, which can only happen when $E$ is totally real and $\dim_{E}\mathsf{P}_{B,b}=4$ . See [Moonen, Prop. 6.4(iii)] and its proof.

If $\mathrm{Mon}(\mathsf{V}_{B},b)$ is not connected, we say that $(\mathcal{X}/S,\boldsymbol{\xi})$ is in case $?$ for $?$ = (R1), (R2), (CM) or (R2’) if it is in case $?$ up to replacing $S$ by a connected étale cover $S^{\prime}$ and $b$ by a lift such that $\mathrm{Mon}(\mathsf{V}_{B},b)$ becomes connected. The definition is clearly independent of these choices.

Proposition (2.2.5).

Suppose that $\mathsf{V}$ has non-maximal monodromy, or equivalently $(\mathcal{X}/S,\boldsymbol{\xi})$ belongs to case (R2’). Then for a general $s\in S$ and $X:=\mathcal{X}_{s}$ , the Kodaira-Spencer map $\nabla_{s}:T_{s}S\to\mathrm{Hom}(\mathrm{H}^{1}(\Omega^{1}_{X}),\mathrm{H}^{2}(\mathcal{O}_{X}))$ has rank $1$ .

Proof.

We may assume that $\mathrm{Mon}(\mathsf{V}_{B},b)$ is conncted, so that there is a decomposition $\mathbf{1}_{S}^{\oplus\rho}\oplus\mathsf{P}$ as above. The rank of $\nabla_{s}$ achieves its maximum on an open dense $U\subseteq S$ . Choose some $s\in U$ . By [Voisin, Thm 3.5] (cf. [Moonen, Prop. 6.4]), for some point $z$ in a small analytic neighborhood of $s$ , $\mathsf{P}_{B,z}^{(0,0)}$ contains a nonzero class $\zeta$ . Let $Z^{\prime}\subseteq S$ be the irreducible component of the Noether-Lefschetz loci defined by $(z,\zeta)$ and $Z$ be its smooth locus. Up to moving $z$ along $Z^{\prime}$ a little bit, we may assume that $z$ lies in $Z$ , is Hodge-generic for the VHS $\mathsf{P}|_{Z}$ , and $\mathrm{rank\,}\nabla_{s}=\mathrm{rank\,}\nabla_{z}$ . Let $W$ be the Hodge structure of K3-type defined by the fiber of $\mathsf{P}$ at $z$ and let $T$ be its transcendental part. Then $\dim_{E}T<\dim_{E}W=4$ . Set $F:=\mathrm{End}_{\mathrm{Hdg}}T$ , which contains $E$ .

By assumption on a $\heartsuit$ -family, $\mathrm{rank\,}\nabla_{s}>0$ . Suppose by way of contradiction that $\mathrm{rank\,}\nabla_{s}>1$ . Then as $Z\subseteq S$ has codimension $1$ (cf. [Voisin, Lem. 3.1]), $\nabla_{z}$ does not vanish on $T_{z}Z$ . This implies that $\mathrm{Mon}^{\circ}(\mathsf{P}_{B}|_{Z},z)\neq 1$ and $\dim_{E}T=3$ by [Moonen, Prop. 6.4] and its proof. Indeed, if $F$ is CM, then $\dim_{F}T\geq 2$ , so that $\dim_{E}T\geq 4$ , which is impossible. Therefore, $F$ is totally real and $\dim_{F}T\geq 3$ . This forces $E=F$ and $\dim_{E}T=3$ . Now let $\mathcal{T}$ be the $E$ -bilinear lift of $T$ . Then by [Zarhin, Thm 2.2.1] $\mathrm{MT}(T)=\mathrm{Res}_{E/\mathbb{Q}}\mathrm{SO}(\mathcal{T})$ . As this group is simple, $\mathrm{Mon}^{\circ}(\mathsf{P}_{B}|_{Z},z)\,{\cong}\,\mathrm{Res}_{E/\mathbb{Q}}\mathrm{SO}(\mathcal{T})$ . On the other hand, [Moonen, §8.1] tells us that $\mathrm{Mon}^{\circ}(\mathsf{P}_{B},s)\,{\cong}\,\mathrm{Res}_{E/\mathbb{Q}}L$ for some $E$ -form $L$ of $\mathrm{SL}_{2}$ . However, as $\mathrm{SL}_{2}$ and $\mathrm{SO}(3)$ are not even isomorphic over $\mathbb{C}$ , $\mathrm{Res}_{E/\mathbb{Q}}L$ cannot have a subgroup isomorphic to $\mathrm{Res}_{E/\mathbb{Q}}\mathrm{SO}(\mathcal{T})$ , which constradicts the fact that parallel transport (noncanonically) sends $\mathrm{Mon}^{\circ}(\mathsf{P}_{B}|_{Z},z)$ into $\mathrm{Mon}^{\circ}(\mathsf{P}_{B},s)$ . ∎

(2.2.6)

The following statements are the key inputs we will use from Moonen’s paper [Moonen]. A little adaptation we make is that we will uniformly use the category of motives $\mathsf{Mot}_{\mathrm{AH}}(\mathbb{C})$ with absolute Hodge cycles, whereas Moonen used André’s category with motivated cycles ([AndreMot]). This adaptation makes a difference only for (2.2.7)(d) below. We use $\mathsf{Mot}(\mathbb{C})$ to denote André’s category (for base field $\mathbb{C}$ ) when explaining this difference, but otherwise all motives are considered in $\mathsf{Mot}_{\mathrm{AH}}(\mathbb{C})$ . Note that motivated cycles are automatically absolute Hodge, so that $\mathsf{Mot}(\mathbb{C})$ is a subcategory of $\mathsf{Mot}_{\mathrm{AH}}(\mathbb{C})$ .

Recall our notations in (2.1.4) and (2.2.4) for the theorem below:

Theorem (2.2.7).

Assume that $\mathrm{Mon}(\mathsf{V}_{B},b)$ is connected and let $\mathsf{P}$ and $E$ be as in (2.2.4). Let $s\in S(\mathbb{C})$ be any point and write $\mathfrak{p}_{s}\in\mathsf{Mot}_{\mathrm{AH}}(\mathbb{C})$ for the submotive of $\mathfrak{h}^{2}(\mathcal{X}_{s})(1)$ such that $\omega_{\mathrm{Hdg}}(\mathfrak{p}_{s})=\mathsf{P}_{s}$ . Then the action of $E$ on $\mathsf{P}_{s}$ is absolute Hodge, i.e., induced by an action of $E$ on $\mathfrak{p}_{s}$ . Moreover:

(a)

In case (R1), $\mathrm{Nm}_{E/\mathbb{Q}}(\mathfrak{p}_{s}){\otimes}_{\mathbb{Q}}\det(\mathfrak{p}_{s,(\mathbb{Q})})$ is an object of $\mathsf{Mot}_{\mathsf{Ab}}(\mathbb{C})$ .
(b)

In case (R2), for $\mathbf{1}_{E}:=\mathbf{1}{\otimes}E$ and $\mathfrak{p}_{s}^{\sharp}:=\mathfrak{p}_{s}\oplus\mathbf{1}_{E}$ , $\mathrm{Nm}_{E/\mathbb{Q}}(\mathfrak{p}^{\sharp}_{s}){\otimes}_{\mathbb{Q}}\det(\mathfrak{p}^{\sharp}_{s,(\mathbb{Q})})$ is an object of $\mathsf{Mot}_{\mathsf{Ab}}(\mathbb{C})$ .
(c)

In case (R2’), $\mathrm{Nm}_{E/\mathbb{Q}}(\mathfrak{p}_{s})$ is an object of $\mathsf{Mot}_{\mathsf{Ab}}(\mathbb{C})$ .
(d)

In case (CM), $\mathfrak{p}_{s}$ is an object of $\mathsf{Mot}_{\mathsf{Ab}}(\mathbb{C})$ .

Proof.

The statement that the action of $E$ on $\mathsf{P}_{s}$ is absolute Hodge is implied by [Moonen, Prop. 6.6]. Note that our $\mathfrak{p}_{s}$ is Moonen’s $\boldsymbol{V}_{s}$ , and our $\mathsf{P}$ is Moonen’s $\mathbb{V}$ . Below we view $\mathfrak{p}_{s}$ as an object of $\mathsf{Mot}_{\mathrm{AH}}(\mathbb{C})_{(E)}$ . In this proof, $\underline{\mathrm{End}}$ (or $\underline{\mathrm{Hom}}$ ) always mean internal $\mathrm{End}$ (or $\mathrm{Hom}$ ) in the category $\mathsf{Mot}_{\mathrm{AH}}(\mathbb{C})$ .

(a) We first treat the case (R+). Let $\mathcal{V}$ be the $E$ -bilinear lift of the quadratic form given by $\mathsf{P}_{B,b}$ with its self-dual $E$ -action. By [Moonen, §6.9, 6.10] there is a family of abelian schemes $\mathscr{A}\to S$ with multiplication by $D:=\mathrm{Cl}^{+}_{E/\mathbb{Q}}(\mathcal{V})$ (see (2.1.5)) such that there is an isomorphism

\mathrm{Cl}^{+}_{E/\mathbb{Q}}(\mathfrak{p}_{s})\stackrel{{\scriptstyle\sim}}{{\to}}\underline{\mathrm{End}}_{D}(\mathfrak{h}^{1}(\mathscr{A}_{s}))

(2)

of algebra objects in $\mathsf{Mot}_{\mathrm{AH}}(\mathbb{C})$ . In case (R1), $\dim_{E}\mathcal{V}$ is odd, and by (2.1.6) $\mathrm{Nm}_{E/\mathbb{Q}}(\mathfrak{p}_{s}){\otimes}\det(\mathfrak{p}_{s,(\mathbb{Q})})$ is non-canonically a submotive of $\mathrm{Cl}^{+}_{E/\mathbb{Q}}(\mathfrak{p}_{s})$ , and hence is an object of $\mathsf{Mot}_{\mathsf{Ab}}(\mathbb{C})$ .

(b) In case (R2), (2) still holds, so that $\mathrm{Cl}^{+}_{E/\mathbb{Q}}(\mathfrak{p}_{s})$ is still an object of $\mathsf{Mot}_{\mathsf{Ab}}(\mathbb{C})$ , but a further trick is needed to recover (a variant of) $\mathrm{Nm}_{E/\mathbb{Q}}(\mathfrak{p}_{s})$ from $\mathrm{Cl}^{+}_{E/\mathbb{Q}}(\mathfrak{p}_{s})$ . Recall $\mathcal{E}$ defined in (2.1.5)(d). As in [Moonen, §6.11, 6.12], we consider the VHS $\mathsf{P}^{\sharp}:=\mathsf{P}\oplus(\mathbf{1}_{S}{\otimes}\mathcal{E}_{(\mathbb{Q})})$ , where $\mathbf{1}_{S}$ standards for the unit VHS on $S$ . Let $\mathcal{V}^{\sharp}:=\mathcal{V}\oplus\mathcal{E}$ and $D^{\sharp}:=\mathrm{Cl}^{+}_{E/\mathbb{Q}}(\mathcal{V}^{\sharp})$ . Then by [Moonen, §6.9, 6.10] again, there is an abelian scheme $\mathscr{A}^{\sharp}\to S$ with multiplication by $D^{\sharp}$ such that

\mathrm{Cl}^{+}_{E/\mathbb{Q}}(\mathsf{P}^{\sharp})\,{\cong}\,\underline{\mathrm{End}}_{D^{\sharp}}(\mathsf{H})

(3)

where $\mathsf{H}$ is the VHS given by the first relative cohomology of $\mathscr{A}^{\sharp}$ . It is shown in loc. cit. that the fiber of the isomorphism (3) at every $\mathbb{C}$ -point on $S$ is induced by an absolute Hodge cycle. Here is a summary of the argument in our notations: Choose a point $s_{0}\in S(\mathbb{C})$ such that $\mathsf{P}_{s_{0}}^{(0,0)}\neq 0$ . By the last paragraph of [Moonen, §6.12], there is an isomorphism

\mathrm{Cl}^{+}_{E/\mathbb{Q}}(\mathfrak{p}^{\sharp}_{s_{0}})\,{\cong}\,\mathrm{Cl}^{+}_{E/\mathbb{Q}}(\mathfrak{p}_{s_{0}})^{\oplus 2^{[E:\mathbb{Q}]}}

of objects in $\mathsf{Mot}_{\mathrm{AH}}(\mathbb{C})$ . As $\mathrm{Cl}^{+}_{E/\mathbb{Q}}(\mathfrak{p}_{s_{0}})$ is an object of $\mathsf{Mot}_{\mathsf{Ab}}(\mathbb{C})$ , so is $\mathrm{Cl}^{+}_{E/\mathbb{Q}}(\mathfrak{p}_{s_{0}}^{\sharp})$ . Therefore, by (2.1.2) the isomorphism (3) is absolute Hodge at $s_{0}$ , and hence so at every other $s$ by Deligne’s Principle B. This implies $\mathrm{Cl}^{+}_{E/\mathbb{Q}}(\mathfrak{p}^{\sharp}_{s})\in\mathsf{Mot}_{\mathsf{Ab}}(\mathbb{C})$ , so that (b) follows from (2.1.6) again.

(c) This follows directly from [Moonen, Prop. 8.5].

(d) In case (CM), [Moonen, §7.4] tells us that there exists a motive $\mathfrak{u}$ (denoted by $\boldsymbol{U}$ therein), an abelian variety $A$ over $\mathbb{C}$ and an abelian scheme $\mathscr{B}\to S$ , all equipped with multiplication by $E$ , such that there is an isomorphism

\mathfrak{p}_{s}{\otimes}_{E}\mathfrak{u}\stackrel{{\scriptstyle\sim}}{{\to}}\mathfrak{h}_{s}:=\underline{\mathrm{Hom}}_{E}(\mathfrak{h}^{1}(A),\mathfrak{h}^{1}(\mathscr{B}_{s})).

This implies that $\mathfrak{p}_{s}{\otimes}\mathfrak{u}\in\mathsf{Mot}_{\mathsf{Ab}}(\mathbb{C})$ . In order to show $\mathfrak{p}_{s}\in\mathsf{Mot}_{\mathsf{Ab}}(\mathbb{C})$ , it suffices to argue that $\mathfrak{u}\,{\cong}\,\mathbf{1}_{E}$ in $\mathsf{Mot}_{\mathrm{AH}}(\mathbb{C})$ .

In loc. cit., $\mathfrak{u}$ is in fact constructed as an object $\mathsf{Mot}(\mathbb{C})$ . Moonen remarked that conjecturally there should be an isomorphism $\mathfrak{u}\,{\cong}\,\mathbf{1}_{E}$ in $\mathsf{Mot}(\mathbb{C})$ and proved that $\mathfrak{u}$ indeed has trivial Hodge and $\ell$ -adic realizations. We note that his argument in [Moonen, Lem. 7.5] in fact implies that $\mathfrak{u}\,{\cong}\,\mathbf{1}_{E}$ in $\mathsf{Mot}_{\mathrm{AH}}(\mathbb{C})$ , i.e., $\omega_{B}(\mathfrak{u})$ is spanned by absolute Hodge classes: The idea is to take advantage of the fact that $\mathfrak{u}$ is independent of $s$ , and that for some $s_{0}\in S$ , the algebraic part $\omega_{B}(\mathfrak{p}_{s_{0}})^{(0,0)}$ of $\omega_{B}(\mathfrak{p}_{s_{0}})$ is nonempty. Since every class in $\omega_{B}(\mathfrak{u})$ is of type $(0,0)$ , we have $\omega_{B}(\mathfrak{p}_{s_{0}})^{(0,0)}{\otimes}_{E}\omega_{B}(\mathfrak{u})=\omega_{B}(\mathfrak{h}_{s_{0}})^{(0,0)}$ . By Lefschetz $(1,1)$ -theorem, every class in $\omega_{B}(\mathfrak{p}_{s_{0}})^{(0,0)}$ comes from a line bundle on $\mathcal{X}_{s}$ and hence is absolute Hodge. As classes in $\omega_{B}(\mathfrak{h}_{s_{0}})^{(0,0)}$ are also absolute Hodge, we may now conclude by (2.2.8) below. ∎

Proposition (2.2.8).

Let $E$ be a number field and let $\mathfrak{m},\mathfrak{n}$ be objects of $\mathsf{Mot}_{\mathrm{AH}}(\mathbb{C})$ with $E$ -action. Let $\mathfrak{h}\coloneqq\mathfrak{m}{\otimes}_{E}\mathfrak{n}$ . Let $m\in\omega_{B}(\mathfrak{m}),n\in\omega_{B}(\mathfrak{n})$ be nonzero Hodge cycles and define $h\in\omega_{B}(\mathfrak{h})$ to be $m{\otimes}_{E}n$ . If $h$ and $m$ are both absolute Hodge, then so is $n$ .

Proof.

Recall notations in (2.1.3). We need to show that for every $\sigma\in\mathrm{Aut}(\mathbb{C})$ , the image $n^{\prime}$ of $n{\otimes}1$ under the canonical isomorphisms

\omega_{B}(\mathfrak{n})\otimes(\mathbb{C}\otimes\mathbb{A}_{f})\cong\omega_{\mathrm{dR}}(\mathfrak{n})\times\omega_{\mathbb{A}_{f}}(\mathfrak{n})\,{\cong}\,_{\mathrm{bc}}\omega_{\mathrm{dR}}(\mathfrak{n}^{\sigma})\times\omega_{\mathbb{A}_{f}}(\mathfrak{n}^{\sigma})\cong\omega_{B}(\mathfrak{n}^{\sigma})\otimes(\mathbb{C}\otimes\mathbb{A}_{f}).

is contained in $\omega_{B}(\mathfrak{n}^{\sigma})$ (i.e., is of the form $n^{\sigma}{\otimes}1$ for some $n^{\sigma}\in\omega_{B}(\mathfrak{n}^{\sigma})$ ). Since $m$ and $h$ are absolute Hodge, we know that $m^{\sigma}\in\omega_{B}(\mathfrak{m}^{\sigma})$ and $h^{\sigma}=m^{\sigma}\otimes n^{\prime}\in\omega_{B}(\mathfrak{h}^{\sigma})$ . Then apply (2.2.9) with $k=\mathbb{Q}$ , $R=\mathbb{A}_{f}\times\mathbb{C}$ , $M=\omega_{B}(\mathfrak{m}^{\sigma})$ and $N=\omega_{B}(\mathfrak{n}^{\sigma})$ to deduce that $n^{\prime}\in\omega_{B}(\mathfrak{n}^{\sigma})$ . ∎

Lemma (2.2.9).

Let $E/k$ be a field extension, $R$ be any $k$ -algebra and $M,N$ be finite dimensional $E$ -vector spaces. Let $H\coloneqq M{\otimes}_{E}N$ and $h\in H{\otimes}_{k}R$ be a nonzero element of the form $m{\otimes}n$ under the canonical isomorphism

H{\otimes}_{k}R\,{\cong}\,(M{\otimes}_{k}R){\otimes}_{E{\otimes}_{k}R}(N{\otimes}_{k}R),

where $m\in M{\otimes}_{k}R$ and $n\in N{\otimes}_{k}R$ . If $h\in H$ and $m\in M$ , then $n\in N$ .

Proof.

Let $\alpha=\dim_{E}M$ and $\beta=\dim_{E}N$ . By choosing bases of $M$ and $N$ , we may assume $M=E^{\alpha}$ and $N=E^{\beta}$ . Identify $H$ with the space of $(\alpha\times\beta)$ -matrices over $E$ and $h$ with $m\cdot n^{T}$ . We denote by $(m_{i}),(n_{j}),(h_{i,j})$ the respective $E{\otimes}_{k}R$ -coordinates of $m,n$ and $h$ and fix $i$ such that $m_{i}\not=0$ . Then for every $j$ , $h_{i,j}=m_{i}\cdot n_{j}$ and hence $n_{j}=m_{i}^{-1}h_{i,j}\in E$ because $m_{i},h_{i,j}\in E\subseteq E{\otimes}_{k}R$ by assumption. ∎

3 Moduli Interpretations of Shimura Varieties

In this section, we first recall the moduli description for the canonical models of some Shimura varieties of abelian type from [YangSystem]. Then we give some prelimary results on those of orthogonal type over totally real fields and review their integral models when the reflex field is $\mathbb{Q}$ .

3.1 Systems of realizations

Definition (3.1.1).

Let $k$ be a subfield of $\mathbb{C}$ and $S$ be a smooth $k$ -variety. By a system of realizations we mean a tuple $\mathsf{V}=(\mathsf{V}_{B},\mathsf{V}_{\mathrm{dR}},\mathsf{V}_{\mathrm{{\acute{e}}t}},i_{\mathrm{dR}},i_{\mathrm{{\acute{e}}t}})$ where

•

$\mathsf{V}_{B}$ is a $\mathbb{Q}$ -local system over $S_{\mathbb{C}}:=S{\otimes}_{k}\mathbb{C}$ ;
•

$\mathsf{V}_{\mathrm{dR}}$ is a filtered flat vector bundle over $S$ ;
•

$\mathsf{V}_{\mathrm{{\acute{e}}t}}$ is an étale local system of $\mathbb{A}_{f}$ -coefficients over $S$ ;
•

$i_{\mathrm{dR}}:(\mathsf{V}_{B}{\otimes}\mathcal{O}^{\mathrm{an}}_{S_{\mathbb{C}}},\mathrm{id}{\otimes}d)\stackrel{{\scriptstyle\sim}}{{\to}}(\mathsf{V}_{\mathrm{dR}}|_{S_{\mathbb{C}}})^{\mathrm{an}}$ is an isomorphism of flat (holomorphic) vector bundles such that $(\mathsf{V}_{B},\mathsf{V}_{\mathrm{dR}}|_{S_{\mathbb{C}}})$ is a polarizable VHS;
•

$i_{\mathrm{{\acute{e}}t}}:\mathsf{V}_{B}{\otimes}\mathbb{A}_{f}\stackrel{{\scriptstyle\sim}}{{\to}}\mathsf{V}_{\mathrm{{\acute{e}}t}}|_{S_{\mathbb{C}}}$ is an isomorphism between (the pro-étale sheaf associated to) $\mathsf{V}_{B}{\otimes}\mathbb{A}_{f}$ and $\mathsf{V}_{\mathrm{{\acute{e}}t}}|_{S_{\mathbb{C}}}$ .

We may often omit $(i_{\mathrm{dR}},i_{\mathrm{{\acute{e}}t}})$ in the notation and write $\mathsf{V}_{B}{\otimes}\mathbb{A}_{f}=\mathsf{V}_{\mathrm{{\acute{e}}t}}|_{S_{\mathbb{C}}}$ and $\mathsf{V}_{B}{\otimes}\mathcal{O}_{S_{\mathbb{C}}}^{\mathrm{an}}=\mathsf{V}_{\mathrm{dR}}|_{S_{\mathbb{C}}}$ a little abusively. Let $\mathsf{R}(S)$ denote the category of systems of realizations over $S$ , with morphisms defined in the obvious way. Then $\mathsf{R}(S)$ is a naturally a Tannakian category. Let $\mathbf{1}_{S}$ be its unit object and for each $n\in\mathbb{Z}$ let $\mathbf{1}_{S}(n)$ be the Tate object. For every $\mathsf{V}\in\mathsf{R}(S)$ , set $\mathsf{V}(n):=\mathsf{V}{\otimes}\mathbf{1}_{S}(n)$ . Note that if $k=\mathbb{C}$ , then $\mathsf{R}(S)$ is naturally identified with the category of polarizable VHS over $S$ . We write $\mathrm{H}^{0}(\mathsf{V})$ for $\mathrm{Hom}(\mathbf{1},\mathsf{V})$ .

Our definition is a little different from [FuMoonen, §6.1] because we fixed an embedding $k\hookrightarrow\mathbb{C}$ , but see [YangSystem, (3.1.3)] for a comparison. Also, recall our convention in (1.0.5)(e) for VHS. We remind the reader that since $S$ is defined over $k\subseteq\mathbb{C}$ , when we write $s\in S(\mathbb{C})$ , we mean a $k$ -linear morphism $\mathrm{Spec\,}(\mathbb{C})\to S$ , which we view simultaneously as a closed point on $S_{\mathbb{C}}=S{\otimes}_{k}\mathbb{C}$ .

Next, we recall how to define a $\mathsf{K}$ -level structure in this context. Interestingly this definition can be leveraged to define additional structures on $\mathsf{V}$ , as (3.1.3) below shows.

Definition (3.1.2).

Let $G$ be a reductive group over $\mathbb{Q}$ , let $V\in\mathrm{Rep}(G)$ be a finite dimensional $G$ -representation and $\mathsf{K}\subseteq G(\mathbb{A}_{f})$ be a compact open subgroup. For any subfield $k\subseteq\mathbb{C}$ , smooth $k$ -variety $S$ and any system of realizations $\mathsf{V}=(\mathsf{V}_{B},\mathsf{V}_{\mathrm{dR}},\mathsf{V}_{\mathrm{{\acute{e}}t}})$ on $S$ we make the following definitions.

(a)

A $(G,V,\mathsf{K})$ -level structure on $\mathsf{V}_{\mathrm{{\acute{e}}t}}$ , or simply a $\mathsf{K}$ -level structure, is a global section $[\eta]$ of $\mathsf{K}\backslash\underline{\mathrm{Isom}}(V{\otimes}\mathbb{A}_{f},\mathsf{V}_{\mathrm{{\acute{e}}t}})$ . Here $\underline{\mathrm{Isom}}(V{\otimes}\mathbb{A}_{f},\mathsf{V}_{\mathrm{{\acute{e}}t}})$ denotes the pro-étale sheaf consisting of isomorphisms $V{\otimes}\mathbb{A}_{f}\stackrel{{\scriptstyle\sim}}{{\to}}\mathsf{V}_{\mathrm{{\acute{e}}t}}$ , and $\mathsf{K}$ acts by pre-composition through its image in $\mathrm{GL}(V{\otimes}\mathbb{A}_{f})$ . For each geometric point $s\to S$ , write the $\mathsf{K}$ -orbit of isomorphisms $V{\otimes}\mathbb{A}_{f}\stackrel{{\scriptstyle\sim}}{{\to}}\mathsf{V}_{\mathrm{{\acute{e}}t},s}$ determined by $[\eta]$ as $[\eta]_{s}$ .
(b)

If $[\eta]$ is a $(G,V,\mathsf{K})$ -level structure on $\mathsf{V}_{\mathrm{{\acute{e}}t}}$ , every element $v\in I:=(V^{\otimes})^{G}$ gives rise to a global section of $\mathsf{V}_{\mathrm{{\acute{e}}t}}$ , which we denote by $\eta(v)_{\mathrm{{\acute{e}}t}}$ , such that for every geometric point $s\to S$ , every representative of $[\eta]_{s}$ sends $v{\otimes}1$ to $\eta(v)_{\mathrm{{\acute{e}}t},s}$ . We say that $[\eta]$ is $\mathsf{V}$ -rational if (i) for all $v\in I$ , there exists a global section $\mathbf{v}=(\mathbf{v}_{B},\mathbf{v}_{\mathrm{dR}},\mathbf{v}_{\mathrm{{\acute{e}}t}})\in\mathrm{H}^{0}(\mathsf{V}^{\otimes})$ such that $\mathbf{v}_{\mathrm{{\acute{e}}t}}=\eta(v)_{\mathrm{{\acute{e}}t}}$ , and (ii) for every $s\in S(\mathbb{C})$ , $(\mathsf{V}_{B,s},\{\mathbf{v}_{B,s}\}_{v\in I})\,{\cong}\,(V,\{v\}_{v\in I})$ .
(c)

Let $\Omega$ be a $G(\mathbb{R})$ -conjugacy class of morphisms $\mathbb{S}\to G_{\mathbb{R}}$ . When $V$ is faithful, we say that a $\mathsf{V}$ -rational $\mathsf{K}$ -level structure is of type $\Omega$ if for every $s\in S(\mathbb{C})$ , under some (and hence every) isomorphism $(\mathsf{V}_{B,s},\{\mathbf{v}_{B,s}\}_{v\in I})\,{\cong}\,(V,\{v\}_{v\in I})$ the Hodge structure on $\mathsf{V}_{B,s}$ is defined by an element of $\Omega$ .
(d)

If there is a morphism $G^{\prime}\to G$ from another reductive subgroup $G^{\prime}$ , and $\mathsf{K}^{\prime}\subseteq G^{\prime}(\mathbb{A}_{f})$ is a compact open subgroup whose image is contained in $\mathsf{K}$ , then we say that a $\mathsf{K}^{\prime}$ -level structure $[\eta^{\prime}]$ on $\mathsf{V}_{\mathrm{{\acute{e}}t}}$ refines $[\eta]$ if $[\eta]$ is induced by $[\eta^{\prime}]$ via the natural (forgetful) map $\mathsf{K}^{\prime}\backslash\underline{\mathrm{Isom}}(V{\otimes}\mathbb{A}_{f},\mathsf{V}_{\mathrm{{\acute{e}}t}})\to\mathsf{K}\backslash\underline{\mathrm{Isom}}(V{\otimes}\mathbb{A}_{f},\mathsf{V}_{\mathrm{{\acute{e}}t}})$ .

The definitions (b) and (c) above are used to simplify the formalism of the moduli interpretation of Shimura varieties. To explain this, we give a PEL-type example:

Example (3.1.3).

Let $(B,\ast,(V,\psi))$ be a simple PEL-datum (i.e., $B$ is a simple $\mathbb{Q}$ -algebra with positive involution $\ast$ of type $A$ or $C$ and $(V,\psi)$ is a symplectic $B$ -module). We denote by $G\coloneqq\mathrm{GSp}_{B}(V)$ the $\mathbb{Q}$ -group of $B$ -linear similitudes and fix an open compact subgroup $\mathsf{K}\subset G(\mathbb{A}_{f})$ . Then there exists a unique $G(\mathbb{R})$ -conjugacy class $\Omega$ such that $(G,\Omega)$ is a Shimura datum and $\mathrm{Sh}_{\mathsf{K}}(G,\Omega)(\mathbb{C})$ is in canonical bijection with the set $\mathcal{I}$ of isomorphism classes of tuples $(\mathsf{V},\lambda,i,[\eta])$ where

•

$\mathsf{V}$ is a $\mathbb{Q}$ -Hodge structure $(\mathsf{V}_{B},\mathsf{V}_{\mathrm{dR}})$ of type $\{(1,0),(0,1)\}$ ,
•

$i\colon B\hookrightarrow\mathrm{End}(\mathsf{V})$ is an algebra morphism,
•

$\lambda$ is a $\mathbb{Q}^{\times}$ -equivalence class of $B$ -linear symplectic pairing of $\mathsf{V}$ and
•

$[\eta]$ is a $\mathsf{K}$ -orbit of a $B$ -linear similitude $V\otimes\mathbb{A}_{f}\to\mathsf{V}_{\mathrm{{\acute{e}}t}}$

such that $(\spadesuit)$ there exists a $B$ -linear similitude $V\,{\cong}\,\mathsf{V}_{B}$ through which the Hodge structure on $\mathsf{V}$ is defined by an element of $\Omega$ (see e.g. [MilIntro, Prop. 8.14, Thm. 8.17]). Here we implicitly applied the equivalence $A\mapsto\mathrm{H}^{1}(A,\mathbb{Q})$ between the category of complex abelian varieties up to isogeny and that of polarizable Hodge structures of type $\{(1,0),(0,1)\}$ .

Note that an object of $\mathsf{R}(\mathbb{C}):=\mathsf{R}(\mathrm{Spec\,}(\mathbb{C}))$ is nothing but a polarizable Hodge structure. We can more concisely define $\mathcal{I}$ as the isomorphism classes of pairs $(\mathsf{V},[\eta])$ where

•

$\mathsf{V}\in\mathsf{R}(\mathbb{C})$ and
•

$[\eta]$ is a $\mathsf{V}$ -rational $(G,V,\mathsf{K})$ -level structure on $\mathsf{V}_{\mathrm{{\acute{e}}t}}$ of type $\Omega$ .

Indeed, if we view each $b\in B$ as a tensor in $\mathrm{End}_{\mathbb{Q}}(V)=V^{\vee}\otimes V$ and $c\coloneqq\mathbb{Q}^{\times}\cdot\psi$ as a tensor in $V^{2}\otimes(V^{\vee})^{2}$ (cf. [Kim:RZ, Ex. 2.1.6]), we have $i(b){\otimes}\mathbb{A}_{f}=\eta(b)_{\mathrm{{\acute{e}}t}}$ and $\lambda{\otimes}\mathbb{A}_{f}=\eta(c)_{\mathrm{{\acute{e}}t}}$ in the notation of (3.1.2)(b). In particular, the datum of $i$ and $\lambda$ is remembered by the condition that $\eta(b)_{\mathrm{{\acute{e}}t}}$ and $\eta(c)_{\mathrm{{\acute{e}}t}}$ come from global sections (i.e., elements) of $\mathsf{V}_{B}^{\otimes}$ of Hodge type $(0,0)$ . Since $G$ is the stabilizer of $B$ and $c$ in $\mathrm{GL}(V)$ , we may equivalently say that this is true for all $v\in(V^{\otimes})^{G}$ . Therefore, the existence of $i$ and $\lambda$ such that there exists a $B$ -linear similitude $V\,{\cong}\,\mathsf{V}_{B}$ is equivalent to $[\eta]$ being $\mathsf{V}$ -rational. Then the condition $(\spadesuit)$ can be simply stated as “ $[\eta]$ is of type $\Omega$ ” in the sense of (3.1.2)(c).

Now we introduce some notation to keep track of Galois descent data in a system of realizations:

Notation (3.1.4).

Let $S$ be a smooth variety over a subfield $k$ of $\mathbb{C}$ . Let $s\in S(\mathbb{C})$ and $\sigma\in\mathrm{Aut}(\mathbb{C}/k)$ be any elements. Denote by $\sigma(s)\in S(\mathbb{C})$ the point given by pre-composing the $k$ -linear morphism $s:\mathrm{Spec\,}(\mathbb{C})\to S$ with $\mathrm{Spec\,}(\sigma)$ . Given $(\mathsf{V}_{B},\mathsf{V}_{\mathrm{{\acute{e}}t}},\mathsf{V}_{\mathrm{dR}})\in\mathsf{R}(S)$ , we write $\sigma_{\mathsf{V}_{\mathrm{{\acute{e}}t}},s}:\mathsf{V}_{B,s}{\otimes}\mathbb{A}_{f}\stackrel{{\scriptstyle\sim}}{{\to}}\mathsf{V}_{B,\sigma(s)}{\otimes}\mathbb{A}_{f}$ the natural isomorphism induced by $\mathsf{V}_{\mathrm{{\acute{e}}t}}$ , viewed as a descent of $\mathsf{V}_{B}{\otimes}\mathbb{A}_{f}$ from $S_{\mathbb{C}}$ to $S$ ; similarly, we write $\sigma_{\mathsf{V}_{\mathrm{dR}},s}:\mathsf{V}_{B,s}{\otimes}\mathbb{C}\stackrel{{\scriptstyle\sim}}{{\to}}\mathsf{V}_{B,\sigma(s)}{\otimes}\mathbb{C}$ for the natural $\sigma$ -linear isomorphism induced by the descent $\mathsf{V}_{\mathrm{dR}}$ of (the algebraization of) $\mathsf{V}_{B}{\otimes}\mathcal{O}_{S_{\mathbb{C}}}^{\mathrm{an}}$ .

To clarify the meaning of $\sigma_{\mathsf{V}_{\mathrm{{\acute{e}}t}},s}$ , we remark that $s$ and $\sigma(s)$ are usually different closed points on $S_{\mathbb{C}}$ , and they are equal if and only if $\sigma$ fixes the residue field $k(s_{0})$ , where $s_{0}\in S$ is the image of $s$ . The collection of $\sigma_{\mathsf{V}_{\mathrm{{\acute{e}}t}},s}$ as $\sigma$ runs through $\mathrm{Aut}(k(s)/k(s_{0}))=\mathrm{Aut}(\mathbb{C}/k(s_{0}))$ is nothing but the Galois action on the stalk $\mathsf{V}_{B,s}{\otimes}\mathbb{A}_{f}=\mathsf{V}_{\mathrm{{\acute{e}}t},s}$ .

Using the notations of (2.1.3) and (3.1.4), we define:

Definition (3.1.5).

We say that a system of realizations $\mathsf{V}$ is weakly abelian-motivic (weakly AM) if for every $s\in S(\mathbb{C})$ , there exists $M\in\mathsf{Mot}_{\mathsf{Ab}}(\mathbb{C})$ such that $\omega_{\mathrm{Hdg}}(M)\,{\cong}\,(\mathsf{V}_{B,s},\mathsf{V}_{\mathrm{dR},s})$ ; moreover, for any such isomorphism $\gamma:\omega_{\mathrm{Hdg}}(M)\stackrel{{\scriptstyle\sim}}{{\to}}(\mathsf{V}_{B,s},\mathsf{V}_{\mathrm{dR},s})$ and $\sigma\in\mathrm{Aut}(\mathbb{C}/k)$ , there exists an isomorphism $\gamma^{\sigma}:\omega_{\mathrm{Hdg}}(M^{\sigma})\stackrel{{\scriptstyle\sim}}{{\to}}(\mathsf{V}_{B,\sigma(s)},\mathsf{V}_{\mathrm{dR},\sigma(s)})$ such that the Betti components $\gamma_{B},\gamma^{\sigma}_{B}$ of $\gamma,\gamma^{\sigma}$ fit into commutative diagrams:

and

(4)

We note that $\gamma^{\sigma}$ is uniquely determined by $\gamma$ provided that it exists. Denote the full subcategory of $\mathsf{R}(S)$ given by these objects by $\mathsf{R}^{*}_{\mathrm{am}}(S)$ . It is easy to check that if $T\to S$ is a morphism between smooth $k$ -varieties, the natural pullback functor $\mathsf{R}(S)\to\mathsf{R}(T)$ sends $\mathsf{R}^{*}_{\mathrm{am}}(S)$ to $\mathsf{R}^{*}_{\mathrm{am}}(T)$ .

Lemma (3.1.6).

([YangSystem, (3.4.1)]) Let $S$ be a smooth variety over $k\subseteq\mathbb{C}$ and take $\mathsf{V},\mathsf{W}\in\mathsf{R}_{\mathrm{am}}^{*}(S)$ . Let $\varphi_{\mathbb{C}}$ be a morphism $\mathsf{V}|_{S_{\mathbb{C}}}\to\mathsf{W}|_{S_{\mathbb{C}}}$ . Then $\varphi_{\mathbb{C}}$ descends to a morphism $\mathsf{V}\to\mathsf{W}$ if and only if either the étale or the de Rham component of $\varphi_{\mathbb{C}}$ descends to $S$ .

When applied to the case $\mathsf{V}=\mathbf{1}_{S}$ , the above lemma says that for $\mathsf{W}\in\mathsf{R}^{*}_{\mathrm{am}}(S)$ , if the étale realization of a global section of $\mathsf{W}_{B}|_{S_{\mathbb{C}}}$ which is everywhere of Hodge type $(0,0)$ descends to $S$ , then so does the de Rham realization, and vice versa. This is a global version of the following statement: Suppose that $M\in\mathsf{Mot}_{\mathrm{AH}}(k)$ and $v\in\omega_{B}(M_{\mathbb{C}})$ is absolute Hodge. Then $v{\otimes}\mathbb{A}_{f}\in\omega_{\mathrm{{\acute{e}}t}}(M_{\mathbb{C}})$ descends to $k$ (i.e., is $\mathrm{Aut}(\mathbb{C}/k)$ -invariant) if and only if $v{\otimes}\mathbb{C}\in\omega_{\mathrm{dR}}(M_{\mathbb{C}})$ descends to $\omega_{\mathrm{dR}}(M)$ . Note that if $M\in\mathsf{Mot}_{\mathsf{Ab}}(k)$ , then any Hodge cycle $v\in\omega_{B}(M_{\mathbb{C}})$ is automatically absolute Hodge.

For future reference we give a handy lemma on (b) and (c) in (3.1.2).

Lemma (3.1.7).

Suppose that in (3.1.2)(a), the representation $V\in\mathrm{Rep}(G)$ is faithful, $S$ is geometrically connected as a $k$ -variety, and for some $b\in S(\mathbb{C})$ there is an isomorphism $\eta_{b}:V\stackrel{{\scriptstyle\sim}}{{\to}}\mathsf{V}_{B,b}$ such that $[\eta]_{b}=\mathsf{K}\cdot(\eta_{b}{\otimes}\mathbb{A}_{f})$ ; moreover, for some $\Omega$ as in (3.1.2)(c), the Hodge structure on $\mathsf{V}_{B,b}$ is defined by an element of $\Omega$ via $\eta_{b}$ .

Then, assuming either $k=\mathbb{C}$ or $\mathsf{V}\in\mathsf{R}^{*}_{\mathrm{am}}(S)$ , $[\eta]$ is $\mathsf{V}$ -rational and of type $\Omega$ .

Proof.

Note that by assumption $S_{\mathbb{C}}$ is connected, and the fact that $\eta_{b}{\otimes}\mathbb{A}_{f}$ represents a $\mathsf{K}$ -level structure at $b$ implies that its $\mathsf{K}$ -orbit is $\pi^{\mathrm{{\acute{e}}t}}_{1}(S,b)$ -stable. In particular, for each $v\in(V^{\otimes})^{G}$ , $\pi_{1}(S_{\mathbb{C}},b)$ fixes $\eta_{b}(v)$ , so there exists a $\mathbf{v}_{B}\in\mathrm{H}^{0}(\mathsf{V}_{B}^{\otimes})$ such that $\mathbf{v}_{B,b}=\eta_{b}(v)$ . Likewise, there exists $\mathbf{v}_{\mathrm{{\acute{e}}t}}\in\mathrm{H}^{0}(\mathsf{V}_{\mathrm{{\acute{e}}t}}^{\otimes})$ such that $\mathbf{v}_{\mathrm{{\acute{e}}t},b}=\eta_{b}(v){\otimes}1$ , and $\mathbf{v}_{\mathrm{{\acute{e}}t}}$ is precisely the $\eta(v)_{\mathrm{{\acute{e}}t}}$ in (3.1.2)(b).

Note that $\mathbf{v}_{B}$ is necessarily of Hodge type $(0,0)$ at $b$ , because the Mumford-Tate group of the Hodge structure $(\mathsf{V}_{B,b},\mathsf{V}_{\mathrm{dR},b})$ is contained in $G$ via $\eta_{b}$ . This implies that $\mathbf{v}_{B}$ is of Hodge type $(0,0)$ everywhere by the theorem of the fixed part. Let $\mathbf{v}_{\mathrm{dR},\mathbb{C}}$ be the global section in $(\mathsf{V}_{B}{\otimes}\mathcal{O}_{S_{\mathbb{C}}}^{\mathrm{an}})^{\otimes}$ induced by $\mathbf{v}_{B}$ . Then $\mathbf{v}_{\mathrm{dR},\mathbb{C}}$ algebraizes to a global section of $(\mathsf{V}_{\mathrm{dR}}|_{S_{\mathbb{C}}})^{\otimes}$ (cf. [DelVB, II Thm 5.9]).

If $k=\mathbb{C}$ (so that $S=S_{\mathbb{C}}$ ), then we have already shown that $[\eta]$ is $\mathsf{V}$ -rational; moreover, one deduces from the connectedness of $S_{\mathbb{C}}$ and [DelVdShimura, (1.1.12)] that $[\eta]$ remains of type $\Omega$ on $S_{\mathbb{C}}$ . If $k\subseteq\mathbb{C}$ is a general subfield, it remains to show that $\mathbf{v}_{\mathrm{dR},\mathbb{C}}$ descends to $S$ under the additional assumption that $\mathsf{V}\in\mathsf{R}^{*}_{\mathrm{am}}(S)$ . However, as the étale realization of $\mathbf{v}_{B}$ descends to a global section of $\mathsf{V}_{\mathrm{{\acute{e}}t}}^{\otimes}$ (i.e., $\mathbf{v}_{\mathrm{{\acute{e}}t}}$ ), this follows from (3.1.6). ∎

3.2 Shimura varieties

Let $(G,\Omega)$ be a Shimura datum which satisfies the axioms in [Milne:CanonicalModels, II (2.1)]. Let $E(G,\Omega)$ be the reflex field, $Z$ be the center of $G$ and $Z_{s}$ be the maximal anisotropic subtorus of $Z$ that is split over $\mathbb{R}$ . In this paper, we always assume that

\text{the weight is defined over $\mathbb{Q}$ and $Z_{s}$ is trivial}.

(5)

Note that in particular the latter condition ensures that $Z(\mathbb{Q})$ is discrete in $Z(\mathbb{A}_{f})$ ([MilIntro, Rmk 5.27]). We will often drop the Hermitian symmetric domain $\Omega$ from the notation of Shimura varieties when no confusion would arise.

For any compact open subgroup $\mathsf{K}\subseteq G(\mathbb{A}_{f})$ , let $\mathrm{Sh}_{\mathsf{K}}(G)_{\mathbb{C}}$ denote the resulting Shimura variety with a complex uniformization $G(\mathbb{Q})\backslash\Omega\times G(\mathbb{A}_{f})/\mathsf{K}$ and let $\mathrm{Sh}_{\mathsf{K}}(G)$ denote the canonical model over $E(G,\Omega)$ . Let $\mathrm{Sh}(G)$ denote the inverse limit $\varprojlim_{\mathsf{K}}\mathrm{Sh}_{\mathsf{K}}(G)$ as $\mathsf{K}$ runs through all compact open subgroups. Under our assumptions, $\mathrm{Sh}(G)(\mathbb{C})$ is described by $G(\mathbb{Q})\backslash\Omega\times G(\mathbb{A}_{f})$ ([MilIntro, (5.28)]). Note that $\mathrm{Sh}_{\mathsf{K}}(G)=\mathrm{Sh}(G)/\mathsf{K}$ .

(3.2.1)

Let $G\to\mathrm{GL}(V)$ be a representation. For any neat compact open subgroup $\mathsf{K}\subseteq G(\mathbb{A}_{f})$ , we can attach to $\mathrm{Sh}_{\mathsf{K}}(G)_{\mathbb{C}}$ an automorphic VHS $(\mathsf{V}_{B},\mathsf{V}_{\mathrm{dR},\mathbb{C}})$ (cf. [Milne:CanonicalModels, Ch. II 3.3], [Taelman2, §2.2]). In particular, $\mathsf{V}_{B}$ is defined to be the contraction product

\mathsf{V}_{B}:=V\times^{G(\mathbb{Q})}[\Omega\times G(\mathbb{A}_{f})/\mathsf{K}]

(6)

The filtration on $\mathsf{V}_{\mathrm{dR},\mathbb{C}}$ is obtained by descending the filtration on the tautological VHS on $V\times\Omega$ . Analogously, the automorphic étale local system $\mathsf{V}_{\mathrm{{\acute{e}}t}}$ on the proétale site of $\mathrm{Sh}_{\mathsf{K}}(G)$ is defined as the contraction product

\mathsf{V}_{\mathrm{{\acute{e}}t}}:=V_{\mathbb{A}_{f}}\times^{\mathsf{K}}\mathrm{Sh}(G)

(7)

which comes with a comparison isomorphism $\mathsf{V}_{B}\otimes\mathbb{A}_{f}\,{\cong}\,\mathsf{V}_{\mathrm{{\acute{e}}t}}|_{\mathrm{Sh}_{\mathsf{K}}(G)_{\mathbb{C}}}$ . Moreover, by construction $\mathsf{V}_{\mathrm{{\acute{e}}t}}$ over $\mathrm{Sh}_{\mathsf{K}}(G)$ comes with a tautological $\mathsf{K}$ -level structure $[\eta_{V}]$ , i.e., a global section of $\mathsf{K}\backslash\underline{\mathrm{Isom}}(V{\otimes}\mathbb{A}_{f},\mathsf{V}_{\mathrm{{\acute{e}}t}})$ .

Define $[\eta^{\mathrm{an}}_{V}]$ to be the restriction of $[\eta_{V}]$ to ${\mathrm{Sh}_{\mathsf{K}}(G)_{\mathbb{C}}}$ . Using $(\mathsf{V}_{B},\mathsf{V}_{\mathrm{dR},\mathbb{C}})$ , we can already give a moduli interpretation of $\mathrm{Sh}_{\mathsf{K}}(G)_{\mathbb{C}}$ . Below is a reformulation of [Mil94, Prop. 3.10] in our terminology (cf. [YangSystem, (4.1.2)])⁵⁵5Technically, the example discussed in (3.1.3) may not satisfy (5), but it helps illustrate how to compare our formulation and Milne’s..

Theorem (3.2.2).

Assume that $V$ is faithful. For every smooth $\mathbb{C}$ -variety $T$ , let $\mathcal{M}_{V}(T)$ be the groupoid of tuples of the form $(\mathsf{W},[\xi^{\mathrm{an}}])$ where $\mathsf{W}=(\mathsf{W}_{B},\mathsf{W}_{\mathrm{dR}})$ is a VHS over $T$ and $[\xi^{\mathrm{an}}]$ is a $\mathsf{K}$ -level structure on $\mathsf{W}_{B}{\otimes}\mathbb{A}_{f}$ which is $\mathsf{W}$ -rational and is of type $\Omega$ .

Then $(\mathsf{V}_{\mathbb{C}}:=(\mathsf{V}_{B},\mathsf{V}_{\mathrm{dR},\mathbb{C}}),[\eta^{\mathrm{an}}_{V}])$ is an object of $\mathcal{M}_{V}(\mathrm{Sh}_{\mathsf{K}}(G)_{\mathbb{C}})$ and for every object $(\mathsf{W},[\xi^{\mathrm{an}}])\in\mathcal{M}_{V}(T)$ there exists a unique morphism $\rho:T\to\mathrm{Sh}_{\mathsf{K}}(G)_{\mathbb{C}}$ such that $\rho^{*}(\mathsf{V}_{\mathbb{C}},[\eta^{\mathrm{an}}_{V}])\,{\cong}\,(\mathsf{W},[\xi^{\mathrm{an}}])$ .

We remark that as $\mathsf{K}$ is neat, assumption (5) ensures that the objects in $\mathcal{M}_{V}(T)$ above have no nontrivial automorphisms (cf. [Mil94, Rmk 3.11]).

To describe the moduli problem for the canonical model, we restrict to the following subclass of Shimura data, which contains all cases we will consider in the following chapters.

Assumption (3.2.3).

There exists a morphism of Shimura data $(\widetilde{G},\widetilde{\Omega})\to(G,\Omega)$ such that

(i)

$(\widetilde{G},\widetilde{\Omega})$ is of Hodge type and also satisfies assumption (5);
(ii)

$\widetilde{G}\to G$ is surjective and the kernel lies in the center of $\widetilde{G}$ ;
(iii)

the embedding of reflex fields $E(G,\Omega)\subseteq E(\widetilde{G},\widetilde{\Omega})$ is an equality.

Below we assume that $(G,\Omega)$ is a Shimura datum which satisfies the above assumptions.

(3.2.4)

For any representation $G\to\mathrm{GL}(V)$ , and $\mathsf{V}_{B},\mathsf{V}_{\mathrm{dR},\mathbb{C}},\mathsf{V}_{\mathrm{{\acute{e}}t}}$ set up in (3.2.1), there exists a unique descent $\mathsf{V}_{\mathrm{dR}}$ of $\mathsf{V}_{\mathrm{dR},\mathbb{C}}$ to $\mathrm{Sh}_{\mathsf{K}}(G)$ such that $(\mathsf{V}_{B},\mathsf{V}_{\mathrm{dR}},\mathsf{V}_{\mathrm{{\acute{e}}t}})$ is a weakly AM system of realizations (see ([YangSystem, (4.2.2)])). We call $\mathsf{V}:=(\mathsf{V}_{B},\mathsf{V}_{\mathrm{dR}},\mathsf{V}_{\mathrm{{\acute{e}}t}})\in\mathsf{R}^{*}_{\mathrm{am}}(\mathrm{Sh}_{\mathsf{K}}(G))$ the automorphic system (of realizations) on $\mathrm{Sh}_{\mathsf{K}}(G)$ attached to $V\in\mathrm{Rep}(G)$ . The tautological $\mathsf{K}$ -level structure $[\eta_{V}]$ on $\mathsf{V}$ is $\mathsf{V}$ -rational, and is of type $\Omega$ when $V$ is faithful.

Theorem (3.2.5).

([YangSystem, (4.3.1)]) Assume that $V$ is faithful. Let $E^{\prime}\subseteq\mathbb{C}$ be a subfield which contains $E=E(G,\Omega)$ and $T$ be a smooth $E^{\prime}$ -variety. Let $\mathcal{M}_{V,E^{\prime}}(T)$ be the groupoid of pairs $(\mathsf{W},[\xi])$ where $\mathsf{W}=(\mathsf{W}_{B},\mathsf{W}_{\mathrm{dR}},\mathsf{W}_{\mathrm{{\acute{e}}t}})\in\mathsf{R}^{*}_{\mathrm{am}}(T)$ , and $[\xi]$ is a $\mathsf{K}$ -level structure on $\mathsf{W}_{\mathrm{{\acute{e}}t}}$ such that $[\xi]$ is $\mathsf{W}$ -rational and $(\mathsf{W},[\xi])$ is of type $\Omega$ .

Then $(\mathsf{V}=(\mathsf{V}_{B},\mathsf{V}_{\mathrm{dR}},\mathsf{V}_{\mathrm{{\acute{e}}t}}),[\eta_{V}])$ is an object of $\mathcal{M}_{V,E}(\mathrm{Sh}_{\mathsf{K}}(G))$ , and for each $(\mathsf{W},[\xi])\in\mathcal{M}_{V,E^{\prime}}(T)$ , there exists a unique $\rho:T\to\mathrm{Sh}_{\mathsf{K}}(G)_{E^{\prime}}$ such that $(\mathsf{W},[\xi])\,{\cong}\,\rho^{*}(\mathsf{V},[\eta_{V}])$ .

Note that any object in $\mathcal{M}_{V,E^{\prime}}(T)$ above has no nontrivial automorphisms because this is already true when $E^{\prime}=\mathbb{C}$ . This is a key fact that we shall use repeatedly throughout the paper. The above is proved by first constructing a morphism $\rho_{\mathbb{C}}:T_{\mathbb{C}}\to\mathrm{Sh}_{\mathsf{K}}(G)_{\mathbb{C}}$ such that $(\mathsf{W},[\xi])|_{T_{\mathbb{C}}}\,{\cong}\,\rho_{\mathbb{C}}^{*}(\mathsf{V},[\eta_{V}])$ and then showing that the action of $\rho_{\mathbb{C}}$ on the $\mathbb{C}$ -points are $\mathrm{Aut}(\mathbb{C}/E^{\prime})$ -equivariant. This is clearly inspired by the proof of [MPTate, Cor. 5.4]. However, unlike [MPTate, Prop. 5.6(1)], we show that $(\mathsf{W},[\xi])|_{T_{\mathbb{C}}}\,{\cong}\,\rho_{\mathbb{C}}^{*}(\mathsf{V},[\eta_{V}])$ descends over $T$ using a rigidity lemma about weakly-AM systems of realizations (see [YangSystem, (3.4.4), (3.4.5) and (4.3.2)]).

3.3 Orthogonal Shimura Varieties over Totally Real Fields

(3.3.1)

Let $E$ be a totally real number field and let $\mathcal{V}$ be a quadratic form over $E$ which has signature $(2,\dim_{E}\mathcal{V}-2)$ at a unique real place $\tau$ and is negative definitive at every other real place. We set $\mathcal{G}:=\mathrm{Res}_{E/\mathbb{Q}}\mathrm{SO}(\mathcal{V})$ and

\Omega_{\mathcal{V}}\coloneqq\{v\in\mathbb{P}(\mathcal{V}{\otimes}_{\tau}\mathbb{C})\mid\langle v,v\rangle=0,\langle v,\bar{v}\rangle>0\}.

Let $V$ be the transfer $\mathcal{V}_{(\mathbb{Q})}$ (recall (2.1.5)). Set $G:=\mathrm{SO}(V)$ and define $\Omega:=\Omega_{V}$ as above (applied to the $E=\mathbb{Q}$ case). Note that $\mathcal{G}$ is the identity component of the centeralizer of the $E$ -action in $G$ . We view $\Omega$ as a $G(\mathbb{R})$ -conjugacy class of morphisms $\mathbb{S}\to G_{\mathbb{R}}$ such that an element $v\in\Omega$ corresponds to the morphism $h$ which gives $V$ a Hodge structure of K3-type with $v=V_{\mathbb{C}}^{(1,-1)}$ (cf. [CSpin, §3.1]). Likewise we view $\Omega_{\mathcal{V}}$ as a $\mathcal{G}(\mathbb{R})$ -conjugacy class of morphisms $\mathbb{S}\to\mathcal{G}_{\mathbb{R}}$ , which are those whose composition with $\mathcal{G}_{\mathbb{R}}\to G_{\mathbb{R}}$ defines a Hodge structure on $V$ which is preserved by the $E$ -action on $V=\mathcal{V}_{(\mathbb{Q})}$ . It is well-known that $(\mathcal{G},\Omega_{\mathcal{V}})$ is a Shimura datum with reflex field $E$ , viewed as a subfield of $\mathbb{C}$ under $\tau$ .

Let $\widetilde{G}$ denote $\mathrm{CSpin}(V)$ . Then $(\widetilde{G},\Omega)$ is a Shimura datum of Hodge type with reflex field $\mathbb{Q}$ which admits a natural morphism to $(G,\Omega)$ .

Lemma (3.3.2).

$(\mathcal{G},\Omega_{\mathcal{V}})$ satisfies Assumption (3.2.3).

Proof.

Consider $\widetilde{\mathcal{G}}^{\prime}:=\mathrm{Res}_{E/\mathbb{Q}}\mathrm{CSpin}(\mathcal{V})$ . It is well known that $(\widetilde{\mathcal{G}}^{\prime},\Omega_{\mathcal{V}})$ is a Shimura datum with reflex field $E$ . Unfortunately, it does not satisfy condition (5) unless $E=\mathbb{Q}$ as the center $Z^{\prime}$ of $\widetilde{\mathcal{G}}^{\prime}$ has identity component $\mathrm{Res}_{E/\mathbb{Q}}\mathbb{G}_{m,E}$ . Thus we modify this approach by dividing out the maximal anisotropic torus of $\mathrm{Res}_{E/\mathbb{Q}}\mathbb{G}_{m,E}$ ; explicitly we consider ( $\mathrm{Nm}=$ the norm map)

\widetilde{\mathcal{G}}\coloneqq\widetilde{\mathcal{G}}^{\prime}/\ker(\mathrm{Nm}\colon\mathrm{Res}_{E/\mathbb{Q}}\mathbb{G}_{m,E}\to\mathbb{G}_{m})).

Note that $\ker(\widetilde{\mathcal{G}^{\prime}}\to\mathcal{G})=\mathrm{Res}_{E/\mathbb{Q}}\mathbb{G}_{m,E}$ , so we obtain a morphism $(\widetilde{\mathcal{G}},\Omega_{\mathcal{V}})\to(\mathcal{G},\Omega_{\mathcal{V}})$ of Shimura data. It remains to check (i) and (iii) in (3.2.3). First, note that $\widetilde{\mathcal{G}}$ can be canonically identified with the fiber product $\mathcal{G}\times_{G}\widetilde{G}$ (see [Moonen, §4.2, 4.3]), so that there is a diagram with exact rows

Now $Z_{s}(\widetilde{\mathcal{G}})=1$ is clear. As $(\widetilde{G},\Omega)$ is of Hodge type and $(\widetilde{\mathcal{G}},\Omega_{\mathcal{V}})$ embeds into $(\widetilde{G},\Omega)$ , $(\widetilde{\mathcal{G}},\Omega_{\mathcal{V}})$ is also of Hodge type. As the reflex fields of $(\mathcal{G},\Omega_{\mathcal{V}})$ and $(\widetilde{\mathcal{G}}^{\prime},\Omega_{\mathcal{V}})$ are both well known to be $E$ , the same must be true for $(\widetilde{\mathcal{G}},\Omega_{\mathcal{V}})$ . ∎

(3.3.3)

We do some preparations to future reference. Below for any $\mathbb{Q}$ -linear Tannakian category $\mathscr{C}$ , we write $\mathfrak{N}:\mathscr{C}_{(E)}\to\mathscr{C}$ for either one of the following two functors: $M\mapsto\mathrm{Nm}_{E/\mathbb{Q}}(M)$ or $M\mapsto\mathrm{Nm}_{E/\mathbb{Q}}(M){\otimes}_{\mathbb{Q}}\det(M_{(\mathbb{Q})})$ (see notations in (2.1.4)). Any conclusion applies to both functors.

Recall the notations from (2.1.5), and write $Z^{1}_{E}(\mathcal{V})$ and $\mathcal{H}(\mathcal{V})$ simply as $Z^{1}_{E}$ and $\mathcal{H}$ . Then $N:=\mathfrak{N}(\mathcal{V})$ is a faithful representation of $\mathcal{H}$ , as the composite $\mathcal{G}\to\mathrm{Res}_{E/\mathbb{Q}}\mathrm{GL}(\mathcal{V})\to\mathrm{GL}(N)$ has kernel exactly $Z^{1}_{E}$ . Let $\mathcal{K}\subseteq\mathcal{G}(\mathbb{A}_{f})$ and $\mathcal{C}\subseteq\mathcal{H}(\mathbb{A}_{f})$ be compact open subgroups such that $\mathcal{C}$ contains the image of $\mathcal{K}$ . For each prime $\ell$ , let $\mathcal{K}_{\ell}$ be the image of $\mathcal{K}$ under the projection $\mathcal{G}(\mathbb{A}_{f})\to\mathcal{G}(\mathbb{Q}_{\ell})$ .

Let $\Omega_{\mathcal{H}}$ be the $\mathcal{H}(\mathbb{R})$ -conjugacy class of morphisms $\mathbb{S}\to\mathcal{H}_{\mathbb{R}}$ which contains the image of $\Omega_{\mathcal{V}}$ . Then $(\mathcal{H},\Omega_{\mathcal{H}})$ is a Shimura datum which admits a natural morphism $(\mathcal{G},\Omega_{\mathcal{V}})\to(\mathcal{H},\Omega_{\mathcal{H}})$ . Since $Z^{1}_{E}$ lies in the center of $\mathcal{G}$ and is discrete, by [DeligneTdShimura, Prop. 3.8] $(\mathcal{H},\Omega_{\mathcal{H}})$ has the same reflex field as $(\mathcal{G},\Omega_{\mathcal{V}})$ . Moreover, one easily checks that $(\mathcal{H},\Omega_{\mathcal{H}})$ satisfies (3.2.3) using that $(\mathcal{G},\Omega_{\mathcal{V}})$ does.

Remark (3.3.4).

Note that if $\dim_{E}\mathcal{V}$ is odd, then $Z^{1}_{E}$ is trivial; in this case, the reader should read the content below with $(\mathcal{G},\Omega_{\mathcal{V}},\mathcal{K})=(\mathcal{H},\Omega_{\mathcal{H}},\mathcal{C})$ in mind. The case when $\dim_{E}\mathcal{V}$ is even will only be used for §4.3.

(3.3.5)

Let $k\subseteq\mathbb{C}$ be a subfield and $T$ be a smooth $k$ -variety. Note that the identification $V=\mathcal{V}_{(\mathbb{Q})}$ gives $V$ an $E$ -action, which commutes with $\mathcal{G}$ . Suppose that $\mathsf{W}\in\mathsf{R}(T)$ is a system equipped with a $(\mathcal{G},V,\mathcal{K})$ -level structure $[\mu]$ . Then $[\mu]$ transports the $E$ -action on $V{\otimes}\mathbb{A}_{f}$ to one on $\mathsf{W}_{\mathrm{{\acute{e}}t}}$ , through which we may view $[\mu]$ as a global section of $\mathcal{K}\backslash\underline{\mathrm{Isom}}_{E}(\mathcal{V}{\otimes}_{\mathbb{Q}}\mathbb{A}_{f},\mathsf{W}_{\mathrm{{\acute{e}}t}})$ . Moreover, there is a natural map

\mathcal{K}\backslash\underline{\mathrm{Isom}}_{E}(\mathcal{V}{\otimes}_{\mathbb{Q}}\mathbb{A}_{f},\mathsf{W}_{\mathrm{{\acute{e}}t}})\to\mathcal{C}\backslash\underline{\mathrm{Isom}}(\mathfrak{N}(\mathcal{V}){\otimes}\mathbb{A}_{f},\mathfrak{N}(\mathsf{W}_{\mathrm{{\acute{e}}t}}))

through which $[\mu]$ defines a $(\mathcal{H},N,\mathcal{C})$ -level structure on $\mathfrak{N}(\mathsf{W}_{\mathrm{{\acute{e}}t}})$ , which we denote by $\mathfrak{N}([\mu])$ .

Lemma (3.3.6).

Let $T$ , $\mathsf{W}$ and $[\mu]$ be as above. Let $\mathsf{W}^{\prime}\in\mathsf{R}(T)$ be another system with $\mathcal{K}$ -level structure $[\mu^{\prime}]$ . Suppose that there is an isomorphism $\gamma_{\mathrm{{\acute{e}}t}}:(\mathsf{W}_{\mathrm{{\acute{e}}t}},[\mu])|_{T_{\mathbb{C}}}\,{\cong}\,(\mathsf{W}_{\mathrm{{\acute{e}}t}}^{\prime},[\mu^{\prime}])|_{T_{\mathbb{C}}}$ such that $\mathfrak{N}(\gamma_{\mathrm{{\acute{e}}t}})$ descends to an isomorphism $(\mathfrak{N}(\mathsf{W}_{\mathrm{{\acute{e}}t}}),\mathfrak{N}([\mu]))\,{\cong}\,(\mathfrak{N}(\mathsf{W}_{\mathrm{{\acute{e}}t}}^{\prime}),\mathfrak{N}([\mu^{\prime}]))$ over $T$ .

If $\ell$ is a prime such that $\mathcal{K}_{\ell}\cap Z^{1}_{E}(\mathbb{Q}_{\ell})=1$ , then the $\ell$ -adic component $\gamma_{\ell}$ of $\gamma_{\mathrm{{\acute{e}}t}}$ descends to an isomorphism $\mathsf{W}_{\ell}\,{\cong}\,\mathsf{W}_{\ell}^{\prime}$ over $T$ .

Proof.

We may assume that $T$ is connected. Choose a base point $b\in T(\mathbb{C})$ . Set $\gamma:=\gamma_{\ell,b}$ and let $g\in\pi_{1}^{\mathrm{{\acute{e}}t}}(T,b)$ be any element. Our goal is to show that $\delta:=g^{-1}\gamma^{-1}g\gamma=1$ . Note that as $\mathfrak{N}(\gamma_{\mathrm{{\acute{e}}t}})$ descends to $T$ , we already know that $\mathfrak{N}(\delta)=1$ . Let $\mu:\mathcal{V}{\otimes}_{\mathbb{Q}}\mathbb{A}_{f}\stackrel{{\scriptstyle\sim}}{{\to}}\mathsf{W}_{\mathrm{{\acute{e}}t},b}$ be a representative of $[\mu]_{b}$ . Then $\mu^{-1}g\mu\in\mathcal{K}_{\ell}$ , because the $\mathcal{K}$ -orbit $[\mu]_{b}$ is $\pi_{1}^{\mathrm{{\acute{e}}t}}(S,b)$ -stable. Set $\mu^{\prime}:=\gamma\mu$ . Then $\mu^{\prime}$ represents $[\mu^{\prime}]_{b}$ , so that $(\mu^{\prime})^{-1}g\mu^{\prime}\in\mathcal{K}_{\ell}$ . Now we have

\mu^{-1}\delta\mu=[\mu^{-1}g^{-1}\mu][(\mu^{\prime})^{-1}g\mu^{\prime}]\in\mathcal{K}_{\ell}.

Note that $\mathfrak{N}(\mu^{-1}\delta\mu)=1\in\mathrm{GL}(N{\otimes}\mathbb{Q}_{\ell})$ . However, as the kernel of the $\mathcal{K}_{\ell}$ -action on $N{\otimes}\mathbb{Q}_{\ell}$ lies in $Z^{1}_{E}(\mathbb{Q}_{\ell})$ , we must have $\mu^{-1}\delta\mu=1$ , i.e., $\delta=1$ . ∎

(3.3.7)

We will often consider the following diagram of Shimura data:

Let $\mathsf{K}\subseteq G(\mathbb{A}_{f}),\mathcal{K}\subseteq\mathcal{G}(\mathbb{A}_{f})$ and $\mathcal{C}\subseteq\mathcal{H}(\mathbb{A}_{f})$ be neat compact open subgroups such that $\mathcal{K}\subseteq\mathsf{K}$ and $\mathcal{C}$ contains the image of $\mathcal{K}$ . Then we have Shimura morphisms $i\colon\mathrm{Sh}_{\mathcal{K}}(\mathcal{G})\to\mathrm{Sh}_{\mathsf{K}}(G)_{E}$ and $\pi\colon\mathrm{Sh}_{\mathcal{K}}(\mathcal{G})\to\mathrm{Sh}_{\mathcal{C}}(\mathcal{H})$ defined over $E$ . Let $\mathsf{V}$ (resp. $\widetilde{\mathsf{V}}$ ) be the automorphic system on $\mathrm{Sh}_{\mathsf{K}}(G)$ (resp. $\mathrm{Sh}_{\mathcal{K}}(\mathcal{G})$ ) attached to the standard representation $V$ of $G$ (resp. $\mathcal{G}$ ) and $[\eta_{V}]$ (resp. $[\eta_{\mathcal{V}}]$ ) be the tautological $\mathsf{K}$ -level structure (resp. $\mathcal{K}$ -level structure), as defined in (3.2.4). Then there is a natural identification $\widetilde{\mathsf{V}}=i^{*}(\mathsf{V})$ and $[\eta_{\mathcal{V}}]$ refines $i^{*}([\eta_{V}])$ in the sense of (3.1.2)(d).

Note that $[\eta_{\mathcal{V}}]$ endows $\widetilde{\mathsf{V}}$ with a canonical $E$ -action. A priori it only defines an $E$ -action on $\widetilde{\mathsf{V}}_{\mathrm{{\acute{e}}t}}$ , but since $[\eta_{\mathcal{V}}]$ is $\widetilde{\mathsf{V}}$ -rational, its restriction to $\mathrm{Sh}_{\mathcal{K}}(\mathcal{G})_{\mathbb{C}}$ comes from an $E$ -action on $\widetilde{\mathsf{V}}_{B}$ . Then by (3.1.6), one deduces that the resulting $E$ -action on $\widetilde{\mathsf{V}}_{\mathrm{dR}}|_{\mathrm{Sh}_{\mathcal{K}}(\mathcal{G})_{\mathbb{C}}}$ via the Riemann-Hilbert correspondence descends to $\mathrm{Sh}_{\mathcal{K}}(\mathcal{G})$ . Therefore, it makes sense to form $\mathfrak{N}(\widetilde{\mathsf{V}})$ . Since $\widetilde{\mathsf{V}}$ is weakly AM in the sense of (3.1.5), so is $\mathfrak{N}(\widetilde{\mathsf{V}})$ . This is simply because we may apply the functor $\mathfrak{N}(-)$ to $\mathsf{Mot}_{\mathsf{Ab}}(\mathbb{C})_{(E)}$ and it commutes with the cohomological realizations.

Now let $\mathsf{N}$ be the automorphic system on $\mathrm{Sh}_{\mathcal{C}}(\mathcal{H})$ attached to $N\in\mathrm{Rep}(\mathcal{H})$ and let $[\eta_{N}]$ be its tautological $\mathcal{C}$ -level structure. We claim that there is an (necessarily unique) isomorphism

\pi^{*}(\mathsf{N},[\eta_{N}])\,{\cong}\,(\mathfrak{N}({\widetilde{\mathsf{V}}}),\mathfrak{N}([\eta_{\mathcal{V}}])).

(8)

Indeed, to verify this isomorphism one first checks its implications on the automorphic VHS and étale local systems, which follow from the explicit descriptions in (3.2.1), then applies (3.1.6).

3.4 Integral Model

Let $V$ be a quadratic form over $\mathbb{Q}$ and suppose that there is a self-dual $\mathbb{Z}_{(p)}$ -lattice $L_{(p)}\subseteq V$ for a prime $p>2$ . Then $G:=\mathrm{SO}(V)$ extends to the reductive $\mathbb{Z}_{(p)}$ -group $\mathrm{SO}(L_{(p)})$ , which we still write as $G$ by abuse of notation. Let $\mathsf{K}\subseteq G(\mathbb{A}_{f})$ be a neat compact open subgroup of the form $\mathsf{K}_{p}\mathsf{K}^{p}$ with $\mathsf{K}_{p}=G(\mathbb{Z}_{p})$ and $\mathsf{K}^{p}\subseteq G(\mathbb{A}^{p}_{f})$ . Then by [KisinInt] and [CSpin], $\mathrm{Sh}_{\mathsf{K}}(G)$ admits a canonical integral model $\mathscr{S}_{\mathsf{K}}(G)$ over $\mathbb{Z}_{(p)}$ .

The Shimura variety $\mathscr{S}_{\mathsf{K}}(G)$ is typically studied via the corresponding spinor Shimura variety, which is of Hodge type. Let $\widetilde{G}\coloneqq\mathrm{CSpin}(L_{(p)})$ . Set $\mathbb{K}_{p}$ to be $\widetilde{G}(\mathbb{Z}_{p})$ , $\mathbb{K}^{p}\subseteq\widetilde{G}(\mathbb{A}^{p}_{f})$ to be a small enough compact open subgroup whose image in $G(\mathbb{A}^{p}_{f})$ is contained in $\mathsf{K}^{p}$ , and $\mathbb{K}$ to be the product $\mathbb{K}_{p}\mathbb{K}^{p}$ . The reflex field of $(\widetilde{G},\Omega)$ is $\mathbb{Q}$ and by [KisinInt] there is an canonical integral model $\mathscr{S}_{\mathbb{K}}(\widetilde{G})$ over $\mathbb{Z}_{(p)}$ . There is a suitable sympletic space $(H,\psi)$ and a Siegel half space $\mathcal{H}^{\pm}$ such that there is an embedding of Shimura data $(\widetilde{G},\Omega)\hookrightarrow(\mathrm{GSp}(\psi),\mathcal{H}^{\pm})$ which eventually equips $\mathscr{S}_{\mathbb{K}}(\widetilde{G})$ with a universal abelian scheme $\mathscr{A}$ .⁶⁶6Technically, in [KisinInt] and [CSpin], $\mathscr{A}$ is only defined as a sheaf of abelian schemes up to prime-to- $p$ quasi-isogeny. However, for $\mathbb{K}^{p}$ sufficiently small, we can take $\mathscr{A}$ to be an actual abelian scheme (cf. [KisinInt, (2.1.5)]). Let $a:\mathscr{A}\to\mathscr{S}_{\mathbb{K}}(\widetilde{G})$ be the structural morphism. Define the sheaves $\mathbf{H}_{B}:=R^{1}a_{\mathbb{C}*}\mathbb{Z}_{(p)}$ , $\mathbf{H}_{\ell}:=R^{1}a_{*}\underline{\mathbb{Q}}_{\ell}$ ( $\ell\neq p$ ), $\mathbf{H}_{p}:=R^{1}a_{\mathbb{Q}*}\underline{\mathbb{Z}}_{p}$ , $\mathbf{H}_{\mathrm{dR}}:=R^{1}a_{*}\Omega^{\bullet}_{\mathscr{A}/\mathscr{S}_{\mathbb{K}}(\widetilde{G})}$ and $\mathbf{H}_{\mathrm{cris}}:=R^{1}\bar{a}_{\mathrm{cris}*}\mathcal{O}_{\mathscr{A}_{\mathbb{F}_{p}}/\mathbb{Z}_{p}}$ ( $\bar{a}:=a{\otimes}\mathbb{F}_{p}$ ). The abelian scheme $\mathscr{A}$ is equipped with a “CSpin-structure”: a $\mathbb{Z}/2\mathbb{Z}$ -grading, $\mathrm{Cl}(L)$ -action and an idempotent projector $\boldsymbol{\pi}_{?}:\mathrm{End}(\mathbf{H}_{?})\to\mathrm{End}(\mathbf{H}_{?})$ for $?=B,\ell,p,\mathrm{dR},\mathrm{cris}$ on (various applicable fibers of) $\mathscr{S}_{\mathbb{K}}(\widetilde{G})$ . We use $\mathbf{L}_{?}$ to denote the images of $\boldsymbol{\pi}_{?}$ , and recall the definition of special endomorphisms ([CSpin, Def. 5.2, see also Lem. 5.4, Cor. 5.22]):

Definition (3.4.1).

For any $\mathscr{S}_{\mathbb{K}}(\widetilde{G})$ -scheme $T$ , $f\in\mathrm{End}(\mathscr{A}_{T})$ is called a special endomorphism if for some (and hence all) $\ell\in\mathcal{O}_{T}^{\times}$ , the $\ell$ -adic realization of $f$ lies in $\mathbf{L}_{\ell}|_{T}\subseteq\mathrm{End}(\mathbf{H}_{\ell}|_{T})$ ; if $\mathcal{O}_{T}=k$ for a perfect field $k$ in characteristic $p$ , then equivalently $f$ is called a special endomorphism if the crystalline realization of $f$ lies in $\mathbf{L}_{\mathrm{cris},T}$ . We write the submodule of $\mathrm{End}(\mathscr{A}_{T})$ consisting of special endomorphisms as $\mathrm{L}(\mathscr{A}_{T})$ .

(3.4.2)

It is explained in [CSpin, §5.24] that the sheaves $\mathbf{L}_{?}$ on (applicable fibers of) $\mathscr{S}_{\mathbb{K}}(\widetilde{G})$ in fact descend to the corresponding fibers of $\mathscr{S}_{\mathsf{K}}(G)$ . We denote the descent of these sheaves by the same letters. It is not hard to see that $\mathbf{H}_{B}[1/p]$ together with the restrictions of $\prod_{\ell\neq p}\mathbf{H}_{\ell}\times\mathbf{H}_{p}[1/p]$ and $\mathbf{H}_{\mathrm{dR}}$ to $\mathrm{Sh}_{\mathbb{K}}(\widetilde{G})$ is nothing but the automorphic system attached to $H\in\mathrm{Rep}(\widetilde{G})$ , in our terminology (3.2.4). Similarly, $\mathbf{L}_{B}[1/p]$ together with the restrictions of $\prod_{\ell\neq p}\mathbf{L}_{\ell}\times\mathbf{L}_{p}[1/p]$ and $\mathbf{L}_{\mathrm{dR}}$ to $\mathrm{Sh}_{\mathbb{K}}(\widetilde{G})$ (or $\mathrm{Sh}_{\mathsf{K}}(G)$ ) is precisely the automorphic system attached to $V\in\mathrm{Rep}(\widetilde{G})$ (or $\mathrm{Rep}(G)$ ). Readers who wish to check this can look at how the automorphic systems are constructed in [YangSystem, §4.2], which is essentially a generalization of [CSpin, §5.24]. The construction of the $\mathbf{L}_{?}$ -sheaves are also summarized in more detail in [Yang, (3.1.3)]. In particular, there are natural identifications of $\mathsf{V}_{B}$ with $\mathbf{L}_{B}[1/p]$ , and $\mathsf{V}_{\mathrm{{\acute{e}}t}}$ (resp. $\mathsf{V}_{\mathrm{dR}}$ ) with the restriction of $\prod_{\ell\neq p}\mathbf{L}_{\ell}\times\mathbf{L}_{p}[1/p]$ (resp. $\mathbf{L}_{\mathrm{dR}}$ ) to $\mathrm{Sh}_{\mathsf{K}}(G)$ , where $(\mathsf{V}_{B},\mathsf{V}_{\mathrm{dR}},\mathsf{V}_{\mathrm{{\acute{e}}t}})$ was the notation we used in (3.3.7).

(3.4.3)

Let $\mathrm{Sh}_{\mathsf{K}_{p}}(G)$ be the limit $\varprojlim_{\mathsf{K}^{p}}\mathrm{Sh}_{\mathsf{K}_{p}\mathsf{K}^{\prime p}}(G)$ as $\mathsf{K}^{\prime p}$ runs through the compact open subgroups of $G(\mathbb{A}^{p}_{f})$ and define $\mathscr{S}_{\mathsf{K}_{p}}(G)$ similarly. The canonical extension property of $\mathscr{S}_{\mathsf{K}_{p}}(G)$ is that for any regular, formally smooth $\mathbb{Z}_{(p)}$ -scheme $S$ , every morphism $S_{\mathbb{Q}}\to\mathscr{S}_{\mathsf{K}_{p}}(G)$ extends to $S$ . We give an extension property for finite level, which is certainly well known to experts:

Theorem (3.4.4).

Let $S$ be a smooth $\mathbb{Z}_{(p)}$ -scheme which admits a morphism $\rho:S_{\mathbb{Q}}\to\mathrm{Sh}_{\mathsf{K}}(G)$ . If $\rho^{*}\mathsf{V}_{\ell}$ extends to a local system $\mathsf{W}_{\ell}$ over $S$ for every prime $\ell\neq p$ , then $\rho$ extends to a morphism $S\to\mathscr{S}_{\mathsf{K}}(G)$ , through which $\mathsf{W}_{\ell}$ is identified with the pullback of $\mathbf{L}_{\ell}$ .

Proof.

To simplify notation, let $\mathsf{V}_{\mathrm{{\acute{e}}t}}^{(p)}\coloneqq\prod_{\ell\neq p}\mathsf{V}_{\ell}$ and $\mathsf{W}_{\mathrm{{\acute{e}}t}}^{(p)}:=\prod_{\ell\neq p}\mathsf{W}_{\ell}$ . Note that $\mathsf{W}_{\mathrm{{\acute{e}}t}}^{(p)}|_{S_{\mathbb{Q}}}=\rho^{*}(\mathsf{V}_{\mathrm{{\acute{e}}t}}^{(p)})$ . Now the prime-to- $p$ part of the tautological level structure $[\eta_{V}]$ gives us a section $[\eta_{V}^{(p)}]:\mathrm{Sh}_{\mathsf{K}}(G)\to\mathsf{K}^{p}\backslash\underline{\mathrm{Isom}}(V{\otimes}\mathbb{A}^{p}_{f},\mathsf{V}_{\mathrm{{\acute{e}}t}}^{(p)})$ , where the target is viewed as a pro-étale cover of $\mathrm{Sh}_{\mathsf{K}}(G)$ . By [stacks-project, 0BQM], $\rho^{\ast}[\eta^{(p)}_{V}]$ extends to a morphism $S\to\mathsf{K}^{p}\backslash\underline{\mathrm{Isom}}(V\otimes\mathbb{A}_{f}^{p},\mathsf{W}_{\mathrm{{\acute{e}}t}}^{(p)})$ , where the target is a pro-étale cover of $S$ . We define $\widetilde{S}$ as the pull-back

Note that since $\widetilde{S}\to S$ is a $\mathsf{K}^{p}$ -torsor for the proétale topology, $\widetilde{S}$ is representable by a scheme. As $\mathrm{Sh}_{\mathsf{K}_{p}}(G)\to\mathrm{Sh}_{\mathsf{K}}(G)$ can be defined by an analogous construction, $\rho$ lifts to a $\mathsf{K}^{p}$ -equivariant morphism $\widetilde{\rho}\colon\widetilde{S}_{\mathbb{Q}}\to\mathrm{Sh}_{\mathsf{K}_{p}}(G)$ . The canonical extension property allows us to extend it to a (necessarily $\mathsf{K}^{p}$ -equivariant) morphism $\widetilde{S}\to\mathscr{S}_{\mathsf{K}_{p}}(G)$ . Now the $\mathsf{K}^{p}$ -action defines an étale descend datum which yields the desired morphism $S\to\mathscr{S}_{\mathsf{K}}(G)$ . ∎

4 Period Morphisms

4.1 The Basic Set-up

We first state a few basic definitions and results which will be needed to construct the period morphism.

Definition (4.1.1).

Let $T$ be a connected noetherian normal scheme with geometric point $t$ , $\ell$ be a prime with $\ell\in\mathcal{O}_{T}^{\times}$ and $\mathsf{W}_{\ell}$ be an étale $\mathbb{Q}_{\ell}$ -local system over $T$ . We denote by $\mathrm{Mon}(\mathsf{W}_{\ell},t)$ the Zariski closure of the image of $\pi_{1}^{\mathrm{{\acute{e}}t}}(T,t)$ in $\mathrm{GL}(\mathsf{W}_{\ell,t})$ , and denote by $\mathrm{Mon}^{\circ}(\mathsf{W}_{\ell},t)$ the identity component of $\mathrm{Mon}(\mathsf{W}_{\ell},t)$ .

If $\mathsf{W}_{\ell}^{\prime}$ is another $\mathbb{Q}_{\ell}$ -local system, then we say that $\mathsf{W}_{\ell}$ and $\mathsf{W}_{\ell}^{\prime}$ are étale locally isomorphic if they are isomorphic over some finite connected étale cover $T^{\prime}$ of $T$ ; or equivalently, there is an isomorphism $\mathsf{W}_{\ell,t}\stackrel{{\scriptstyle\sim}}{{\to}}\mathsf{W}^{\prime}_{\ell,t}$ which is equivariant under an open subgroup of $\pi^{\mathrm{{\acute{e}}t}}_{1}(T,t)$ .

Lemma (4.1.2).

Let $T$ be a noetherian integral normal scheme with generic point $\eta$ . Let $f:\mathcal{Y}\to T$ be a smooth proper morphism.

(a)

The natural map $\mathrm{Pic}\,(\mathcal{Y})\to\mathrm{Pic}\,(\mathcal{Y}_{\eta})$ is surjective with kernel $\mathrm{Pic}\,(T)$ .
(b)

If for some geometric point $b$ over $\eta$ and prime $\ell\in\mathcal{O}_{T}^{\times}$ , $\mathrm{Mon}(R^{2}f_{*}\mathbb{Q}_{\ell},b)$ is connected, then the natural map $\mathrm{NS}(\mathcal{Y}_{\eta})_{\mathbb{Q}}\to\mathrm{NS}(\mathcal{Y}_{b})_{\mathbb{Q}}$ is an isomorphism.

Proof.

(a) We may always extend a line bundle on $\mathcal{Y}_{\eta}$ to $\mathcal{Y}_{U}$ for some open dense subscheme $U\subseteq T$ , and then to a line bundle on $\mathcal{Y}$ (use e.g., [Hartshorne, Prop. II.6.5]). But any two extensions to $\mathcal{Y}$ differ by an element of $\mathrm{Pic}\,(T)$ by [EGAIV4, Err_IV Cor. 21.4.13].

(b) Let $\bar{\eta}$ be the geometric point over $\eta$ obtained by taking the separable closure of $k(\eta)$ in $k(b)$ . Then every class of $\mathrm{NS}(\mathcal{Y}_{b})_{\mathbb{Q}}$ descends to $\bar{\eta}$ . As the natural morphism $\mathrm{Gal}(\bar{\eta}/\eta)=\pi^{\mathrm{{\acute{e}}t}}_{1}(\eta,b)\to\pi^{\mathrm{{\acute{e}}t}}_{1}(T,b)$ is surjective, and $\mathrm{Gal}(\bar{\eta}/\eta)$ acts on $\mathrm{NS}(\mathcal{Y}_{\bar{\eta}})_{\mathbb{Q}}$ through a finite quotient, the connectedness assumption on $\mathrm{Mon}(R^{2}f_{*}\mathbb{Q}_{\ell},b)$ implies that $\mathrm{Gal}(\bar{\eta}/\eta)$ in fact acts trivially. This implies that every class in $\mathrm{NS}(\mathcal{Y}_{\bar{\eta}})_{\mathbb{Q}}$ descends to $\eta$ . ∎

Now we state the set-up we will work with for the entire section.

Set-up (4.1.3).

Let $F\subseteq\mathbb{C}$ be a subfield finitely generated over $\mathbb{Q}$ and let $S$ be a connected smooth $F$ -variety with generic point $\eta$ . Let $f\colon\mathcal{X}\to S$ be a $\heartsuit$ -family with a relatively ample line bundle $\boldsymbol{\xi}$ . We fix a subspace $\Lambda\subseteq\mathrm{NS}(\mathcal{X}_{\eta})_{\mathbb{Q}}$ containing the class of $\boldsymbol{\xi}_{\eta}$ . Now choose an $F$ -linear embedding $k(\eta)\hookrightarrow\mathbb{C}$ and let $b\in S(\mathbb{C})$ be the resulting point, which we also view as a closed point on $S_{\mathbb{C}}$ . Let $S^{\circ}\subset S_{\mathbb{C}}$ denote the connected component containing $b$ . We assume that for some (and hence every, see (4.1.4) below) prime $\ell$ , $\mathrm{Mon}(R^{2}f_{*}\mathbb{Q}_{\ell},b)$ is connected.

Define a pairing on $\mathrm{H}^{2}_{\mathrm{dR}}(\mathcal{X}/S):=R^{2}f_{*}\Omega^{\bullet}_{\mathcal{X}/S}$ by $(x,y)\mapsto x\cup y\cup c_{1}(\boldsymbol{\xi})^{d-2}$ for local sections $x,y$ and $d=\dim\mathcal{X}/S$ . By (4.1.2), every line bundle on $\mathcal{X}_{\eta}$ extends to a relative line bundle on $\mathcal{X}/S$ . By taking Chern classes, we obtain a well defined embedding $\underline{\Lambda}\hookrightarrow\mathrm{H}^{2}_{\mathrm{dR}}(\mathcal{X}/S)$ , where $\underline{\Lambda}$ is the constant sheaf with fiber $\Lambda$ . Define $\mathsf{P}_{\mathrm{dR}}$ to be the orthogonal complement of $\underline{\Lambda}$ in $\mathrm{H}^{2}_{\mathrm{dR}}(\mathcal{X}/S)(1)$ . We define the primitive Betti cohomology $\mathsf{P}_{B}$ over $S_{\mathbb{C}}$ and étale cohomology $\mathsf{P}_{\mathrm{{\acute{e}}t}}$ over $S$ analogously. Then $\mathsf{P}:=(\mathsf{P}_{B},\mathsf{P}_{\mathrm{dR}},\mathsf{P}_{\mathrm{{\acute{e}}t}})\in\mathsf{R}(S)$ is a system of realizations over $S$ in the sense of (3.1.1). Note that we applied a Tate twist so that the VHS $\mathsf{P}|_{S_{\mathbb{C}}}$ has weight $0$ . Since we assumed $\mathrm{Mon}(R^{2}f_{*}\mathbb{Q}_{\ell},b)$ is connected for some $\ell$ , $\mathrm{NS}(\mathcal{X}_{\eta})_{\mathbb{Q}}=\mathrm{NS}(\mathcal{X}_{b})_{\mathbb{Q}}$ by (4.1.2). This implies that $\mathsf{P}$ orthogonally decomposes into $(\Lambda^{\perp}{\otimes}\mathbf{1}_{S})\oplus\mathsf{P}_{0}$ where $\Lambda^{\perp}$ is the orthogonal complement of $\Lambda$ in $\mathrm{NS}(\mathcal{X}_{\eta})_{\mathbb{Q}}$ and $\mathsf{P}_{0}=(\mathsf{P}_{0,B},\mathsf{P}_{0,\mathrm{dR}},\mathsf{P}_{0,\mathrm{{\acute{e}}t}})$ is another object in $\mathsf{R}(S)$ . Moreover, there are no nonzero $(0,0)$ -classes in $\mathsf{P}_{0,B,b}$ . Note that $\mathsf{P}_{0}$ is nothing but $\mathsf{P}$ when $\Lambda=\mathrm{NS}(\mathcal{X}_{\eta})_{\mathbb{Q}}$ .

Lemma (4.1.4).

In the above set-up, $\mathrm{Mon}(R^{2}f_{*}\mathbb{Q}_{\ell},b)$ is connected for every prime $\ell$ , and $b$ is a Hodge-generic point for the VHS $\mathsf{P}|_{S^{\circ}}$ .

Proof.

The first statement follows from the assumption that $F$ is finitely generated over $\mathbb{Q}$ and [Larsen-Pink, Prop. 6.14], which implies that the étale group scheme of connected components of $\mathrm{Mon}(R^{2}f_{*}\mathbb{Q}_{\ell},b)$ is independent of $\ell$ . The second statement follows from the main theorem of [Moonen].⁷⁷7The main theorem of [Moonen] is an overkill for the purpose here and only used to avoid a case by case discussion of Mumford-Tate groups. Since the statement only concerns $S^{\circ}$ , we may replace $S$ by $S^{\circ}$ and $F$ by its field of definition and thus assume that $S$ is geometrically connected. Then we may make use of the notion of Galois-generic points ([Moonen-Fom, Def. 4.2.1]). As the Momford-Tate conjecture is known for $H^{2}$ of the fibers of $\mathcal{X}/S$ and $b$ lies above $\eta$ , the fact that $\eta$ is Galois-generic implies that $b$ is Hodge-generic. ∎

(4.1.5)

Let $V$ be a quadratic form over $\mathbb{Q}$ which is isomorphic to $\mathsf{P}_{B,b}$ and fix an isometry $\mu_{b}:V\stackrel{{\scriptstyle\sim}}{{\to}}\mathsf{P}_{B,b}$ . Let $G:=\mathrm{SO}(V)$ . Let $V_{0}:=\mu_{b}^{-1}(\mathsf{P}_{0,B,b})$ . Note that via $\mu_{b}$ , the monodromy representation of $\pi^{\mathrm{{\acute{e}}t}}_{1}(S,b)$ takes values in $G(\mathbb{A}_{f})$ . We say that $\mathsf{K}$ is admissible if the image of $\pi_{1}^{\mathrm{{\acute{e}}t}}(S,b)$ in $\mathrm{GL}(\mathsf{P}_{B,b}{\otimes}\mathbb{A}_{f})$ lies in $\mathsf{K}$ via $\mu_{b}$ , or equivalently, $\mu_{b}$ extends to a $(G,V,\mathsf{K})$ -level structure $[\mu]$ on $\mathsf{P}_{\mathrm{{\acute{e}}t}}$ with $[\mu]_{b}=\mathsf{K}\cdot(\mu_{b}{\otimes}\mathbb{A}_{f})$ . Define $\Omega:=\{v\in\mathbb{P}(V{\otimes}\mathbb{C})\mid\langle v,v\rangle=0,\langle v,\bar{v}\rangle>0\}$ as in (3.3.3). Then $(G,\Omega)$ is a Shimura datum of abelian type with reflex field $\mathbb{Q}$ . Again let $\mathsf{V}=(\mathsf{V}_{B},\mathsf{V}_{\mathrm{dR}},\mathsf{V}_{\mathrm{{\acute{e}}t}})$ be the automorphic system on $\mathrm{Sh}_{\mathsf{K}}(G)$ attached to $V\in\mathrm{Rep}(G)$ , and let $[\eta_{V}]$ be the tautological $\mathsf{K}$ -level structure on $\mathsf{V}_{\mathrm{{\acute{e}}t}}$ (see (3.2.4)).

(4.1.6)

Suppose that $(\mathcal{X}/S,\boldsymbol{\xi})|_{S^{\circ}}$ belongs to case (R+) or (R2’) described in (2.2.4), i.e., the endomorphism field $E$ of the Hodge structure on $\mathsf{P}_{0,B,b}$ is totally real. By (2.2.2), the $E$ -action on $\mathsf{P}_{0,B,b}$ is self-adjoint. Let $E$ act on $V_{0}$ through $\mu_{b}$ . Recall the notations in (2.1.5). Let $\mathcal{V}$ be the $E$ -bilinear lift of $V_{0}$ , i.e., $V_{0}=\mathcal{V}_{(\mathbb{Q})}$ and set $\mathcal{G}:=\mathrm{Res}_{E/\mathbb{Q}}\mathrm{SO}(\mathcal{V})$ , $\mathcal{H}:=\mathcal{G}/Z_{E}^{1}$ . In addition, set $\mathcal{V}^{\sharp}:=\mathcal{V}\oplus\mathcal{E}$ and $\mathcal{G}^{\sharp}:=\mathrm{Res}_{E/\mathbb{Q}}\mathrm{SO}(\mathcal{V}^{\sharp})$ . Embed $\mathrm{SO}(\mathcal{V})$ into $\mathrm{SO}(\mathcal{V}^{\sharp})$ by acting trivially on $\mathcal{E}$ . This induces an embedding $\mathcal{G}\hookrightarrow\mathcal{G}^{\sharp}$ .

We say that a neat compact open subgroup $\mathsf{K}\subset G(\mathbb{A}_{f})$ satisfies condition $(\sharp)$ depending on the particular case ( $\mathcal{K}:=\mathsf{K}\cap\mathcal{G}(\mathbb{A}_{f})$ below):

(R1)

Always.
(R2)

If the image of $\mathcal{K}$ in $\mathcal{G}^{\sharp}(\mathbb{A}_{f})$ lies in some neat compact open subgroup.
(R2’)

If the image of $\mathcal{K}$ in $\mathcal{H}(\mathbb{A}_{f})$ lies in some neat compact open subgroup.

In case (R2’), we say that for a prime $\ell_{0}$ , $\mathsf{K}_{\ell_{0}}$ is sufficiently small if $\mathcal{K}_{\ell_{0}}\cap Z_{E}^{1}(\mathbb{Q}_{\ell_{0}})=1$ . Here $\mathsf{K}_{\ell_{0}}$ is the image of $\mathsf{K}$ under the projection $G(\mathbb{A}_{f})\to G(\mathbb{Q}_{\ell_{0}})$ and $\mathcal{K}_{\ell_{0}}$ is defined similarly.

(4.1.7)

For any subfield $F^{\prime}\subseteq\mathbb{C}$ which contains $F$ and $s\in S(F^{\prime})$ , we denote by $\mathfrak{p}_{s}\in\mathsf{Mot}_{\mathrm{AH}}(F^{\prime})$ the submotive of $\mathfrak{h}^{2}(\mathcal{X}_{s})(1)$ such that $\omega_{?}(\mathfrak{p}_{s})=\mathsf{P}_{?,s}$ for $?=B$ (when $F^{\prime}=\mathbb{C}$ ), or $\mathrm{dR},\mathrm{{\acute{e}}t}$ . Note that for any $\sigma\in\mathrm{Aut}(\mathbb{C}/F)$ and $s\in S(\mathbb{C})$ , there is a natural isomorphism $(\mathfrak{p}_{s})^{\sigma}\,{\cong}\,\mathfrak{p}_{\sigma(s)}$ in $\mathsf{Mot}_{\mathrm{AH}}(\mathbb{C})$ , which we shall label as $\sigma_{\mathfrak{p},s}$ ; moreover, the following diagrams tautologically commute (recall the notations in (2.1.3) and (3.1.4)):

(4.1.8)

For the rest of section 4, we will put a standing assumption that $S$ is geometically connected as an $F$ -variety, so that $S_{\mathbb{C}}=S^{\circ}$ .

Let $\mathsf{K}\subseteq G(\mathbb{A}_{f})$ be a neat compact open subgroup and assume that it is also admissible, so that $\mu_{b}$ extends to a $\mathsf{K}$ -level structure $[\mu]$ on $\mathsf{P}$ with $[\mu]_{b}=\mathsf{K}(\mu_{b}{\otimes}\mathbb{A}_{f})$ . Now (3.1.7) guarantees that the restriction of $[\mu]$ to $S_{\mathbb{C}}$ is $(\mathsf{P}|_{S_{\mathbb{C}}})$ -rational and is of type $\Omega$ . Therefore, (3.2.2) gives us a unique morphism $\rho_{\mathbb{C}}:S_{\mathbb{C}}\to\mathrm{Sh}_{\mathsf{K}}(G)_{\mathbb{C}}$ such that

(\rho_{\mathbb{C}})^{*}(\mathsf{V},[\eta_{V}])\,{\cong}\,(\mathsf{P},[\mu])|_{S_{\mathbb{C}}}.

(9)

Note that as $\mathsf{K}$ is neat, the above isomorphism is unique; we denote its étale component by $\alpha^{\circ}_{\mathrm{{\acute{e}}t}}:\rho_{\mathbb{C}}^{*}(\mathsf{V}_{\mathrm{{\acute{e}}t}})\stackrel{{\scriptstyle\sim}}{{\to}}\mathsf{P}_{\mathrm{{\acute{e}}t}}|_{S_{\mathbb{C}}}$ and let $\alpha^{\circ}_{\ell}$ be its $\ell$ -adic component for a prime $\ell$ .

Theorem (4.1.9).

In the notations above, assume either

(a)

$\mathfrak{p}_{s}\in\mathsf{Mot}_{\mathsf{Ab}}(\mathbb{C})$ for every $s\in S_{\mathbb{C}}$ (e.g., when the family $(\mathcal{X}/S,\boldsymbol{\xi})|_{S_{\mathbb{C}}}$ belongs to case (CM)), or
(b)

the family $(\mathcal{X}/S,\boldsymbol{\xi})|_{S_{\mathbb{C}}}$ belongs to case (R+) = (R1) + (R2) and $\mathsf{K}$ satisfies condition $(\sharp)$ as defined in (4.1.6).

Then $\rho_{\mathbb{C}}$ descends to a morphism $\rho:S\to\mathrm{Sh}_{\mathsf{K}}(G)_{F}$ over $F$ . Moreover, $\alpha^{\circ}_{\mathrm{{\acute{e}}t}}$ descends to an isomorphism $\alpha_{\mathrm{{\acute{e}}t}}:\rho^{*}(\mathsf{V}_{\mathrm{{\acute{e}}t}})\stackrel{{\scriptstyle\sim}}{{\to}}\mathsf{P}_{\mathrm{{\acute{e}}t}}$ over $S$ .

Case (a) is easy: The diagrams in (4.1.7) readily imply that $\mathsf{P}$ over $S$ is weakly AM, so that the conclusion follows from (3.2.5). In fact, we have that the de Rham component of the isomorphism (9) also descends to $S$ . Note that case (a) in particular covers the case when $(\mathcal{X}/S,\boldsymbol{\xi})|_{S_{\mathbb{C}}}$ belongs to case (CM) by (2.2.7)(d). The proof of case (b) is the content of §4.2 below. If $(\mathcal{X}/S,\boldsymbol{\xi})|_{S_{\mathbb{C}}}$ belongs to case (R2’) and it is not known that $\mathfrak{p}_{s}\in\mathsf{Mot}_{\mathsf{Ab}}(\mathbb{C})$ for every $s\in S_{\mathbb{C}}$ , a slightly weaker version of the above theorem holds:

Theorem (4.1.10).

Assume that the family $(\mathcal{X}/S,\boldsymbol{\xi})|_{S^{\circ}}$ belongs to case (R2’), $\Lambda=\Lambda_{0}$ , $\mathsf{K}$ satisfies condition $(\sharp)$ , and for a prime $\ell_{0}$ , $\mathsf{K}_{\ell_{0}}$ is sufficiently small as defined in (4.1.6).

Then $\rho_{\mathbb{C}}:S_{\mathbb{C}}\to\mathrm{Sh}_{\mathsf{K}}(G)_{\mathbb{C}}$ descends to a morphism $\rho:S_{F^{\prime}}\to\mathrm{Sh}_{\mathsf{K}}(G)_{F^{\prime}}$ for a finite extension $F^{\prime}/F$ in $\mathbb{C}$ . Moreover, $\alpha^{\circ}_{\ell_{0}}$ descends to an isomorphism $\alpha_{\ell_{0}}:\rho^{*}\mathsf{V}_{\ell_{0}}\stackrel{{\scriptstyle\sim}}{{\to}}\mathsf{P}_{\ell_{0}}|_{S_{F^{\prime}}}$ , and $\rho^{*}\mathsf{V}_{\ell}$ is étale locally isomorphic to $\mathsf{P}_{\ell}|_{S_{F^{\prime}}}$ for every other $\ell$ in the sense of (4.1.1).

Remark (4.1.11).

In literature, to define period morphisms to orthogonal Shimura varieties, one usually keeps track of a trivialization of the determinants (cf. [MPTate, Prop. 4.3]). Such a trivialization (i.e., an isometry $\underline{\det(V)}\stackrel{{\scriptstyle\sim}}{{\to}}\det(\mathsf{P}_{B})$ in our notations) is implicit in the statement that (the restriction to $S_{\mathbb{C}}$ of) $[\mu]$ is $(\mathsf{P}|_{S_{\mathbb{C}}})$ -rational as a $(G,V,\mathsf{K})$ -level structure, because $\det(V)$ is $G$ -invariant. More concretely, one obtains this trivialization by globalizing $\det(\mu_{b})$ , using that $\pi_{1}(S_{\mathbb{C}},b)$ fixes $\det(\mathsf{P}_{B,b})$ (cf. the proof of (3.1.7)). However, we remark that if $S$ were not geometrically connected, we would not be able to show that $[\mu]$ is $(\mathsf{P}|_{S_{\mathbb{C}}})$ -rational over the connected components of $S_{\mathbb{C}}$ other than $S^{\circ}$ , unless we know $\det(\mathsf{P}_{B,b})$ is spanned by an absolute Hodge class (e.g., in case (a) of (4.1.9)).

On the other hand, for the purpose of putting level structures, the lack of absolute-Hodgeness of $\det(\mathsf{P}_{B,b})$ is partially remedied by independence-of- $\ell$ type results on algebraic monodromy (e.g., [Saitodisc, Lem. 3.2], cf. [Taelman2, Cor. 5.9]) which implies that if for some $\ell$ , $\det(\mathsf{P}_{B,b}){\otimes}\mathbb{Q}_{\ell}$ is $\pi_{1}^{\mathrm{{\acute{e}}t}}(S,b)$ -invariant, then the same is true for all $\ell$ . In (4.1.4) we used Larsen-Pink’ result [Larsen-Pink, Prop. 6.14] to achieve a similar effect. Later in (4.2.4) this is used to overcome a similar difficulty: We do not know that the tensors which cut out $\mathrm{Res}_{E/\mathbb{Q}}\mathrm{SO}(\mathcal{V})$ from $\mathrm{Res}_{E/\mathbb{Q}}\mathrm{O}(\mathcal{V})$ are given by absolute-Hodge tensors on $\mathsf{P}_{B,b}$ via $\mu_{b}$ . As far as we are aware of, this cannot be deduced from (2.2.7). However, we can put $\mathcal{K}$ -level structures on $\mathsf{P}_{\mathrm{{\acute{e}}t}}$ in question and proceed.

4.2 Case (R+): Maximal Monodromy

Lemma (4.2.1).

It suffices to prove (4.1.9) when $\Lambda=\mathrm{NS}(\mathcal{X}_{\eta})_{\mathbb{Q}}$ .

Proof.

Let $\Lambda^{\perp}$ be the orthogonal complement of $\Lambda$ in $\Lambda_{0}:=\mathrm{NS}(\mathcal{X}_{\eta})_{\mathbb{Q}}$ and set $M=\mu_{b}^{-1}(\Lambda^{\perp})$ . Then $V=V_{0}\oplus M$ and we view $G_{0}$ as the stabilizer of $M$ in $G$ . The image of $\pi_{1}^{\mathrm{{\acute{e}}t}}(S,b)$ in $G(\mathbb{A}_{f})$ via $\mu_{b}$ actually lies in $G_{0}(\mathbb{A}_{f})$ . Therefore, $\mathsf{K}_{0}=\mathsf{K}\cap G_{0}(\mathbb{A}_{f})$ satisfies condition $(\sharp)$ for $\Lambda_{0}$ .

Let $\mathsf{V}_{0}$ be the automorphic system on $\mathrm{Sh}_{0}:=\mathrm{Sh}_{\mathsf{K}_{0}}(G_{0})$ given by $V_{0}\in\mathrm{Rep}(G_{0})$ , $[\eta_{V_{0}}]$ be the $\mathsf{K}_{0}$ -level structure on $\mathsf{V}_{0}$ , and $[\mu_{0}]$ be the $\mathsf{K}_{0}$ -level structure on $\mathsf{P}_{0}$ defined by $\mu_{0,b}:=\mu_{b}|_{V_{0}}$ . As in the paragraph above (4.1.9), we obtain a morphism $\rho_{0,\mathbb{C}}:S_{\mathbb{C}}\to\mathrm{Sh}_{0,\mathbb{C}}$ such that $(\mathsf{V}_{0},[\eta_{V_{0}}])|_{S_{\mathbb{C}}}\,{\cong}\,(\mathsf{P}_{0},[\mu_{0}])|_{S_{\mathbb{C}}}$ . Define $\alpha^{\circ}_{0,\mathrm{{\acute{e}}t}}$ accordingly.

Consider the Shimura morphism $j:\mathrm{Sh}_{0}\to\mathrm{Sh}:=\mathrm{Sh}_{\mathsf{K}}(G)$ . Then we have natural identifications

j^{*}(\mathsf{V})=\mathsf{V}_{0}\oplus(M{\otimes}\mathbf{1}_{\mathrm{Sh}_{0}}),\text{ and }\mathsf{P}=\mathsf{P}_{0}\oplus(\Lambda^{\perp}{\otimes}\mathbf{1}_{S})

(10)

Moreover, the level structure $[\eta_{V_{0}}]$ (resp. $[\mu_{0}]$ ) refines $j^{*}([\eta_{V}])$ (resp. $[\mu]$ ) in the sense of (3.1.2)(d). Therefore, one easily checks that

(j_{\mathbb{C}}\circ\rho_{0,\mathbb{C}})^{*}(\mathsf{V},[\eta_{V}])\,{\cong}\,(\mathsf{P},[\mu])|_{S_{\mathbb{C}}}.

By the uniqueness statement in (3.2.2), this implies that $\rho_{\mathbb{C}}=j_{\mathbb{C}}\circ\rho_{0,\mathbb{C}}$ .

Assume now that $\rho_{0,\mathbb{C}}$ descends to $\rho_{0}$ over $F$ , and $\alpha^{\circ}_{0,\mathrm{{\acute{e}}t}}$ descends to $\alpha_{0,\mathrm{{\acute{e}}t}}$ over $S$ . Then $\rho_{\mathbb{C}}$ descends to $j_{F}\circ\rho_{0}$ , and $\alpha^{\circ}_{\mathrm{{\acute{e}}t}}$ descends to an isomorphism $\alpha_{0,\mathrm{{\acute{e}}t}}\oplus(\mathrm{id}_{\Lambda^{\perp}}{\otimes}\underline{\mathbb{A}}_{f})$ over $S$ . ∎

(4.2.2)

In this section we prove (4.1.9). By (4.2.1), we may assume that $\Lambda=\Lambda_{0}$ , so that $V=V_{0}$ and $\mathsf{P}=\mathsf{P}_{0}$ . Let $E,\mathcal{G},\mathcal{V},\mathcal{G}^{\sharp},\mathsf{K},\mathcal{K}$ be as introduced in (4.1.6). In particular, $E$ is the endomorphism field of the Hodge structure $\mathsf{P}_{b}$ , and $\mathcal{V}$ is the $E$ -bilinear lift of $V$ , which carries an $E$ -action via the isometry $\mu_{b}:V\stackrel{{\scriptstyle\sim}}{{\to}}\mathsf{P}_{B,b}$ fixed in (4.1.5).

Note that by (2.2.7) each $e\in E$ , viewed as an element of $\mathrm{End}(\mathsf{P}_{B,b})=\mathrm{End}(\omega_{B}(\mathfrak{p}_{b}))$ , is absolute Hodge, so its image in $\mathrm{End}(\mathsf{P}_{\mathrm{{\acute{e}}t},b})$ is stabilized by an open subgroup of $\pi_{1}^{\mathrm{{\acute{e}}t}}(S,b)$ . Since we assumed that $\mathrm{Mon}(\mathsf{P}_{\ell},b)$ is connected for every $\ell$ , the $E$ -action on $\mathsf{P}_{\mathrm{{\acute{e}}t},b}$ must already be $\pi_{1}^{\mathrm{{\acute{e}}t}}(S,b)$ -equivariant. This has the following consequence:

Lemma (4.2.3).

The action of $E$ on $\mathsf{P}_{B,b}$ extends to an action on $\mathsf{P}$ . If $\tau:E\hookrightarrow\mathbb{C}$ is the embedding induced by the action of $E$ on $\mathrm{Fil}^{1}\mathsf{P}_{\mathrm{dR},b}$ , or equivalently the unique indefinite real place of $\mathcal{V}$ , then $\tau(E)\subseteq F$ .

Proof.

For the first statement, it is clear that the $E$ -action on $\mathsf{P}_{B,b}$ (resp. $\mathsf{P}_{\mathrm{{\acute{e}}t},b}$ ) extends (necessarily uniquely) to $\mathsf{P}_{B}$ and (resp. $\mathsf{P}_{\mathrm{{\acute{e}}t}}$ ) because it is $\pi_{1}(S_{\mathbb{C}},b)$ (resp. $\pi_{1}^{\mathrm{{\acute{e}}t}}(S,b)$ )-equivariant. It remains to show that the $E$ -action on $\mathsf{P}_{\mathrm{dR}}|_{S_{\mathbb{C}}}$ , obtained via the Riemann-Hilbert correspondence, descends to an action on $\mathsf{P}_{\mathrm{dR}}$ . This follows from the fact that the de Rham realization of every $e\in E$ , as an element of $\mathsf{P}_{\mathrm{dR},b}^{\otimes}$ , descends to $\mathsf{P}_{\mathrm{dR},\eta}^{\otimes}$ : Since $e$ is absolute Hodge and its étale component is $\pi^{\mathrm{{\acute{e}}t}}_{1}(\eta,b)$ -invariant, its de Rham component descends to $\eta$ (cf. the argument for [KisinInt, (2.2.2)]).

We remind the reader that by $S(\mathbb{C})$ we mean the set of $F$ -linear morphisms $\mathrm{Spec\,}(\mathbb{C})\to S$ . The first statement implies that for every $s\in S(\mathbb{C})$ , $\mathsf{P}_{B,s}$ carries an action of $E$ , which is self-adjoint by (2.2.2); moreover, for every $\sigma\in\mathrm{Aut}(\mathbb{C}/F)$ , the $\sigma$ -linear isomorphism $\sigma_{\mathsf{P}_{\mathrm{dR}},s}:\mathsf{P}_{\mathrm{dR},s}\,{\cong}\,\mathsf{P}_{\mathrm{dR},\sigma(s)}$ of filtered vector spaces is $E$ -equivariant (see (3.1.4) for this notation). Therefore, if we let $\tau_{s}:E\to\mathbb{R}$ be the place through which $E$ acts on $\mathrm{Fil}^{1}\mathsf{P}_{\mathrm{dR},s}$ , then $\tau_{\sigma(s)}=\sigma\circ\tau_{s}$ . Now we use that $\tau_{s}$ can also be characterized as the unique real place of $E$ such that $\mathsf{P}_{B,s}{\otimes}_{\tau_{s}}\mathbb{R}$ is indefinite. Parallel transport implies that $\tau_{s}$ is constant on $S(\mathbb{C})$ , which by assumption is connected. As $\tau=\tau_{b}$ and $\sigma(b)\in S(\mathbb{C})$ , $\sigma\circ\tau=\tau$ for every $\sigma\in\mathrm{Aut}(\mathbb{C}/F)$ . This implies that $\tau(E)\subseteq F$ . ∎

(4.2.4)

Below we shall view $E$ as a subfield of $F$ (and hence of $\mathbb{C}$ ) via $\tau$ as above and drop $\tau$ from the notation. Now we recall the discussion in (3.3.7): Let $\Omega_{\mathcal{V}}\subseteq\Omega$ be the Hermitian symmetric subdomain $\{w\in\mathbb{P}(\mathcal{V}{\otimes}_{E}\mathbb{C})\mid\langle w,\bar{w}\rangle>0,\langle w,w\rangle=0\}$ . Then $(\mathcal{G},\Omega_{\mathcal{V}})$ is a Shimura subdatum of $(G,\Omega)$ with reflex field $E$ . Therefore, there is a Shimura morphism $i_{\mathbb{C}}:\mathrm{Sh}_{\mathcal{K}}(\mathcal{G})_{\mathbb{C}}\to\mathrm{Sh}_{\mathsf{K}}(G)_{\mathbb{C}}$ which descends to $i:\mathrm{Sh}_{\mathcal{K}}(\mathcal{G})\to\mathrm{Sh}_{\mathsf{K}}(G)_{E}$ over $E$ . Let $\widetilde{\mathsf{V}}$ be the automorphic system on $\mathrm{Sh}_{\mathcal{K}}(\mathcal{G})$ defined by $V$ and let $[\eta_{\mathcal{V}}]$ be its tautological $\mathcal{K}$ -structure. Recall that $\widetilde{\mathsf{V}}$ is identified with $i^{*}\mathsf{V}$ , and $[\eta_{\mathcal{V}}]$ refines $i^{*}([\eta_{V}])$ .

Consider the situation in (4.1.8). Recall that we assumed that $\mathrm{Mon}(\mathsf{P}_{\ell},b)$ is connected for every $\ell$ . Since the centralizer of the $E$ -action in $\mathrm{O}(V)$ can be identified with $\mathrm{Res}_{E/\mathbb{Q}}\mathrm{O}(\mathcal{V})$ , which contains $\mathcal{G}$ as the identity component, the monodromy action of $\pi_{1}^{\mathrm{{\acute{e}}t}}(S,b)$ must take values in $\mathcal{K}:=\mathsf{K}\cap\mathcal{G}(\mathbb{A}_{f})$ via $\mu_{b}$ . Therefore, there exists a $(\mathcal{G},V,\mathcal{K})$ -level structure $[\widetilde{\mu}]$ on $\mathsf{P}_{\mathrm{{\acute{e}}t}}$ such that $[\widetilde{\mu}]_{b}=\mathcal{K}\cdot(\mu_{b}{\otimes}\mathbb{A}_{f})$ . As the Hodge structure on $\mathsf{P}_{B,b}$ is defined by a point on $\Omega_{\mathcal{V}}$ via $\mu_{b}$ , by (3.1.7) the restriction of $[\widetilde{\mu}]$ to $S_{\mathbb{C}}$ is $(\mathsf{P}|_{S_{\mathbb{C}}})$ -rational and of type $\Omega_{\mathcal{V}}$ . One easily checks that:

Lemma (4.2.5).

Let $\varrho_{\mathbb{C}}:S_{\mathbb{C}}\to\mathrm{Sh}_{\mathcal{K}}(\mathcal{G})_{\mathbb{C}}$ be the unique morphism such that

(\varrho_{\mathbb{C}})^{*}({\widetilde{\mathsf{V}}},[\eta_{\mathcal{V}}])\,{\cong}\,(\mathsf{P},[\widetilde{\mu}])|_{S_{\mathbb{C}}}

(11)

given by (3.2.2). Then $\rho_{\mathbb{C}}=i_{\mathbb{C}}\circ\varrho_{\mathbb{C}}$ .

Hence we reduce (4.1.9) to:

Theorem (4.2.6).

Under the hypothesis of (4.1.9) and notations above, the morphism $\varrho_{\mathbb{C}}$ descends to an $F$ -morphism $\varrho:S\to\mathrm{Sh}_{\mathcal{K}}(\mathcal{G})_{F}$ ; moreover, the étale component $\alpha^{\circ}_{\mathrm{{\acute{e}}t}}:\varrho_{\mathbb{C}}^{*}\widetilde{\mathsf{V}}_{\mathrm{{\acute{e}}t}}\stackrel{{\scriptstyle\sim}}{{\to}}\mathsf{P}_{\mathrm{{\acute{e}}t}}|_{S_{\mathbb{C}}}$ of (11) descends to an isomorphism of $\alpha_{\mathrm{{\acute{e}}t}}:\varrho^{*}{\widetilde{\mathsf{V}}}_{\mathrm{{\acute{e}}t}}\stackrel{{\scriptstyle\sim}}{{\to}}\mathsf{P}_{\mathrm{{\acute{e}}t}}$ over $S$ .

Note that $\alpha_{\mathrm{{\acute{e}}t}}^{\circ}$ above agrees with the one in (4.1.9) because ${\widetilde{\mathsf{V}}}_{\mathrm{{\acute{e}}t}}=i^{*}\mathsf{V}_{\mathrm{{\acute{e}}t}}$ .

Below for any $\mathbb{Q}$ -linear Tannakian category $\mathscr{C}$ , we write $\mathfrak{N}(-)$ for the functor $\mathscr{C}_{(E)}\to\mathscr{C}$ which sends every $M\in\mathscr{C}_{(E)}$ to $\mathrm{Nm}_{E/\mathbb{Q}}(M){\otimes}_{\mathbb{Q}}\det(M_{(\mathbb{Q})})$ (cf. (2.1.4)). Recall that in (3.3.5) we explained how to apply $\mathfrak{N}(-)$ to a $\mathcal{K}$ -level structure on a system of realizations.

(4.2.7)

We first treat the case when $m:=\dim_{E}\mathcal{V}$ is odd. In this case, $N:=\mathfrak{N}(\mathcal{V})\in\mathsf{Rep}(\mathcal{G})$ is faithful. Let $\mathsf{N}$ be the automorphic system on $\mathrm{Sh}_{\mathsf{K}}(\mathcal{G})$ associated to $N$ . Let $[\eta_{N}]$ be the tautological $\mathcal{K}$ -level structure on $\mathsf{N}$ . Then by (3.3.7), there is a unique isomorphism

(\mathfrak{N}({\widetilde{\mathsf{V}}}),\mathfrak{N}(\eta_{\mathcal{V}}))\,{\cong}\,(\mathsf{N},[\eta_{N}]).

(12)

As remarked in (3.3.4), one should read (3.3.7) with $(\mathcal{H},\mathcal{C},\Omega_{\mathcal{H}})=(\mathcal{G},\mathcal{K},\Omega_{\mathcal{V}})$ .

Lemma (4.2.8).

$\mathfrak{N}(\mathsf{P})\in\mathsf{R}(S)$ is weakly AM in the sense of (3.1.5).

Proof.

As the formation of $\mathfrak{N}(-)$ on $\mathsf{Mot}_{\mathrm{AH}}(\mathbb{C})_{(E)}$ commutes with cohomological realizations, for every $s\in S(\mathbb{C})$ , $\mathfrak{N}((\mathsf{P}_{B,s},\mathsf{P}_{\mathrm{dR},s}))=\omega_{\mathrm{Hdg}}(\mathfrak{N}(\mathfrak{p}_{s}))$ . By (2.2.7)(a), $\mathfrak{N}(\mathfrak{p}_{s})\in\mathsf{Mot}_{\mathsf{Ab}}(\mathbb{C})$ . Therefore, in (3.1.5) we may take $M=\mathfrak{N}(\mathfrak{p}_{s})$ , so that $M^{\sigma}=(\mathfrak{N}(\mathfrak{p}_{s}))^{\sigma}=\mathfrak{N}((\mathfrak{p}_{s})^{\sigma})$ . By appling $\mathfrak{N}(-)$ to the objects in the diagram (4.1.7), one checks that the diagrams in (3.1.5) commute for $\mathfrak{N}(\mathsf{P})$ . ∎

Lemma (4.2.9).

There exists a unique morphism $\rho_{N}:S\to\mathrm{Sh}_{\mathcal{K}}(\mathcal{G})_{F}$ such that

\rho_{N}^{*}(\mathsf{N},[\eta_{N}])\,{\cong}\,(\mathfrak{N}(\mathsf{P}),\mathfrak{N}([\widetilde{\mu}])).

(13)

Moreover, $\varrho_{\mathbb{C}}=\rho_{N}|_{S_{\mathbb{C}}}$ .

Proof.

One easily checks from (3.3.5) that $\mathfrak{N}([\widetilde{\mu}])=\mathcal{K}\cdot(\mathfrak{N}(\mu_{b}){\otimes}\mathbb{A}_{f})$ , so by (3.1.7) $\mathfrak{N}([\widetilde{\mu}])$ is $\mathfrak{N}(\mathsf{P})$ -rational. Moreover, as $[\widetilde{\mu}]$ is of type $\Omega_{\mathcal{V}}$ , so is $\mathfrak{N}([\widetilde{\mu}])$ . As $\mathfrak{N}(\mathsf{P})$ is weakly AM, (3.2.5) gives the $\rho_{N}$ for which (13) holds. On the other hand, by appling $\mathfrak{N}(-)$ to (11), we see that

(\varrho_{\mathbb{C}})^{*}(\mathsf{N},[\eta_{\mathfrak{N}}])\,{\cong}\,(\mathfrak{N}(\mathsf{P}),\mathfrak{N}([\widetilde{\mu}]))|_{S_{\mathbb{C}}}.

(14)

By the uniquness statement in (3.2.2), this implies that $\varrho_{\mathbb{C}}=\rho_{N}|_{S_{\mathbb{C}}}$ . ∎

Proof of (4.2.6) for $m$ odd: To affirm the first statement, set $\varrho:=\rho_{N}$ . The second statement now follows from (3.3.6). Indeed, comparing (12), (13) and (14), we see that $\mathfrak{N}(\alpha^{\circ}_{\mathrm{{\acute{e}}t}})$ descends to $S$ . However, as $\dim_{E}\mathcal{V}$ is odd, $Z^{1}_{E}(\mathcal{V})=1$ (see (2.1.5)). By (3.3.6), $\alpha^{\circ}_{\mathrm{{\acute{e}}t}}$ descends to $S$ . ∎

(4.2.10)

Recall that in (4.1.6) $\mathcal{V}^{\sharp}=\mathcal{V}\oplus\mathcal{E}$ for $\mathcal{E}$ defined in (2.1.5). If $m=\dim_{E}\mathcal{V}$ is even, we let $\mathcal{V}^{\sharp}$ play the role of $\mathcal{V}$ in the above proof. Recall that in (4.1.9) we assumed that $\mathsf{K}$ satisfies condition $(\sharp)$ and we are currently in situation $V=V_{0}$ , so that $\mathcal{K}=\mathsf{K}\cap\mathcal{G}(\mathbb{A}_{f})\subseteq\mathcal{U}$ for some neat compact open $\mathcal{U}\subseteq\mathcal{G}^{\sharp}(\mathbb{A}_{f})$ . Define the Hermitian symmetric domain $\Omega_{\mathcal{V}^{\sharp}}$ with $\mathcal{V}$ replaced by $\mathcal{V}^{\sharp}$ in $\Omega_{\mathcal{V}}$ . Then we obtain an embedding of Shimura data $(\mathcal{G},\Omega_{\mathcal{V}})\hookrightarrow(\mathcal{G}^{\sharp},\Omega_{\mathcal{V}^{\sharp}})$ . By [DeligneTdShimura, (1.15)], for some $\mathcal{U}^{\prime}\supseteq\mathcal{K}$ , the Shimura morphism $\mathrm{Sh}_{\mathcal{K}}(\mathcal{G})\to\mathrm{Sh}_{\mathcal{U}^{\prime}}(\mathcal{G}^{\sharp})$ is an embedding. Replacing $\mathcal{U}$ by $\mathcal{U}\cap\mathcal{U}^{\prime}$ if necessary, we may assume that the Shimura morphism $\mathrm{Sh}_{\mathcal{K}}(\mathcal{G})\to\mathrm{Sh}_{\mathcal{U}}(\mathcal{G}^{\sharp})$ is also an embedding. Below we write this embedding simply as $j:\mathrm{Sh}_{\mathcal{K}}\to\mathrm{Sh}_{\mathcal{U}}$ . Let $\mathsf{W}$ be the automorphic system of realizations on $\mathrm{Sh}_{\mathcal{U}}$ given by $W:=(\mathcal{V}^{\sharp})_{(\mathbb{Q})}\in\mathsf{Rep}(\mathcal{G}^{\sharp})$ and let $[\eta_{W}]$ be the tautological $\mathcal{U}$ -level structure on $\mathsf{W}$ .

The reader should now apply the discussion in (3.3.7) with $(\mathcal{G},\Omega_{\mathcal{V}})$ replaced by $(\mathcal{G}^{\sharp},\Omega_{V^{\sharp}})$ , $\widetilde{\mathsf{V}}$ replaced by $\mathsf{W}$ , $(\mathcal{H},\mathcal{C},\Omega_{\mathcal{H}})=(\mathcal{G}^{\sharp},\mathcal{U},\Omega_{\mathcal{V}^{\sharp}})$ and $\pi=\mathrm{id}$ (cf. (3.3.4)). In particular, $\mathsf{W}$ is equipped with a natural $E$ -action. This time we set $N=\mathfrak{N}(\mathcal{V}^{\sharp})\in\mathrm{Rep}(\mathcal{G}^{\sharp})$ . Let $\mathsf{N}$ be the automorphic system on $\mathrm{Sh}_{\mathcal{U}}$ given by $N$ and let $[\eta_{N}]$ be the tautological $\mathcal{U}$ -level structure. Note that $N\in\mathrm{Rep}(\mathcal{G}^{\sharp})$ is faithful as $\dim_{E}\mathcal{V}^{\sharp}$ is odd. Now (3.3.7) tells us that

(\mathsf{N},[\eta_{N}])=(\mathfrak{N}(\mathsf{W}),\mathfrak{N}([\eta_{W}]))

(15)

It is not hard to see that the restriction of $\mathsf{W}$ to $\mathrm{Sh}_{\mathcal{K}}$ is naturally identified with ${\widetilde{\mathsf{V}}}\oplus(\mathcal{E}{\otimes}\mathbf{1}_{\mathrm{Sh}_{\mathcal{K}}})$ . Correspondingly, we set $\mathsf{Q}:=\mathsf{P}\oplus(\mathcal{E}{\otimes}\mathbf{1}_{S})$ and define $\nu_{b}:W\stackrel{{\scriptstyle\sim}}{{\to}}\mathsf{Q}_{B,b}$ by $\mu_{b}\oplus\mathrm{id}_{\mathcal{E}}$ . Then $\nu_{b}$ defines a $\mathcal{U}$ -level structure $[\nu]$ with $[\nu]_{b}=\mathcal{U}\cdot(\nu_{b}{\otimes}\mathbb{A}_{f})$ . Define $\beta_{\mathbb{C}}:=j_{\mathbb{C}}\circ\varrho_{\mathbb{C}}$ . Then we have:

(\beta_{\mathbb{C}})^{*}(\mathsf{W},[\eta_{W}])\,{\cong}\,(\mathsf{Q},[\nu])|_{S_{\mathbb{C}}}

(16)

Lemma (4.2.11).

There exists a unique morphism $\beta_{N}:S\to(\mathrm{Sh}_{\mathcal{U}})_{F}$ such that

\beta_{N}^{*}(\mathsf{N},[\eta_{N}])\,{\cong}\,(\mathfrak{N}(\mathsf{Q}),\mathfrak{N}([\nu])).

(17)

Moreover, $\beta_{\mathbb{C}}=\beta_{N}|_{S_{\mathbb{C}}}$ .

Proof.

By (2.2.7)(b) and a slight variant of the argument for (4.2.8), $\mathfrak{N}(\mathsf{Q})$ is weakly AM. As $\mathfrak{N}([\nu])_{b}=\mathcal{U}\cdot(\mathfrak{N}(\mu_{b}){\otimes}\mathbb{A}_{f})$ , (3.1.7) says that the $(\mathcal{G}^{\sharp},N,\mathcal{U})$ -level structure $\mathfrak{N}([\nu])$ is $\mathfrak{N}(\mathsf{Q})$ -rational. As $[\widetilde{\mu}]$ is of type $\Omega_{\mathcal{V}}$ , $[\nu]$ is of type $\Omega_{\mathcal{V}^{\sharp}}$ , so that $\mathfrak{N}([\nu])$ is also of type $\Omega_{\mathcal{V}^{\sharp}}$ . Applying (3.2.5) to the faithful representation $N$ of $\mathcal{G}^{\sharp}$ , we obtain the desired map $\beta_{N}$ such that (17) holds. By the uniqueness statement in (3.2.2), to show $\beta_{\mathbb{C}}=\beta_{N}|_{S_{\mathbb{C}}}$ it suffices to observe that

\beta_{\mathbb{C}}^{*}(\mathfrak{N}(\mathsf{W}),\mathfrak{N}([\eta_{W}]))\,{\cong}\,(\mathfrak{N}(\mathsf{Q}),\mathfrak{N}([\nu]))|_{S_{\mathbb{C}}}.

One checks this by applying $\mathfrak{N}(-)$ to (16). ∎

Proof of (4.2.6) for $m$ even: The above implies that $\beta_{\mathbb{C}}$ descends to $\beta_{N}$ over $F$ . Recall that $\beta_{\mathbb{C}}=j_{\mathbb{C}}\circ\varrho_{\mathbb{C}}$ and $j_{\mathbb{C}}$ is an embedding. As the actions on $\mathbb{C}$ -points of both $\beta_{\mathbb{C}}$ and $j_{\mathbb{C}}$ are $\mathrm{Aut}(\mathbb{C}/F)$ -equivariant, the same is true for $\varrho_{\mathbb{C}}$ , so that $\varrho_{\mathbb{C}}$ descends to a morphism $\varrho$ over $F$ with $\beta_{N}=j_{F}\circ\varrho$ .

Let $\lambda^{\circ}_{\mathrm{{\acute{e}}t}}:(\mathsf{W}_{\mathrm{{\acute{e}}t}},[\eta_{W}])|_{S_{\mathbb{C}}}\,{\cong}\,(\mathsf{Q}_{\mathrm{{\acute{e}}t}},[\nu])|_{S_{\mathbb{C}}}$ be the étale component of (16). Then $\mathfrak{N}(\lambda^{\circ}_{\mathrm{{\acute{e}}t}})$ is the étale component of (17) restricted to $S_{\mathbb{C}}$ . Therefore, $\mathfrak{N}(\lambda^{\circ}_{\mathrm{{\acute{e}}t}})$ descends to $S$ . Applying (3.3.6) to $\mathcal{V}^{\sharp}$ , we have that $\lambda^{\circ}_{\mathrm{{\acute{e}}t}}$ descends to an isomorphism $\lambda_{\mathrm{{\acute{e}}t}}:\beta_{N}^{*}\mathsf{W}_{\mathrm{{\acute{e}}t}}\,{\cong}\,\mathsf{Q}_{\mathrm{{\acute{e}}t}}$ over $S$ . Note the decompositions $\beta_{N}^{*}\mathsf{W}=\varrho^{*}{\widetilde{\mathsf{V}}}\oplus(\mathcal{E}{\otimes}\mathbf{1}_{S})$ and $\mathsf{Q}=\mathsf{P}\oplus(\mathcal{E}{\otimes}\mathbf{1}_{S})$ . Since $\lambda_{\mathrm{{\acute{e}}t}}^{\circ}$ respects these decompositions over $S_{\mathbb{C}}$ by construction, its descent $\lambda_{\mathrm{{\acute{e}}t}}$ over $S$ must also respect these decompositions, which are defined over $S$ . Hence $\lambda_{\mathrm{{\acute{e}}t}}$ restricts to the sought after $\alpha_{\mathrm{{\acute{e}}t}}:\varrho^{*}{\widetilde{\mathsf{V}}}_{\mathrm{{\acute{e}}t}}\stackrel{{\scriptstyle\sim}}{{\to}}\mathsf{P}_{\mathrm{{\acute{e}}t}}$ in (4.2.6). ∎

4.3 Case (R2’): non-maximal monodromy

By (2.2.5), we expect this case to be rare in practice. Readers who are not particularly interested in this case might skip to the next section.

Lemma (4.3.1).

Let $k\subseteq\mathbb{C}$ be a subfield and $A,B$ be $k$ -varieties with $A$ being geometrically connected. Suppose that there is a morphism $f:A_{\mathbb{C}}\to B_{\mathbb{C}}$ over $\mathbb{C}$ , and an étale morphism $g:B\to C$ over $k$ such that for some $h:A\to C$ , $g_{\mathbb{C}}\circ f=h_{\mathbb{C}}$ . Then $f$ descends to a subfield $k^{\prime}$ of $\mathbb{C}$ which is finite over $k$ such that $g_{k^{\prime}}\circ f=h_{k^{\prime}}$ .

Proof.

We assume without loss of generality that $k$ is algebraically closed. The graph $\Gamma_{f}\colon A_{\mathbb{C}}\to(A\times_{C}B)_{\mathbb{C}}$ of $f$ (as a morphism between $C_{\mathbb{C}}$ -schemes) defines a section of the étale morphism $(g\times_{C}h)_{\mathbb{C}}\colon(A\times_{C}B)_{\mathbb{C}}\to A_{\mathbb{C}}$ . Hence $\Gamma_{f}$ maps $A_{\mathbb{C}}$ isomorphically onto a connected component $D_{\mathbb{C}}$ of $(A\times_{C}B)_{\mathbb{C}}$ . Since $k$ is algebraically closed, $D_{\mathbb{C}}$ comes from an extension of scalars of a connected component $D\subset(A\times_{C}B)$ . As the natural projection $D\to A$ is defined over $k$ and its base change to $\mathbb{C}$ is the inverse of $\Gamma_{f}$ , we must have that $\Gamma_{f}$ is also defined over $k$ , and hence so is $f$ . ∎

Below for any $\mathbb{Q}$ -linear Tannakian category $\mathscr{C}$ , we write $\mathfrak{N}(-)$ for the functor $\mathrm{Nm}_{E/\mathbb{Q}}:\mathscr{C}_{(E)}\to\mathscr{C}$ (cf. (2.1.4)).
Proof of (4.1.10). As before, set $\mathcal{K}:=\mathsf{K}\cap\mathcal{G}(\mathbb{A}_{f})$ and let $i:\mathrm{Sh}_{\mathcal{K}}(\mathcal{G})\to\mathrm{Sh}_{\mathsf{K}}(G)_{E}$ be the Shimura morphism. Our discussions in (4.2.2) up to (4.2.5) apply without any change in the (R2’) case, so that $\rho_{\mathbb{C}}$ factors through a morphism $\varrho_{\mathbb{C}}:S_{\mathbb{C}}\to\mathrm{Sh}_{\mathcal{K}}(\mathcal{G})_{\mathbb{C}}$ such that

\varrho_{\mathbb{C}}^{*}({\widetilde{\mathsf{V}}},[\eta_{\mathcal{V}}])\,{\cong}\,(\mathsf{P},[\widetilde{\mu}])|_{S_{\mathbb{C}}},

(18)

and we still have $E\subseteq F$ . We first show that $\varrho_{\mathbb{C}}$ descends to a morphism $\varrho:S_{F^{\prime}}\to\mathrm{Sh}_{\mathcal{K}}(\mathcal{G})_{F^{\prime}}$ for some finite extension $F^{\prime}/F$ in $\mathbb{C}$ .

Set $N:=\mathfrak{N}(\mathcal{V})$ . Then $N$ is a faithful representation of $\mathcal{H}=\mathcal{H}(\mathcal{V})$ (2.1.5). Since in (4.1.10) we assumed that $\mathsf{K}$ satisfies condition $(\sharp)$ , there exists a neat $\mathcal{C}\subseteq\mathcal{H}(\mathbb{A}_{f})$ such that the image of $\mathcal{K}$ lies in $\mathcal{C}$ . Now recall our discussion in (3.3.7) and the notations therein. Let $\pi:\mathrm{Sh}_{\mathcal{K}}(\mathcal{G})\to\mathrm{Sh}_{\mathcal{C}}(\mathcal{H})$ be the natural Shimura morphism over $E$ . Let $\mathfrak{N}([\widetilde{\mu}])$ denote the $\mathcal{C}$ -level structure on $\mathfrak{N}(\mathsf{P}_{\mathrm{{\acute{e}}t}})$ such that $\mathfrak{N}([\widetilde{\mu}])_{b}=\mathcal{C}\cdot(\mathfrak{N}(\widetilde{\mu}_{b}){\otimes}\mathbb{A}_{f})$ . By applying $\mathfrak{N}(-)$ to the diagrams in (4.1.7), (2.2.7)(c) implies that $\mathfrak{N}(\mathsf{P})$ is weakly AM. One checks using (3.1.7) that $\mathfrak{N}(\widetilde{\mu})$ is $\mathfrak{N}(\mathsf{P})$ -rational, and is of type $\Omega_{\mathcal{H}}$ . Then by (3.2.5), we obtain a morphism $\rho_{N}:S\to\mathrm{Sh}_{\mathcal{C}}(\mathcal{H})_{F}$ such that

(\rho_{N})^{*}(\mathsf{N},[\eta_{N}])\,{\cong}\,(\mathfrak{N}(\mathsf{P}),\mathfrak{N}([\widetilde{\mu}])).

(19)

By applying $\mathfrak{N}(-)$ to (18) and comparing with (8) in (3.3.7), for $\beta_{\mathbb{C}}:=\pi_{\mathbb{C}}\circ\varrho_{\mathbb{C}}$ we obtain

\beta_{\mathbb{C}}^{*}(\mathsf{N},[\eta_{N}])=\varrho_{\mathbb{C}}^{*}\pi_{\mathbb{C}}^{*}(\mathsf{N},[\eta_{N}])\,{\cong}\,\varrho_{\mathbb{C}}^{*}(\mathfrak{N}({\widetilde{\mathsf{V}}}),\mathfrak{N}([\eta_{\mathcal{V}}]))\,{\cong}\,(\mathfrak{N}(\mathsf{P}),\mathfrak{N}([\widetilde{\mu}]))|_{S_{\mathbb{C}}}.

Therefore, by the uniqueness statement in (3.2.2), $\beta_{\mathbb{C}}=\rho_{N}|_{S_{\mathbb{C}}}$ , i.e., $\beta_{\mathbb{C}}$ is defined over $F$ . As $\pi$ is étale and is defined over $E\subseteq F$ . By (4.3.1), $\varrho_{\mathbb{C}}$ descends to a morphism $\varrho$ over some finite extension $F^{\prime}/F$ in $\mathbb{C}$ such that $(\rho_{N})_{F^{\prime}}=\pi_{F^{\prime}}\circ\varrho$ .

The above gives the first statement of (4.1.10) and we now turn to the second. Recall that we defined $\alpha^{\circ}_{\mathrm{{\acute{e}}t}}:\rho_{\mathbb{C}}^{*}(\mathsf{V}_{\mathrm{{\acute{e}}t}},[\eta_{V}])\stackrel{{\scriptstyle\sim}}{{\to}}(\mathsf{P},[\mu])|_{S_{\mathbb{C}}}$ in (4.1.8). Since $\widetilde{\mathsf{V}}_{\mathrm{{\acute{e}}t}}=i^{*}\mathsf{V}_{\mathrm{{\acute{e}}t}}$ and $\rho_{\mathbb{C}}=i_{\mathbb{C}}\circ\varrho_{\mathbb{C}}$ , we may alternatively view $\alpha^{\circ}_{\mathrm{{\acute{e}}t}}$ as the étale component of (18), i.e., an isomorphism $\varrho_{\mathbb{C}}^{*}\widetilde{\mathsf{V}}_{\mathrm{{\acute{e}}t}}\stackrel{{\scriptstyle\sim}}{{\to}}\mathsf{P}_{\mathrm{{\acute{e}}t}}|_{S_{\mathbb{C}}}$ , which sends $\varrho_{\mathbb{C}}^{*}[\eta_{\mathcal{V}}]$ to $[\widetilde{\mu}]$ . As $(\rho_{N})_{F^{\prime}}=\pi_{F^{\prime}}\circ\varrho$ , (19) gives us an isomorphism

\varrho^{*}(\mathfrak{N}(\widetilde{\mathsf{V}}_{\mathrm{{\acute{e}}t}}),\mathfrak{N}([\eta_{\mathcal{V}}]))=\varrho^{*}\pi^{*}(\mathsf{N}_{\mathrm{{\acute{e}}t}},[\eta_{N}])\,{\cong}\,(\mathfrak{N}(\mathsf{P}_{\mathrm{{\acute{e}}t}}),\mathfrak{N}([\widetilde{\mu}]))|_{S_{F^{\prime}}}

whose restriction to $S_{\mathbb{C}}$ is $\mathfrak{N}(\alpha^{\circ}_{\mathrm{{\acute{e}}t}})$ . This implies that $\mathfrak{N}(\alpha^{\circ}_{\mathrm{{\acute{e}}t}})$ descends to $S_{F^{\prime}}$ . As we assumed that $\mathcal{K}_{\ell_{0}}\cap Z^{1}_{E}(\mathbb{Q}_{\ell_{0}})=1$ , (3.3.6) tells us that the $\ell_{0}$ -adic component $\alpha^{\circ}_{\ell_{0}}:{\widetilde{\mathsf{V}}}_{\ell_{0}}|_{S_{\mathbb{C}}}\stackrel{{\scriptstyle\sim}}{{\to}}\mathsf{P}_{\ell_{0}}|_{S_{\mathbb{C}}}$ descends to $S_{F^{\prime}}$ . For every other $\ell$ , $\mathcal{K}_{\ell}$ still contains an open subgroup $\mathcal{K}^{\prime}_{\ell}$ such that $\mathcal{K}^{\prime}_{\ell}\cap Z^{1}_{E}(\mathbb{Q}_{\ell})=1$ . Hence (3.3.6) implies that $\varrho^{*}{\widetilde{\mathsf{V}}}_{\ell}$ is étale-locally isomorphic to $\mathsf{P}_{\ell}|_{S_{F^{\prime}}}$ . ∎

Remark (4.3.2).

We remark that (4.1.10) is slightly weaker than (4.1.9) (e.g., one cannot descend $\rho_{\mathbb{C}}$ to $F$ but only to some finite extension) fundamentally because the representation $\mathcal{G}\to\mathrm{GL}(N)$ is not faithful, but has a finite kernel. In the (R2) case, this was avoided because we worked with $\mathcal{V}^{\sharp}$ instead.

5 Proof of Theorem B

5.1 A specialization lemma for monodromy

Definition (5.1.1).

Let $S$ be a noetherian integral normal scheme. Let $\ell\in\mathcal{O}_{S}^{\times}$ be a prime and $\mathsf{W}_{\ell}$ be an étale $\mathbb{Q}_{\ell}$ -local system. We denote by $\lambda(\mathsf{W}_{\ell})$ the dimension $\dim\varinjlim_{U}\mathsf{W}_{\ell,s}^{U}$ where $s$ is a geometric point on $S$ and $U$ runs through open subgroups of $\pi_{1}^{\mathrm{{\acute{e}}t}}(S,s)$ .

It is clear that the definition is independent of the choice of $s$ .

Definition (5.1.2).

Let $T$ be a noetherian base scheme and $S$ be a smooth $T$ -scheme of finite type.

(a)

Let $\ell\in\mathcal{O}_{T}^{\times}$ be a prime and $\mathsf{W}_{\ell}$ be an étale $\mathbb{Q}_{\ell}$ -local system. We say that $\mathsf{W}_{\ell}$ has constant $\lambda^{\mathrm{geo}}$ if there exists a number $\lambda$ such that for every geometric point $t\to T$ and every connected component $S^{\circ}$ of $S_{t}$ , $\lambda(\mathsf{W}_{\ell}|_{S^{\circ}})=\lambda$ . When this condition is satisfied, write $\lambda^{\mathrm{geo}}(\mathsf{W}_{\ell})$ for $\lambda$ .
(b)

We say that $\overline{S}$ is a good relative compactification of $S$ if $\overline{S}$ is a smooth proper $T$ -scheme and there exists a relative normal crossing divisor $D$ of $\overline{S}$ such that $S=\overline{S}-D$ .

Lemma (5.1.3).

Let $T$ be a DVR with special point $t$ and generic point $\eta$ . Assume $\mathrm{char\,}k(\eta)=0$ and $\ell\in\mathcal{O}_{T}^{\times}$ . Let $S\to T$ be a smooth morphism of finite type with $S$ being connected. Let $\mathsf{W}_{\ell}$ be a $\mathbb{Q}_{\ell}$ -local system over $S$ . If $S$ admits a good relative compactification $\overline{S}$ over $T$ , then $\mathsf{W}_{\ell}$ has constant $\lambda^{\mathrm{geo}}$ over $T$ .

Proof.

Let $\widetilde{T}$ be the strict Henselianization of $T$ and let $\widetilde{t}$ and $\widetilde{\eta}$ be the special and generic point of $\widetilde{T}$ . Let $S^{\prime}$ be a connected component of $S_{\widetilde{T}}$ . Then [stacks-project, 055J] tells us that $S^{\prime}_{\widetilde{\eta}}$ is connected. As $S^{\prime}_{\widetilde{\eta}}$ necessarily contains a $k(\widetilde{\eta})$ -rational point, $S^{\prime}_{\widetilde{\eta}}$ is geometrically connected. Now by applying [stacks-project, 0E0N] to $\overline{S}$ , $S_{\widetilde{\eta}}$ and $S_{\widetilde{t}}$ have the same number of geometric connected components, so $S^{\prime}_{\widetilde{t}}$ must be connected. To prove the lemma we may replace $T$ by $\widetilde{T}$ and $S$ by $S^{\prime}$ , so that $S_{t}$ and $S_{\eta}$ are both geometrically connected. Let $\bar{\eta}$ be the geometric point over $\eta$ defined by a chosen algebraic closure of $k(\eta)$ .

Choose a section $\sigma:T\to S$ and set $a=\sigma(t),b=\sigma(\bar{\eta})$ . Note that $\sigma$ provides an étale path between $a$ and $b$ , through which we identify $\mathsf{W}_{\ell,a}$ with $\mathsf{W}_{\ell,b}$ and $\pi^{\mathrm{{\acute{e}}t}}_{1}(S,a)$ with $\pi^{\mathrm{{\acute{e}}t}}_{1}(S,b)$ . Let $\rho:\pi^{\mathrm{{\acute{e}}t}}_{1}(S,a)\to\mathrm{GL}(\mathsf{W}_{\ell,a})$ be the monodromy representation. For any group $G$ with a morphism $G\to\pi^{\mathrm{{\acute{e}}t}}_{1}(S,a)$ implicitly understood, write $\rho(G)^{\circ}$ for the identity component of the Zariski closure of the image of $G$ in $\mathrm{GL}(\mathsf{W}_{\ell,a})$ . Clearly, $\rho(G)^{\circ}$ remains unchanged if we replace $G$ by a finite index subgroup. It suffices to show that $\rho(\pi_{1}^{\mathrm{{\acute{e}}t}}(S_{t},a))^{\circ}=\rho(\pi_{1}^{\mathrm{{\acute{e}}t}}(S_{\bar{\eta}},b))^{\circ}$ , i.e., $\mathrm{Mon}^{\circ}(\mathsf{W}_{\ell}|_{S_{t}},a)=\mathrm{Mon}^{\circ}(\mathsf{W}_{\ell}|_{S_{\bar{\eta}}},b)$ in the notation in (4.1.1).

Take a sequence $\mathcal{F}_{n}$ of locally constant free $\mathbb{Z}/\ell^{n}\mathbb{Z}$ -modules over $S$ such that $\mathsf{W}_{\ell}\,{\cong}\,(\varprojlim_{n}\mathcal{F}_{n}){\otimes}\mathbb{Q}_{\ell}$ . As $\mathrm{char\,}k(\eta)=0$ , each $\mathcal{F}_{n}$ is tamely ramified over $\overline{S}$ by Abhyankar’s lemma [SGA1, XIII App. Prop. 5.5]⁸⁸8There is a typo in the statement: $Y$ should be $\mathrm{Supp}(D)$ , not $X-\mathrm{Supp}(D)$ .. Let us use a superscript “ $t$ ” to indicate tame fundamental group. Then we know that $\rho(\pi^{\mathrm{{\acute{e}}t}}_{1}(S_{?},a))^{\circ}=\rho(\pi_{1}^{\mathrm{{\acute{e}}t}}(S_{?},a)^{t})^{\circ}$ for $?=\emptyset,t$ . By [SGA1, XIII Ex. 2.10], the natural map $\pi_{1}^{\mathrm{{\acute{e}}t}}(S_{t},a)^{t}\to\pi_{1}^{\mathrm{{\acute{e}}t}}(S,a)^{t}$ is an isomorphism, so it remains to show that $\rho(\pi_{1}^{\mathrm{{\acute{e}}t}}(S_{\bar{\eta}},b))^{\circ}=\rho(\pi_{1}^{\mathrm{{\acute{e}}t}}(S,b)^{t})^{\circ}$ .

The section $\sigma(\eta)$ induces an isomorphism $\pi^{\mathrm{{\acute{e}}t}}_{1}(S,b)=\pi^{\mathrm{{\acute{e}}t}}_{1}(S_{\bar{\eta}},b)\rtimes\mathrm{Gal}_{k(\eta)}$ . As $\sigma^{*}(\mathsf{W}_{\ell})$ is necessarily trivial, the subgroup $\mathrm{Gal}_{k(\eta)}\subseteq\pi^{\mathrm{{\acute{e}}t}}_{1}(S,b)$ acts trivially on $\mathsf{W}_{\ell,b}$ . Therefore, $\rho(\pi^{\mathrm{{\acute{e}}t}}_{1}(S_{\bar{\eta}},b))^{\circ}=\rho(\pi^{\mathrm{{\acute{e}}t}}_{1}(S_{\eta},b))^{\circ}=\rho(\pi^{\mathrm{{\acute{e}}t}}_{1}(S,b)^{t})$ as desired. Note that the second equality follows from the simple fact that $\pi_{1}^{\mathrm{{\acute{e}}t}}(S_{\eta},b)$ maps surjectively to $\pi_{1}^{\mathrm{{\acute{e}}t}}(S,b)$ , and $\pi^{\mathrm{{\acute{e}}t}}_{1}(S,b)^{t}$ is a quotient of $\pi_{1}^{\mathrm{{\acute{e}}t}}(S,b)$ . ∎

Proposition (5.1.4).

Let $S$ be a connected smooth $\mathbb{C}$ -variety and $f:\mathcal{X}\to S$ be a $\heartsuit$ -family. Let $s$ be any Hodge-generic point on $S$ . Then for any prime $\ell$ and $\mathsf{V}_{\ell}:=R^{2}f_{*}\mathbb{Q}_{\ell}$ , $\dim\mathrm{NS}(\mathcal{X}_{s})_{\mathbb{Q}}=\lambda(\mathsf{V}_{\ell})$ .

Proof.

Up to replacing $S$ by a connected étale cover, assume that $M_{\ell}:=\mathrm{Mon}(\mathsf{V}_{\ell},s)$ is connected (see (4.1.1)). Set $\rho:=\mathrm{NS}(\mathcal{X}_{s})_{\mathbb{Q}}$ . By (4.1.2), $\rho=\mathrm{NS}(\mathcal{X}_{\eta})_{\mathbb{Q}}$ . As $\mathsf{M}_{\ell}$ is unchanged if we replace $S$ by a further connected étale cover, $\dim\mathsf{V}_{\ell,s}^{M_{\ell}}=\lambda(\mathsf{V}_{\ell})$ . Let $\mathsf{V}=(\mathsf{V}_{B},\mathsf{V}_{\mathrm{dR}})$ be the VHS $R^{2}f_{*}\mathbb{Q}(1)$ . Then $\mathsf{V}$ splits into $\mathbf{1}_{S}^{\oplus\rho}\oplus\mathsf{P}$ for some VHS $\mathsf{P}$ such that $\mathsf{P}_{B,s}^{(0,0)}=0$ . It suffices to argue that $\mathsf{P}_{\ell,s}^{M_{\ell}}=0$ , where $\mathsf{P}_{\ell}=\mathsf{P}_{B}{\otimes}\mathbb{Q}_{\ell}$ .

Set $M:=\mathrm{Mon}(\mathsf{V}_{B},s)$ (see (2.2.3)). Then we have $M_{\ell}=M{\otimes}\mathbb{Q}_{\ell}$ . This implies that $V_{\ell,s}^{M_{\ell}}=V_{B,s}^{M}{\otimes}\mathbb{Q}_{\ell}$ , so we reduce to showing that $\mathsf{P}_{B,s}^{M}=0$ . We recall that Deligne’s theorem of the fixed part [DelHdg, (4.1.2)] says that the subspace $\mathsf{P}_{B,s}^{M}$ has a Hodge structure which is respected by the embedding $\mathsf{P}_{B,s}^{M}\hookrightarrow\mathsf{P}_{B,s}$ . Since the Hodge structure on $\mathsf{P}_{B,s}$ is irreducible ([HuyK3Book, §3 Lem. 2.7]), $\mathsf{P}_{B,s}^{M}$ is either $0$ or $\mathsf{P}_{B,s}$ . But it cannot be $\mathsf{P}_{B,s}$ because $M\neq 1$ by our assumption that $\mathcal{X}/S$ is a $\heartsuit$ -family. ∎

5.2 An Effective Theorem

First, we extend the set-ups in (4.1.3) and (4.1.5) to a family over $\mathbb{Z}_{(p)}$ when $F=\mathbb{Q}$ .

Set-up (5.2.1).

Let $\mathsf{M}$ be a connected separated scheme over $\mathbb{Z}_{(p)}$ which is smooth and of finite type for some prime $p>2$ . Let $(f\colon\mathcal{X}\to\mathsf{M})$ be a smooth projective morphism of relative dimension $d$ such that $\mathcal{X}|_{\mathsf{M}_{\mathbb{Q}}}$ is a $\heartsuit$ -family. Let $\eta$ be the generic point of $\mathsf{M}$ . Let $\boldsymbol{\xi}$ be a relatively ample line bundle on $\mathcal{X}/\mathsf{M}$ , which endows $R^{2}f_{*}\mathbb{A}^{p}_{f}(1)$ and $R^{2}f_{\mathbb{Q}*}\mathbb{Z}_{p}(1)$ a symmetric bilinear pairing. Let $\Lambda\subseteq\Lambda_{0}:=\mathrm{NS}(\mathcal{X}_{\eta})_{\mathbb{Q}}$ be a subspace which contains the class of $\boldsymbol{\xi}_{\eta}$ . Recall that by (4.1.2), $\mathrm{Pic}\,(\mathcal{X}_{\eta})=\mathrm{Pic}\,(\mathcal{X})/\mathrm{Pic}\,(\mathsf{M})$ . By choosing a section $\Lambda_{0}\hookrightarrow\mathrm{Pic}\,(\mathcal{X}_{\eta})_{\mathbb{Q}}$ , we obtain an embedding $\underline{\Lambda}_{0}\to R^{2}f_{*}\mathbb{A}^{p}_{f}(1)$ , and for every field $k$ and $s\in\mathsf{M}(k)$ , $\Lambda_{0}$ (and hence $\Lambda$ ) is naturally a subspace of $\mathrm{NS}(\mathcal{X}_{s})_{\mathbb{Q}}$ . We write $\mathrm{PNS}(\mathcal{X}_{s})_{\mathbb{Q}}$ for the orthogonal complement of $\Lambda$ in $\mathrm{NS}(\mathcal{X}_{s})_{\mathbb{Q}}$ . Note that these definitions are independent of the section $\Lambda_{0}\hookrightarrow\mathrm{Pic}\,(\mathcal{X}_{\eta})_{\mathbb{Q}}$ chosen. Choose a base point $b\in\mathsf{M}(\mathbb{C})$ lying above $\eta$ and let the connected component of $\mathsf{M}_{\mathbb{C}}$ which contains $b$ be $\mathsf{M}^{\circ}$ .

We assume that $\mathrm{Mon}(R^{2}f_{*}\mathbb{Q}_{2},b)$ is connected. Now apply the set-ups in (4.1.3) and (4.1.5) with $S=\mathsf{M}_{\mathbb{Q}}$ and $S^{\circ}=\mathsf{M}^{\circ}$ and define a system of realizations $(\mathsf{P}_{B},\mathsf{P}_{\mathrm{dR}},\mathsf{P}_{\mathrm{{\acute{e}}t}})\in\mathsf{R}(\mathsf{M}_{\mathbb{Q}})$ ; moreover, we fix an isometry $\mu_{b}:V\stackrel{{\scriptstyle\sim}}{{\to}}\mathsf{P}_{B,b}$ and define the Shimura datum $(G,\Omega)$ . Let $(R^{2}f_{\mathbb{Q}*}\mathbb{Z}_{p})_{\mathrm{tf}}$ be the image of $R^{2}f_{\mathbb{Q}*}\mathbb{Z}_{p}$ in $R^{2}f_{\mathbb{Q}*}\mathbb{Q}_{p}$ and define $(R^{2}f_{\mathbb{C}*}\mathbb{Z}_{(p)})_{\mathrm{tf}}$ similarly ( $\mathrm{tf}$ is short for “torsion-free”). Let $\mathbf{P}_{B}:=\mathsf{P}_{B}\cap(R^{2}f_{\mathbb{C}*}\mathbb{Z}_{(p)}(1))_{\mathrm{tf}}$ and $\mathbf{P}_{p}:=\mathsf{P}_{p}\cap(R^{2}f_{\mathbb{Q}*}\mathbb{Z}_{p}(1))_{\mathrm{tf}}$ . For every $\ell\neq p$ , let $\mathbf{P}_{\ell}$ be the orthogonal complement of $\Lambda$ in $R^{2}f_{*}\mathbb{Q}_{\ell}(1)$ , so that $\mathsf{P}_{\ell}$ over $\mathsf{M}_{\mathbb{Q}}$ extends to $\mathbf{P}_{\ell}$ over $\mathsf{M}$ . If $k$ is a perfect field of characteristic $p$ and $W:=W(k)$ , for every point $t\in\mathsf{M}(k)$ , $\boldsymbol{\xi}_{t}$ defines a pairing on the F-isocrystal $\mathrm{H}^{2}_{\mathrm{cris}}(\mathcal{X}_{t}/W)[1/p]$ and we write $\mathbf{P}_{\mathrm{cris},t}[1/p]$ for the orthogonal complement of the classes in $\Lambda\subseteq\mathrm{NS}(\mathcal{X}_{t})_{\mathbb{Q}}$ . Assume that the $\mathbb{Z}_{(p)}$ -pairing on $L_{(p)}:=\mu_{b}^{-1}(\mathbf{P}_{B,b})$ is self-dual. We abusively write the reductive $\mathbb{Z}_{(p)}$ -group $\mathrm{SO}(L_{(p)})$ also as $G$ .

Under the above set-up, we define:

Definition (5.2.2).

Let $\mathsf{K}\subseteq G(\mathbb{A}_{f})$ be a neat compact open subgroup of the form $\mathsf{K}_{p}\mathsf{K}^{p}$ for $\mathsf{K}_{p}=G(\mathbb{Z}_{p})$ and $\mathsf{K}^{p}\subseteq G(\mathbb{A}^{p}_{f})$ . Let $\mathscr{S}_{\mathsf{K}}(G)$ denote the integral model of $\mathrm{Sh}_{\mathsf{K}}(G)$ over $\mathbb{Z}_{(p)}$ . Let $\ell_{0}\neq p$ be a prime. We say that a morphism $\rho:\mathsf{M}\to\mathscr{S}_{\mathsf{K}}(G)$ is an $\ell_{0}$ -admissible period morphism if (recall the notations in (3.4.2))

(a)

there exists an isometry $\alpha_{B}^{\circ}:\rho_{\mathbb{C}}^{*}\mathbf{L}_{B}|_{\mathsf{M}^{\circ}}\stackrel{{\scriptstyle\sim}}{{\to}}\mathbf{P}_{B}|_{\mathsf{M}^{\circ}}$ compatible with the Hodge filtrations (i.e., induces an isomorphism $(\rho_{\mathbb{C}}^{*}\mathsf{V})|_{\mathsf{M}^{\circ}}\stackrel{{\scriptstyle\sim}}{{\to}}\mathsf{P}|_{\mathsf{M}^{\circ}}$ of VHS over $\mathsf{M}^{\circ}$ );
(b)

there is an isometry $\alpha_{\ell_{0}}:\rho^{*}\mathbf{L}_{\ell_{0}}\stackrel{{\scriptstyle\sim}}{{\to}}\mathbf{P}_{\ell_{0}}$ whose restriction to $\mathsf{M}^{\circ}$ agrees with $\alpha_{B}^{\circ}{\otimes}\mathbb{Q}_{\ell_{0}}$ ;
(c)

for every $\ell\neq p$ , $\rho^{*}\mathbf{L}_{\ell}$ is étale-locally isomorphic to $\mathbf{P}_{\ell}$ over $\mathsf{M}$ ;
(d)

$\rho_{\mathbb{Q}}^{*}\mathbf{L}_{p}$ is étale locally isomorphic to $\mathbf{P}_{p}$ over $\mathsf{M}_{\mathbb{Q}}$ .

Note that the isomorphism $\alpha_{\ell_{0}}$ is unique if it exists. If (b) is satisfied for every prime $\ell\neq p$ , then we simply say that $\rho$ is admissible.

Theorem (5.2.3).

Consider the set-up in (5.2.1). Assume that for some prime $\ell_{0}\neq p$

(a)

$\mathbf{P}_{\ell_{0}}$ has constant $\lambda^{\mathrm{geo}}$ over $\mathbb{Z}_{(p)}$ as defined in (5.1.2), and
(b)

there is an $\ell_{0}$ -admissible period morphism $\rho:\mathsf{M}\to\mathscr{S}_{\mathsf{K}}(G)$ for some $\mathsf{K}$ as in (5.2.2).

Then for every $k$ which is finitely generated over $\mathbb{F}_{p}$ and $t:\mathrm{Spec\,}(k)\to\mathsf{M}$ , the fiber $\mathcal{X}_{t}$ satisfies the Tate conjecture in codimension $1$ .

Proof.

Define $\widetilde{G}:=\mathrm{CSpin}(L)$ and $\mathbb{K}_{p}:=\widetilde{G}(\mathbb{Z}_{p})$ as in §3.4. Choose a compact open subgroup $\mathbb{K}^{p}\subseteq\widetilde{G}(\mathbb{A}^{p}_{f})$ whose image is contained in $\mathsf{K}^{p}$ such that $\mathbb{K}:=\mathbb{K}_{p}\mathbb{K}^{p}$ is neat. Up to replacing $\mathsf{M}$ by a further connected étale cover, let us assume that $\rho$ can be lifted to a morphism $\mathsf{M}\to\mathscr{S}:=\mathscr{S}_{\mathbb{K}}(\widetilde{G})$ . Below we shall use $\rho$ to denote this lift. Recall the definition of special endomorphisms in (3.4.1). Under these preparations, we have the following proposition:

Proposition (5.2.4).

For every algebraically closed field $\kappa$ and geometric point $s:\mathrm{Spec\,}(\kappa)\to\mathsf{M}$ , there is an isomorphism $\theta_{s}:\mathrm{L}(\mathscr{A}_{\rho(s)})_{\mathbb{Q}}\stackrel{{\scriptstyle\sim}}{{\to}}\mathrm{PNS}(\mathcal{X}_{s})_{\mathbb{Q}}$ such that the diagram

(20)

commutes, where the vertical arrows are cycle class maps.

Now we prove (5.2.3) assuming the proposition above. First, we introduce a $p$ -adic analogue of $\lambda(-)$ over a closed point: Let $\mathbb{Z}_{p^{r}}:=W(\mathbb{F}_{p^{r}})$ and let $\mathbb{Q}_{p^{r}}$ be its fraction field. Let $\sigma$ denote the Frobenius action on $\mathbb{Q}_{p^{r}}$ . Let $H$ be an $F$ -isocrystal over $\mathbb{Q}_{p^{r}}$ , i.e., a finite dimensional $\mathbb{Q}_{p^{r}}$ -vector space equipped with an isomorphism $F:H{\otimes}_{\sigma}\mathbb{Q}_{p^{r}}\stackrel{{\scriptstyle\sim}}{{\to}}H$ . The $\mathbb{Q}_{p}$ -vector space $H^{F=p^{i}}$ is defined to be $\{h\in H:F(h{\otimes}1)=p^{i}h\}$ . For every $r\mid m$ , one naturally defines an F-isocrystal structure on $H{\otimes}_{\mathbb{Q}_{p^{r}}}\mathbb{Q}_{p^{m}}$ . Now we set

\lambda(H):=\varinjlim_{r\mid m}\dim_{\mathbb{Q}_{p}}(H{\otimes}\mathbb{Q}_{p^{m}})^{F=p}.

Choose a geometric point $\bar{t}$ over $t$ . Below any $\mathsf{M}$ -scheme $T$ is automatically viewed also as an $\mathscr{S}$ -scheme via $\rho$ .

Let us first assume that the field $k$ in (5.2.3) is finite. Then we have

\dim_{\mathbb{Q}_{\ell}}\mathrm{H}^{2}_{\mathrm{{\acute{e}}t}}(\mathcal{X}_{\bar{t}},\mathbb{Q}_{\ell}(1))^{\mathrm{Gal}(\bar{t}/t)}=\dim_{\mathbb{Q}_{p}}(\mathrm{H}^{2}_{\mathrm{cris}}(\mathcal{X}_{t}/W(k))[1/p])^{F=p},

where $F$ is the natural Frobenius action on crystalline cohomology. Indeed, both sides are equal to the order of the pole of the zeta function of $\mathcal{X}_{t}$ at $1$ ([Morrow, Prop. 4.1]). This implies that $\lambda(t^{*}\mathbf{P}_{\ell})=\lambda(\mathbf{P}_{\mathrm{cris},t}[1/p])$ for every prime $\ell\neq p$ . By assumption (5.2.2)(c), we also have $\lambda(t^{*}\mathbf{P}_{\ell})=\lambda(t^{*}\mathbf{L}_{\ell})$ . Similarly, we argue that

\lambda(\mathbf{L}_{\mathrm{cris},t}[1/p])=\lambda(\mathbf{P}_{\mathrm{cris},t}[1/p]).

(21)

Indeed, suppose that $k=\mathbb{F}_{p^{r}}$ . Lift $t$ to some $\widetilde{t}\in\mathsf{M}(\mathbb{Z}_{p^{r}})$ . Take $v$ to be the generic point of $\widetilde{t}$ , and let $\bar{v}$ be a geometric point over $v$ . Using the $p$ -adic comparison isomorphism and the compatibility with cycle class maps (cf. [BMS, Thm 14.3] and [IIK, Cor. 11.6]), as well as [CSpin, Prop. 4.7], we see that

\mathbf{P}_{\mathrm{cris},t}[1/p]\,{\cong}\,(\mathbf{P}_{p,\bar{v}}{\otimes}_{\mathbb{Q}_{p}}B_{\mathrm{cris}})^{\mathrm{Gal}_{\mathbb{Q}_{p^{r}}}}\text{ and }\mathbf{L}_{\mathrm{cris},t}[1/p]\,{\cong}\,(\mathbf{L}_{p,\bar{v}}{\otimes}_{\mathbb{Q}_{p}}B_{\mathrm{cris}})^{\mathrm{Gal}_{\mathbb{Q}_{p^{r}}}}.

Now (21) follows because by assumption (5.2.2)(d) there exists an isomorphism $\mathbf{P}_{p,\bar{v}}\,{\cong}\,\mathbf{L}_{p,\bar{v}}$ which is equivariant under an open subgroup of $\mathrm{Gal}_{\mathbb{Q}_{p^{r}}}$ , so that the F-isocrystals above are isomorphic up to base changing to $\mathbb{Q}_{p^{m}}$ for some $r\mid m$ .

Now for any $k$ (not necessarily finite) and $t$ in (5.2.3), we claim that

\mathrm{L}(\mathscr{A}_{t})_{\mathbb{Q}}{\otimes}\mathbb{Q}_{\ell}=\mathbf{L}_{\ell,\bar{t}}^{\mathrm{Gal}(\bar{t}/t)}

(22)

The above paragraph implies that if $k$ is finite, then every prime $\ell\neq p$ , $\lambda(t^{*}\mathbf{L}_{\ell})=\lambda(\mathbf{L}_{\mathrm{cris},t}[1/p])$ ; that is, assumption [MPTate, (6.2)] is satisfied for $\rho(t)$ , so that (22) is given by [MPTate, Thm 6.4]. When $k$ is not finite, the claim follows from the proof of [MPTate, Cor. 6.11]. Indeed, we may assume that $k$ is the fraction field of some smooth and geometrically connected variety $T$ over a finite extension of $\mathbb{F}_{p}$ , and $t$ extends to a morphism $T\to\mathsf{M}$ . Let us choose a closed point $t_{0}$ on $T$ , which we also view as a $k(t_{0})$ -valued point on $\mathsf{M}$ . Choose a geometric point $\bar{t}_{0}$ over $t_{0}$ , and let $\gamma$ be an étale path connecting $\bar{t}$ and $\bar{t}_{0}$ . By [FC90, I Prop. 2.7], $\mathrm{End}(\mathscr{A}_{T})=\mathrm{End}(\mathscr{A}_{t})$ . This gives us a specialization morphism $\mathrm{L}(\mathscr{A}_{t})\to\mathrm{L}(\mathscr{A}_{t_{0}})$ which fits into a commutative diagram below

(recall the notations in §3.4). We claim that all vertical squares are Cartesian. For the squares on the left and right, this is clear by the definition (3.4.1). For the square at the front, this is obtained by applying (22) to $t_{0}$ . Hence the remaining diagram at the back must also be Cartesian. Therefore, the surjectivity of $\mathrm{End}(\mathscr{A}_{t}){\otimes}\mathbb{Q}_{\ell}\to\mathrm{End}(\mathbf{H}_{\ell,\bar{t}})^{\mathrm{Gal}(\bar{t}/t)}$ ([Zarhin2]) implies the surjectivity of $\mathrm{L}(\mathscr{A}_{t}){\otimes}\mathbb{Q}_{\ell}\to\mathbf{L}_{\ell,\bar{t}}^{\mathrm{Gal}(\bar{t}/t)}$ . Hence we have affirmed (22) for $t$ .

Finally, combining (20), (22), and assumption (5.2.2)(c), we have

\dim\mathrm{PNS}(\mathcal{X}_{\bar{t}})_{\mathbb{Q}}=\lambda(t^{*}\mathbf{P}_{\ell_{0}})=\lambda(t^{*}\mathbf{L}_{\ell_{0}})=\lambda(t^{*}\mathbf{L}_{\ell})=\lambda(t^{*}\mathbf{P}_{\ell})

This implies the Tate conjecture in codimension $1$ for $\mathcal{X}_{t}$ . ∎

Now we prove (5.2.4), which is the key geometric input to (5.2.3).

Proof.

We first prove the statement when $\mathrm{char\,}\kappa=0$ . Without loss of generality, we may assume that $\kappa$ can be embedded to $\mathbb{C}$ ; moreover, as $\mathrm{Aut}(\mathbb{C})$ acts transitively on the set of connected components of $\mathsf{M}_{\mathbb{C}}$ , we may choose an embedding such that the resulting $\mathbb{C}$ -point lies on the distinguished component $\mathsf{M}^{\circ}$ (defined in (5.2.1)) of $\rho$ . To prove the statement we may replace $s$ by this $\mathbb{C}$ -point. Then the statement follows from Hodge theory. Indeed, we obtain a commutative diagram⁹⁹9Note that we are considering every $\mathsf{M}$ -scheme also as an $\mathscr{S}$ -scheme via $\rho$ , so $\mathscr{A}_{s}$ (resp. $\mathbf{L}_{B,s}$ ) is the same as $\mathscr{A}_{\rho(s)}$ (resp. $\mathbf{L}_{B,\rho(s)}$ ). Similar conventions apply below.:

(23)

The vertical maps are again cycle class maps, but this time they are isomorphisms. For the arrow on the right, we are applying the Lefschetz $(1,1)$ -theorem. Since $\alpha_{\ell_{0},s}=\alpha^{\circ}_{B,s}{\otimes}\mathbb{Q}_{\ell_{0}}$ by assumption, we obtain (20).

Now we assume that $\mathrm{char\,}\kappa=p$ . Set $W=W(\kappa)$ . We shall construct $\theta_{s}$ by considering characteristic $0$ liftings of $s$ . Let $\widehat{\mathscr{S}}_{s}$ be the formal completion of $\mathscr{S}_{\mathbb{K}}(\widetilde{G})_{W}$ at $\rho(s)$ . For a special endomorphism $\zeta\in\mathrm{L}(\mathscr{A}_{s})$ consider the following functor:

\mathrm{Def}_{\mathscr{S}}(\zeta,s):R\mapsto\{\widetilde{s}\in\widehat{\mathscr{S}}_{s}(R)\mid\zeta\textit{ deforms to }\mathrm{L}(\mathscr{A}_{\widetilde{s}})\}

(24)

where $R$ runs through all Artin $W$ -algebras. By [CSpin, §5.14], $\mathrm{Def}_{\mathscr{S}}(\zeta,s)$ is represented by a closed formal subscheme of $\widehat{\mathscr{S}}_{s}$ cut out by a single formal power series $f_{\zeta}\in\mathcal{O}_{\widehat{\mathscr{S}}_{s}}$ . Similarly, let $\widehat{\mathsf{M}}_{s}$ be the formal completion of $\mathsf{M}_{W}$ at $s$ . Then $\rho$ restricts to a morphism $\widehat{\mathsf{M}}_{s}\to\widehat{\mathscr{S}}_{s}$ . Consider the pullback of $\mathscr{A}$ to $\mathsf{M}$ and define $\mathrm{Def}_{\mathsf{M}}(\zeta,s)$ to be the functor defined by (24) with $\widehat{\mathscr{S}}_{s}$ replaced by $\widehat{\mathsf{M}}_{s}$ . Then we have a fiber diagram:

(25)

In particular, $\mathrm{Def}_{\mathsf{M}}(\zeta,s)$ is a closed formal subscheme of $\widehat{\mathsf{M}}_{s}$ cut out by the pullback $\rho^{*}(f_{\zeta})$ . Now we prove the key intermediate lemma:

Lemma (5.2.5).

Up to replacing $\zeta$ by a power, $\mathrm{Def}_{\mathsf{M}}(\zeta,s)$ is flat over $W$ .

Proof.

It suffices show that if $\zeta$ does not lie in the image of $\mathrm{L}(\mathscr{A}_{\mathsf{M}})_{\mathbb{Q}}\to\mathrm{L}(\mathscr{A}_{s})_{\mathbb{Q}}$ , then $\mathrm{Def}_{\mathsf{M}}(\zeta,s)$ is flat over $W$ , because otherwise up to replacing $\zeta$ by a power, $\mathrm{Def}_{\mathsf{M}}(\zeta,s)=\widehat{\mathsf{M}}_{s}$ . Suppose by way of contradiction that $\mathrm{Def}_{\mathsf{M}}(\zeta,s)$ is not flat over $W$ , which is equivalent to saying that $\rho^{*}(f_{\zeta})$ vanishes on the entire mod $p$ disk $\widehat{\mathsf{M}}_{\kappa,s}:=\widehat{\mathsf{M}}_{s}{\otimes}_{W}\kappa$ , i.e., the formal completion of $\mathsf{M}_{\kappa}$ at $s$ . Let $\boldsymbol{\zeta}$ be the deformation of $\zeta$ over $\widehat{\mathsf{M}}_{\kappa,s}$ . Let use write $u$ for the generic point of $\widehat{\mathsf{M}}_{\kappa,s}$ . Let $S$ be the connected component of $\mathsf{M}_{\kappa}$ which contains $s$ and let $v$ be its generic point. Since $\widehat{\mathsf{M}}_{\kappa,s}$ is also the completion of $S$ at $s$ , there is natural embedding $k(v)\hookrightarrow k(u)$ of residue fields. Let $\bar{u}$ be the geometric point over $u$ defined by a chosen algebraic closure of $k(u)$ . We view it also as a geometric point over $v$ ..

Recall that we assumed that $\mathrm{Mon}(R^{2}f_{\mathrm{{\acute{e}}t}*}\mathbb{Q}_{2},b)$ is connected in (5.2.1), so that $\mathrm{NS}(\mathcal{X}_{\eta})_{\mathbb{Q}}\hookrightarrow\mathrm{NS}(\mathcal{X}_{b})_{\mathbb{Q}}$ is an isomorphism by (4.1.2). Therefore, $\pi_{1}^{\mathrm{{\acute{e}}t}}(\mathsf{M},b)$ acts trivially on $\mathrm{NS}(\mathcal{X}_{b}){\otimes}\mathbb{Q}_{\ell_{0}}$ . Since $\alpha_{\ell_{0},b}$ is $\pi_{1}^{\mathrm{{\acute{e}}t}}(\mathsf{M},b)$ -equivariant by assumption (5.2.2)(b), $\pi_{1}^{\mathrm{{\acute{e}}t}}(\mathsf{M},b)$ acts trivially on $\mathrm{L}(\mathcal{X}_{b}){\otimes}\mathbb{Q}_{\ell_{0}}$ as well. Hence every element of $\mathrm{L}(\mathscr{A}_{b})_{\mathbb{Q}}$ is defined over $\eta$ . It follows from [FC90, I Prop. 2.7] that $\mathrm{L}(\mathscr{A}_{\eta})=\mathrm{L}(\mathscr{A}_{\mathsf{M}})$ . By (5.1.4), $\dim\mathrm{PNS}(\mathcal{X}_{b})_{\mathbb{Q}}=\lambda^{\mathrm{geo}}(\mathbf{P}_{\ell_{0}})$ . Assumption (5.2.2)(b) implies that $\mathbf{L}_{\ell_{0}}|_{\mathsf{M}}$ also has constant $\lambda^{\mathrm{geo}}$ and $\lambda^{\mathrm{geo}}(\mathbf{L}_{\ell_{0}}|_{\mathsf{M}})=\lambda^{\mathrm{geo}}(\mathbf{P}_{\ell_{0}})$ . Combining these observations, we must have $\dim\mathrm{L}(\mathscr{A}_{\mathsf{M}})_{\mathbb{Q}}=\lambda^{\mathrm{geo}}(\mathbf{L}_{\ell_{0}}|_{\mathsf{M}})$ .

Now consider the subspace $I:=\varinjlim_{U}\mathbf{L}_{\ell_{0},\bar{u}}^{U}$ , as $U$ runs through open subgroups of $\pi^{\mathrm{{\acute{e}}t}}_{1}(v,\bar{u})$ , so that $\dim I=\lambda(\mathbf{L}_{\ell_{0}}|_{S})=\lambda^{\mathrm{geo}}(\mathbf{L}_{\ell_{0}}|_{\mathsf{M}})$ (recall definition (5.1.2)). As the endomorphism scheme of an abelian scheme is representable and unramified over the base, every element in $\mathrm{L}(\mathscr{A}_{\bar{u}})$ is defined over some finite separable extension of $k(v)$ inside $k(\bar{u})$ . Therefore, we obtain a well defined map $\mathrm{L}(\mathscr{A}_{\bar{u}}){\otimes}\mathbb{Q}_{\ell_{0}}\hookrightarrow I$ . However, we note that the composite

\mathrm{L}(\mathscr{A}_{\mathsf{M}}){\otimes}\mathbb{Q}_{\ell_{0}}\to\mathrm{L}(\mathscr{A}_{\bar{u}}){\otimes}\mathbb{Q}_{\ell_{0}}\hookrightarrow I

has to be an isomorphism because $\dim\mathrm{L}(\mathscr{A}_{\mathsf{M}})_{\mathbb{Q}}=\dim I$ . This forces the natural map $\mathrm{L}(\mathscr{A}_{\mathsf{M}})_{\mathbb{Q}}\to\mathrm{L}(\mathscr{A}_{\bar{u}})_{\mathbb{Q}}$ to be an isomorphism. Now note that $\boldsymbol{\zeta}_{\bar{u}}\in\mathrm{L}(\mathscr{A}_{\bar{u}})_{\mathbb{Q}}$ . But as we assumed that $\zeta$ does not come from $\mathrm{L}(\mathscr{A}_{\mathsf{M}})_{\mathbb{Q}}$ , the same has to be true for $\boldsymbol{\zeta}_{\bar{u}}$ . This gives the desired contradiction. ∎

Suppose we have replaced $\zeta$ by a power so that the above lemma holds. Following Deligne’s argument for [Del02, Cor. 1.7], we show that there exists a DVR $V$ finite flat over $W$ such that $\mathrm{Def}_{\mathsf{M}}(\zeta,s)$ admits a $V$ -valued point. Indeed, suppose that $\mathrm{Def}_{\mathsf{M}}(\zeta,s)=\mathrm{Spf}(R)$ for some complete local $W$ -algebra $R$ . As $p$ is not a zero divisor in $R$ , one may complete $p$ into a system of parameters $(p,x_{1},\cdots,x_{d})$ for $R$ , where $d=\dim R-1$ ([stacks-project, 0BWY]). Let $R^{\prime}:=R/(x_{1},\cdots,x_{d})$ . By [stacks-project, 00LJ], $p$ is not a zero divisor in $R^{\prime}$ , so that $R^{\prime}$ remains flat over $W$ . In particular it has zero-dimensional fibres over $W$ and hence is quasi-finite. By [stacks-project, 04GG(13)] we can write $R^{\prime}=A\times B$ where $A$ is a finite $W$ -algebra and $B/pB=0$ . Since $\mathrm{Spec\,}R^{\prime}$ is closed in $\mathrm{Spec\,}R$ we must have $B=0$ , i.e., $R^{\prime}$ is finite over $W$ . Now, by flatness $R^{\prime}$ admits a $W$ -algebra morphism $R^{\prime}\to K$ for some finite extension $K$ of $W[1/p]$ , but then by finiteness $R^{\prime}$ necessarily maps into the integral closure of $W$ in $K$ , which is the desired $V$ .¹⁰¹⁰10We spell out the commutative algebra details because it is a little counterintuitive that this argument does not require $R$ to be reasonably non-singular (cf. (5.2.6)).

Let $\widetilde{s}\in\mathsf{M}_{W}(V)$ denote the point defined by the above morphism $R\to V$ . Choose an algebraic clsoure $\bar{K}$ . As $V$ is a strictly Henselian DVR, it defines an étale path $\gamma_{\widetilde{s}}$ connecting $w:=\widetilde{s}_{\bar{K}}$ and $s$ . There is a compatible specialization map $\mathrm{NS}(\mathcal{X}_{w})\to\mathrm{NS}(\mathcal{X}_{s})$ along $V$ (cf. [MPJumping, Prop. 3.6] and its proof), which we denote by $\mathrm{sp}(\widetilde{s})$ . There is a similar specialization map of special endomorphisms. Now we have the following diagram:

(26)

Note that $\mathrm{char\,}\bar{K}=0$ , so we have shown that $\theta_{w}$ exists. The map $\theta_{s}$ does not exist yet. But by construction of $\widetilde{s}$ , $\zeta$ lifts to some (necessarily unique) $\widetilde{\zeta}$ over $w$ , and we can define $\theta_{s}(\zeta)$ to be $\mathrm{sp}(\widetilde{s})(\theta_{w}(\widetilde{\zeta}))$ . Note that $\theta_{s}(\zeta)$ does not depend on the choice of $\widetilde{s}$ because its class in $\mathbf{P}_{\ell_{0},s}$ is completely determined by the class of $\zeta$ in $\mathbf{L}_{\ell_{0},s}$ via $\alpha_{\ell_{0},s}$ . Repeating this construction for every $\zeta\in\mathrm{L}(\mathscr{A}_{s})$ , we obtain the desired map $\theta_{s}$ . ∎

Remark (5.2.6).

In [Del02], Deligne deduced Cor. 1.7 directly from Thm 1.6—there is no control on how singular the formal scheme $\mathrm{Def}(X_{0},L_{0})$ in Thm 1.6 might be (here $X_{0}$ is a K3 surface over an algebraically closed field $k$ of characteristic $p>0$ and $\mathrm{Def}(X_{0},L_{0})$ is the universal deformation space of the pair $(X_{0},L_{0})$ ). From later work [OgusCrystals, (2.2)], we do know that if $L_{0}$ is not a $p$ -th power of another line bundle and $p>2$ , then $\mathrm{Def}(X_{0},L_{0})$ in Deligne’s Thm 1.6 is indeed always regular. However, as Deligne’s argument shows, this is unnecessary for one to deduce that $\mathrm{Def}(X_{0},L_{0})$ admits a point valued over some finite flat extension of $W(k)$ .

5.3 Proof of Theorem B

(5.3.1)

The reader may have noticed that (4.1.9) and (4.1.10) apply to geometrically connected bases. To make use of these results, we consider the following simple-minded functor, which sometimes goes by the name “Grothendieck restriction”: If $k\subseteq k^{\prime}$ is a field extension, and $T$ is a $k^{\prime}$ -scheme, we write $T_{(k)}$ for the $k$ -scheme obtained by composing the structure morphism of $T$ with the projection $\mathrm{Spec\,}(k^{\prime})\to\mathrm{Spec\,}(k)$ . Then $T\mapsto T_{(k)}$ is the left adjoint to the base change functor from $k$ -schemes to $k^{\prime}$ -schemes.

We will only apply this functor when $k=\mathbb{Q}$ . Let $S$ be a connected smooth $\mathbb{Q}$ -variety. Then its scheme of connected components $\pi_{0}(S)$ is the spectrum of some number field $F$ . The natural morphism $S\to\pi_{0}(S)$ endows $S$ with the structure of an $F$ -variety. We denote this $F$ -variety by ${}^{F}S$ . Note that now $S=({}^{F}S)_{(\mathbb{Q})}$ and ${}^{F}S$ is geometrically connected as an $F$ -variety.

(5.3.2)

Since $S$ and ${}^{F}S$ have the same underlying scheme, étale sheaves or filtered flat vector bundles on $S$ are naturally identified with the corresponding structures on ${}^{F}S$ and vice versa and we do not distinguish them notationally. If $Y$ is a $\mathbb{Q}$ -variety, then there is a canonical bijection between the sets of morphisms $\epsilon:\mathrm{Mor}_{F}({}^{F}S,Y_{F})\stackrel{{\scriptstyle\sim}}{{\to}}\mathrm{Mor}_{\mathbb{Q}}(S,Y)$ . Suppose now that $\rho:{}^{F}S\to Y_{F}$ is a morphism, and $M$ is an étale sheaf or a filtered flat vector bundle on $Y$ , then $\rho^{*}(M|_{Y_{F}})$ is canonically identified with $\epsilon(\rho)^{*}M$ because $\epsilon(\rho)=(Y_{F}\to Y)\circ\rho$ as morphisms of schemes.

Proposition (5.3.3).

Let $U\subseteq\mathrm{Spec}(\mathbb{Z}[2^{-1}])$ be an open subscheme. Let $\mathsf{M}$ be a connected separated smooth $U$ -scheme of finite type with generic point $\eta$ . Let $\mathcal{X}\to S$ be a smooth projective morphism such that $\mathcal{X}|_{\mathsf{M}_{\mathbb{Q}}}$ is a $\heartsuit$ -family.

Let $\boldsymbol{\xi}$ be a relatively ample line bundle of $\mathcal{X}/\mathsf{M}$ and let $\Lambda\subseteq\mathrm{NS}(\mathcal{X}_{\eta})_{\mathbb{Q}}$ be a subspace containing the class of $\boldsymbol{\xi}_{\eta}$ . Let $b\in\mathsf{M}(\mathbb{C})$ be a point lying above $\eta$ and $\mathsf{M}^{\circ}$ be the connected component of $\mathsf{M}_{\mathbb{C}}$ containing $b$ . Assume that $R^{2}f_{*}\mathbb{Q}_{2}$ has constant $\lambda^{\mathrm{geo}}$ , and for every $p\in U$ , the pairing on $\mathrm{PH}^{2}(\mathcal{X}_{b},\mathbb{Z}_{(p)})_{\mathrm{tf}}$ is self-dual. Then we have:

(a)

If $(\mathcal{X}/\mathsf{M},\boldsymbol{\xi})|_{\mathsf{M}^{\circ}}$ has maximal monodromy (see (2.2.3)), then for every $s\in\mathsf{M}$ , $\mathcal{X}_{s}$ satisfies the Tate conjecture in codimension $1$ .
(b)

If $(\mathcal{X}/\mathsf{M},\boldsymbol{\xi})|_{\mathsf{M}^{\circ}}$ belongs to case (R2’) (see (2.2.4)), then up to replacing $U$ by a nonempty open subscheme, the above is true.

Here $\mathrm{PH}^{2}(\mathcal{X}_{b},\mathbb{Z}_{(p)})_{\mathrm{tf}}$ denotes the submodule of $\mathrm{H}^{2}(\mathcal{X}_{b},\mathbb{Z}_{(p)})_{\mathrm{tf}}$ consisting of elements orthogonal to the image of $\Lambda$ in $\mathrm{NS}(\mathcal{X}_{b})_{\mathbb{Q}}$ under the pairing on $\mathrm{H}^{2}(\mathcal{X}_{b},\mathbb{Q})$ induced by $\boldsymbol{\xi}_{b}$ . Recall that the big monodromy case contains (R+) = (R1) + (R2) and (CM).

Proof.

The hypothesis remains unchanged if we replace $\mathsf{M}$ by a connected étale cover (and $b$ by a lift). Therefore, we may assume that $\mathrm{Mon}(R^{2}f_{*}\mathbb{Q}_{2},b)$ is connected. Moreover, to prove (b), we may assume that $\Lambda=\mathrm{NS}(\mathcal{X}_{b})_{\mathbb{Q}}$ . Indeed, replacing $\Lambda$ by $\mathrm{NS}(\mathcal{X}_{b})_{\mathbb{Q}}$ might make $\mathrm{PH}^{2}(\mathcal{X}_{b},\mathbb{Z}_{(p)})_{\mathrm{tf}}$ no longer self-dual only for finitely many $p$ , but we are allowed to shrink $U$ .

Apply the set-ups in (4.1.3) to $S=\mathsf{M}_{\mathbb{Q}}$ and $S^{\circ}=\mathsf{M}^{\circ}$ . Let $V,G=\mathrm{SO}(V)$ and $\mu_{b}:V\stackrel{{\scriptstyle\sim}}{{\to}}\mathsf{P}_{B,b}$ be as defined in (4.1.6). Let $L\subseteq V$ be the $\mathbb{Z}$ -lattice defined by $\mu_{b}^{-1}(\mathsf{P}_{B,b}\cap\mathrm{H}^{2}(\mathcal{X}_{b},\mathbb{Z})_{\mathrm{tf}})$ and choose a $N\gg 0$ such that $\mathsf{K}:=G(\mathbb{A}_{f})\cap\ker(\mathrm{GL}(L{\otimes}\widehat{\mathbb{Z}})\to\mathrm{GL}(L/2^{N}L))$ satisfies condition $(\sharp)$ ; in case (b), we additionally require $\mathsf{K}_{\ell_{0}=2}$ to be sufficiently small (see (4.1.6) for these notions). Note that $\mathsf{K}$ is of the form $\prod_{q}\mathsf{K}_{q}$ , where $q$ runs through all primes and $\mathsf{K}_{q}$ is a compact open subgroup of $G(\mathbb{Q}_{q})$ . By (4.1.4), up to replacing $\mathsf{M}$ by a further finite connected étale cover, we assume that $\mathsf{K}$ is admissible, i.e., the image of $\pi_{1}^{\mathrm{{\acute{e}}t}}(\mathsf{M},b)$ in $\mathrm{GL}(\mathsf{P}_{\mathrm{{\acute{e}}t},b})$ lies in $\mathsf{K}$ via the chosen isometry $\mu_{b}$ . Now we set $F$ to be the field such that $\pi_{0}(S)=\mathrm{Spec\,}(F)$ and recall that the base point $b$ induces an embedding $F\hookrightarrow\mathbb{C}$ through which $S^{\circ}=({}^{F}S)_{\mathbb{C}}$ .

For (a), we may now apply (4.1.9) to ${}^{F}S$ to conclude that there is a period morphism $\rho:{}^{F}S\to\mathrm{Sh}_{\mathsf{K}}(G)_{F}$ together with an isomorphism $\alpha^{\circ}_{B}:\rho_{\mathbb{C}}^{*}(\mathsf{V}_{B})|_{S^{\circ}}\to\mathsf{P}_{B}|_{S^{\circ}}$ which preserves the Hodge filtrations such that $\alpha^{\circ}_{\mathrm{{\acute{e}}t}}:=\alpha^{\circ}_{B}{\otimes}\mathbb{A}_{f}$ descends to an isomorphism $\rho^{*}\mathsf{V}_{\mathrm{{\acute{e}}t}}\stackrel{{\scriptstyle\sim}}{{\to}}\mathsf{P}_{\mathrm{{\acute{e}}t}}$ over ${}^{F}S$ . By (5.3.2) we may view $\rho$ as a morphism $S\to\mathrm{Sh}_{\mathsf{K}}(G)$ over $\mathbb{Q}$ and $\alpha_{\mathrm{{\acute{e}}t}}$ as an isomorphism of étale sheaves over $S$ . Let $p\neq 2$ be a prime in $U$ . Restrict $\mathsf{M}$ to $\mathbb{Z}_{(p)}$ and apply the set-ups in (5.2.1). Note that by construction, $\mathsf{K}_{p}=\mathrm{SO}(L{\otimes}\mathbb{Z}_{p})$ is hyperspecial. Let $\mathscr{S}$ be the integral model of $\mathrm{Sh}_{\mathsf{K}}(G)$ over $\mathbb{Z}_{(p)}$ . Recall that $\mathsf{V}_{\mathrm{{\acute{e}}t}}$ (resp. $\mathsf{P}_{\mathrm{{\acute{e}}t}}$ ) is the restriction of $\prod_{\ell\neq p}\mathbf{L}_{\ell}\times\mathbf{L}_{p}[1/p]$ to $\mathrm{Sh}_{\mathsf{K}}(G)$ (resp. $\prod_{\ell\neq p}\mathbf{P}_{\ell}\times\mathbf{P}_{p}[1/p]$ to $S=\mathsf{M}_{\mathbb{Q}}$ ) (see (3.4.2) and (5.2.1)). The existence of $\alpha_{\mathrm{{\acute{e}}t}}$ allows us to apply (3.4.4) to extend $\rho$ to a morphism $\mathsf{M}_{\mathbb{Z}_{(p)}}\to\mathscr{S}$ , which by construction is admissible in the sense of (5.2.2). Now we conclude by (5.2.3).

For (b) a minor adaptation is needed. Take $\ell_{0}=2$ . By (4.1.10) we still have a period morphism $\rho_{\mathbb{C}}:S^{\circ}\to\mathrm{Sh}_{\mathsf{K}}(G)_{\mathbb{C}}$ equipped with an isomorphism $\alpha^{\circ}_{B}:\rho_{\mathbb{C}}^{*}\mathsf{V}_{B}\stackrel{{\scriptstyle\sim}}{{\to}}\mathsf{P}_{B}|_{S^{\circ}}$ which preserves the Hodge filtrations, but $\rho_{\mathbb{C}}$ does not descend all the way to $F$ . Instead, we only have that for some finite extension $F^{\prime}/F$ in $\mathbb{C}$ , and $\widetilde{S}:=({}^{F}S){\otimes}_{F}F^{\prime}$ , $\rho_{\mathbb{C}}$ descends to a morphism $\rho:\widetilde{S}\to\mathrm{Sh}_{\mathsf{K}}(G)_{F^{\prime}}$ such that

(i)

$\alpha^{\circ}_{\ell_{0}}:=\alpha^{\circ}_{B}{\otimes}\mathbb{Q}_{\ell_{0}}$ descends to an isomorphism $\rho^{*}\mathsf{V}_{\ell_{0}}\stackrel{{\scriptstyle\sim}}{{\to}}\mathsf{P}_{\ell_{0}}|_{\widetilde{S}}$ ;
(ii)

and $\rho^{*}\mathsf{V}_{\ell}$ is étale locally isomorphic to $\mathsf{P}_{\ell}|_{\widetilde{S}}$ for every other $\ell$ .

Set $S^{\prime}:=\widetilde{S}_{(\mathbb{Q})}$ . Then by (5.3.2) again, we may view $\rho$ as a morphism $S^{\prime}\to\mathrm{Sh}_{\mathsf{K}}(G)$ and (i), (ii) above as statements about étale sheaves over $S^{\prime}$ . Note that $S^{\prime}$ is naturally a connected étale over of $S$ . As $L$ is self-dual at every $p\in U$ , there exists a $U$ -scheme $\mathscr{S}$ such that $\mathscr{S}{\otimes}\mathbb{Z}_{(p)}$ is the integral canonical model of $\mathrm{Sh}_{\mathsf{K}}(G)$ over $\mathbb{Z}_{(p)}$ (cf. the first theorem of [Lovering]). By a standard spreading out argument, up to further shrinking $U$ , we may assume that $S^{\prime}$ is the $\mathbb{Q}$ -fiber of a smooth $U$ -scheme $\mathsf{M}^{\prime}$ ; moreover, $S^{\prime}\to S$ extends to an étale covering map $\mathsf{M}^{\prime}\to\mathsf{M}$ and $\rho:S^{\prime}\to\mathrm{Sh}_{\mathsf{K}}(G)$ extends to a morphism $\rho_{U}:\mathsf{M}^{\prime}\to\mathscr{S}$ . The reader readily checks using (i) and (ii) above that for each $p\in U$ the localization $\rho_{U}{\otimes}\mathbb{Z}_{(p)}$ is an $\ell_{0}$ -admissible period morphism in the sense of (5.2.2). Therefore, the conclusion follows from (5.2.3). ∎

We are now ready to prove Theorem B.

Proof.

Note that the conclusion of Theorem B is for $p\gg 0$ . By (5.3.3) it suffices to show that there are exists an open subscheme $U\subseteq\mathrm{Spec\,}(\mathbb{Z}[2^{-1}])$ such that after we restrict $\mathsf{M}$ to $U$ , $R^{2}f_{*}\mathbb{Q}_{2}$ has constant $\lambda^{\mathrm{geo}}$ . By combining Nagata’s compactification and Hironaka’s resolution of singularities in characteristic zero, we can find a compactification $\bar{\mathsf{M}}$ of the generic fiber $\mathsf{M}_{\mathbb{Q}}$ such that the boundary $\mathfrak{D}\coloneqq\bar{\mathsf{M}}-\mathsf{M}_{\mathbb{Q}}$ , equipped with the reduced scheme structure, is a normal crossing divisor. For some open subscheme $U$ of $\mathrm{Spec}(\mathbb{Z}[2^{-1}])$ , $\bar{\mathsf{M}}$ and $\mathfrak{D}$ are defined over $U$ , and $\mathfrak{D}$ becomes a relative normal crossing divisor over $U$ . This implies that for $(p)\in U$ , $\mathsf{M}_{\mathbb{Z}_{(p)}}$ admits a good compactification relative to $\mathbb{Z}_{(p)}$ in the sense of (5.1.2). Now we conclude by (5.1.3). ∎

6 Deforming Curves on Parameter Spaces

6.1 Families of Curves which homogeneously dominate a Variety

Let $k$ be an algebraically closed field. For a morphism $f\colon X\to Y$ betweem $k$ -varieties, we say that $f$ has equi-dimensional fibers if for every two points $y,y^{\prime}\in Y$ , $\dim X_{y}=\dim X_{y^{\prime}}$ . If $f:X\to Y$ is a smooth morphism between $k$ -varieties, denote by $\mathcal{T}(X/Y)$ the relative tangent bundle, i.e., the dual of $\Omega^{1}_{X/Y}$ . If $Y=\mathrm{Spec\,}(k)$ , then we simply write $\mathcal{T}X$ for $\mathcal{T}(X/Y)$ , and for a $k$ -point $x\in X$ , we write $\mathcal{T}_{x}X$ for the tangent space $T_{x}X$ to emphasize that it is a fiber of $\mathcal{T}X$ .

Definition (6.1.1).

Let $S$ and $T$ be two smooth irreducible $k$ -varieties, $g\colon\mathscr{C}\to T$ be a smooth family of connected curves, and $\varphi\colon\mathscr{C}\to S$ be a morphism with equi-dimensional fibers. Let $U$ be the maximal open subvariety of $\mathscr{C}$ on which the composition $\mathcal{T}(\mathscr{C}/T)\hookrightarrow\mathcal{T}\mathscr{C}\to\varphi^{*}(\mathcal{T}S)$ does not vanish. Suppose that

(a)

the induced morphism $\mathscr{C}\to T\times S$ is quasi-finite,
(b)

for every $k$ -point $s\in S$ , $U_{s}:=U\cap\varphi^{-1}(s)$ is dense in $\varphi^{-1}(s)$ ;
(c)

the morphism $U\to\mathbb{P}(\mathcal{T}S)$ has equi-dimensional fibers.

Then we say that the family of curves $\mathscr{C}/T$ homogeneously dominates $S$ (via the morphism $\varphi$ ). If there exists an open dense subvariety $T^{\prime}\subseteq T$ such that the restriction $\mathscr{C}|_{T^{\prime}}$ homogeneously dominates $S$ , then we say that $\mathscr{C}/T$ strongly dominates $S$ .

The natural morphism $U\to\mathbb{P}(\mathcal{T}S)$ is induced by the identification $\mathscr{C}\,{\cong}\,\mathbb{P}(\mathcal{T}(\mathscr{C}/T))$ . Roughly speaking, the family $\mathscr{C}/T$ homogeneously dominates $S$ if there are curves passing through every given point on $S$ in any given direction, and the sub-family of such curves has a fixed dimension.

The notion “ $\mathscr{C}/T$ strongly dominates $S$ ” is only defined for convenience, as sometimes the natural families of curves have some bad locus on $T$ of smaller dimension which does not affect applications.

Lemma (6.1.2).

Let $\mathscr{P}\to S$ be a smooth morphism between smooth $k$ -varieties. Let $\mathcal{X}\subseteq\mathscr{P}$ be a relative effective Cartier divisor whose total space is smooth. If $s\in S(k)$ is a point such that $\mathcal{X}_{s}$ has isolated singularities, then there exists an open dense subvariety $U\subseteq\mathbb{P}(\mathcal{T}_{s}S)$ with the following property: For every unramified morphism $\varphi\colon C\to S$ from a smooth curve $C$ which sends a point $c\in C$ to $s$ , the total space of the pullback family $\mathcal{X}|_{C}$ has no singularity on $\mathcal{X}_{s}$ if $d\varphi(\mathbb{P}(\mathcal{T}_{c}C))\in U$ .

Proof.

Since the question is étale-local in nature, we might as well assume that $S=\mathbb{A}^{m}_{k}$ for $m=\dim S$ , $s=0$ , $\mathcal{X}_{s}$ has a single isolated singularity at a $k$ -point $P\in\mathcal{X}_{s}\subseteq\mathscr{P}_{s}$ , and $\mathscr{P}$ is isomorphic to $\mathbb{A}^{n}_{S}=S\times\mathbb{A}^{n}_{k}$ near $P$ . Let $x_{i}$ ’s and $s_{j}$ ’s be the coordinates on $\mathbb{A}^{n}_{k}$ and $S_{k}$ respectively. Suppose that $\mathcal{X}$ is locally cut out by an equation $F(x_{1},\cdots,x_{n},s_{1},\cdots,s_{m})$ near $P$ . That $P$ is a singularity of the fiber $\mathcal{X}_{s}$ but not of the total space $\mathcal{X}$ implies that $\partial F/\partial s_{j}\neq 0$ at $P$ for some $j$ . One may simply take $U$ to be the open subscheme of $\mathbb{P}(\mathcal{T}_{s}S)\,{\cong}\,\mathbb{P}^{m-1}$ where the coordinate of $\partial_{s_{j}}$ is nonzero. ∎

Definition (6.1.3).

Let $f\colon X\to S$ be a finite type morphism between schemes.

(a)

Let the singular locus $\mathrm{Sing}(f)$ be the reduced closed subscheme of $X$ whose support consists of all points where $f$ fails to be smooth.¹¹¹¹11Note that this definition is different from the one given in [stacks-project, Tag 0C3H].
(b)

If $f$ is in addition proper and flat, we say that the scheme-theoretic image of $\mathrm{Sing}(f)$ is the (generalized) discriminant locus of $f$ , and denote it by $\mathrm{Disc}(f)$ .
(c)

In the above situation, we say that $\mathrm{Disc}(f)$ is mild if it has codimension at least $1$ in $S$ and there exists a dense open subscheme $V\subseteq\mathrm{Disc}(f)$ such that for every geometric point $s$ on $V$ , the fiber $X_{s}$ has only isolated singularities.

Remark (6.1.4).

Note that the properness assumption on $f$ implies that $\mathrm{Disc}(f)$ is closed in $S$ . Moreover, since it is defined to be the scheme-theoretic image of a reduced scheme, it is also reduced. Its formation commutes with flat base change but not arbitrary base change: For any morphism $T\to S$ , $\mathrm{Disc}(f_{T})$ is always the reduced subscheme of $\mathrm{Disc}(f)_{T}$ , so they are equal if and only if the latter is reduced.

Proposition (6.1.5).

Let $\mathcal{X}$ and $S$ be as in (6.1.2). Suppose that

•

the generalized discriminant variety $D\coloneqq\mathrm{Disc}(f)\subseteq S$ is mild;
•

there is a family of smooth curves $g:\mathscr{C}\to T$ which homogeneously dominates $S$ through a morphism $\varphi:\mathscr{C}\to S$ .

Then for a general $k$ -point $t\in T_{k}$ , the total space of the pullback family $\mathcal{X}|_{\mathscr{C}_{t}}$ is smooth.

Proof.

By assumption we have a diagram

Let $\mathcal{Z}\subseteq S\times T$ be the subset of points $(s,t)$ such that $\mathscr{C}_{t}$ passes through $s$ and the total space of $\mathcal{X}|_{\mathscr{C}_{t}}$ has a singularity lying above $\varphi^{-1}_{t}(s)$ . It is easy to see that $\mathcal{Z}$ is constructible: Let $\mathcal{X}|_{\mathscr{C}}$ be the pullback of $\mathcal{X}$ along $\varphi$ . Then we have a natural morphism $\mathcal{X}|_{\mathscr{C}}\to T\times_{k}S$ and $\mathcal{Z}$ is the set-theoretic image of $\mathrm{Sing}(\mathcal{X}|_{\mathscr{C}}\to T)$ . Endow $\mathcal{Z}$ with the structure of a reduced scheme.

It suffices to show that $\dim\mathcal{Z}<\dim T$ . Let $V$ be as in (6.1.3)(c) and for $s\in V(k)$ , let $U_{s}\subseteq\varphi^{-1}(s)$ be as in (6.1.1)(b). Let $t=g(s)$ . By (6.1.2), there exists a proper closed subvariety $U_{s,\mathrm{bad}}\subseteq U$ such that the total space of $\mathcal{X}|_{\mathscr{C}_{t}}$ is not smooth near the fiber $\mathcal{X}_{u}$ only if $u\in U_{s,\mathrm{bad}}$ . Let $\widetilde{\mathcal{Z}}_{s}\subseteq\varphi^{-1}(s)$ be the union of the complement of $U_{s}$ and the Zariski closure of $U_{s,\mathrm{bad}}$ . Then $\widetilde{\mathcal{Z}}_{s}$ is a proper closed subvariety of $\varphi^{-1}(s)$ . Since the morphism $\varphi^{-1}(s)\to T$ is quasi-finite and the fiber $\mathcal{Z}_{s}\subseteq T$ is contained in the image of $\widetilde{\mathcal{Z}}_{s}$ , we have

\dim\mathcal{Z}_{s}<\dim\varphi^{-1}(s)=\dim\mathscr{C}-\dim S.

Since the image of $\mathcal{Z}$ in $S$ is contained in $D$ , and $s$ runs over an open dense subvariety of $D$ , we have that $\dim\mathcal{Z}\leq\dim\mathscr{C}-2=\dim T-1$ as desired. ∎

6.2 Applications of the Baire category theorem

Let $k$ be an algebraically closed field of characteristic $p>0$ . Set $W:=W(k)$ and $K:=W[1/p]$ . Choose an algebraic closure $\bar{K}$ of $K$ .

Lemma (6.2.1).

Suppose that $S\to B$ is a flat morphism between irreducible smooth $W$ -schemes of finite type. Let $N$ be a countable union of closed proper subschemes of $S_{\bar{K}}$ . Let $b\in B(k)$ be any point and $\widehat{B}_{b}$ be the formal completion of $B$ at $b$ . Then the subset of points $\widetilde{b}\in\widehat{B}_{b}(W)$ such that $\mathrm{supp}(S_{\widetilde{b}}{\otimes}\bar{K})$ is not contained in $N$ is analytically dense.

Proof.

Let $U:=\widehat{B}_{b}(W)$ . By taking the union of $N$ with all its Galois conjugates, we may assume that $N$ is defined over $K$ . Let $N_{1},N_{2},\cdots$ be the irreducible components of $N$ . By flatness the morphism $S\to B$ is open, so for each $i$ there exists a proper closed subscheme $Z_{i}\subseteq B$ such that every $z\in B(K)$ satisfies $\mathrm{supp\,}(S_{z})\subseteq N_{i}$ only if $z\in Z_{i}$ . Indeed, one may simply take $Z_{i}$ to be the complement of the image of $S-N_{i}$ . Since each $U-Z_{i}(W)$ is open dense in analytic topology, we may conclude by the Baire category theorem for complete metric spaces that $U-\cup_{i=1}^{\infty}Z_{i}(W)$ is analytically dense. ∎

Lemma (6.2.2).

Let $S$ and $T$ be smooth irreducible $W$ -schemes of finite type and $N$ be a countable union of closed proper subschemes of $S_{\bar{K}}$ . Let $t\in T_{k}$ be a closed point and $\widehat{T}_{t}$ be the formal completion of $T$ at $t$ .

Suppose that $\mathscr{C}$ is a smooth family of geometrically connected curves over $T$ , and there is a morphism $\varphi\colon\mathscr{C}\to S$ such that the family $(\mathscr{C}/T)_{\bar{K}}$ over $\bar{K}$ strongly dominates $S_{\bar{K}}$ . Then the subset of points $\widetilde{t}\in\widehat{T}_{t}(W)$ such that $\varphi(\mathrm{supp\,}(\mathscr{C}_{\widetilde{t}}{\otimes}\bar{K}))$ is not contained $N$ is analytically dense.

Proof.

Again by Galois descent we may assume that $N=\cup_{i=1}^{\infty}N_{i}$ for irreducible closed subschemes $N_{i}$ of $S_{K}$ . Let $M_{i}:=\mathscr{C}_{K}\times_{\varphi}N_{i}$ and $M:=\cup_{i=1}^{\infty}M_{i}$ . The assumption that $(\mathscr{C}/T)_{\bar{K}}$ strongly dominates $S_{\bar{K}}$ implies that each $M_{i}$ is a proper closed subscheme. Now we apply the above lemma with $(S\to B,N)$ replaced by $(\mathscr{C}\to T,M)$ . ∎

7 Elliptic Surfaces with $p_{g}=q=1$

7.1 Generalities on Elliptic Surfaces

In this section we recall some basic facts about elliptic surfaces and describe their moduli. Let $k$ be an algebraically closed field of characteristic $\neq 2,3$ . Let $C$ be a smooth projective curve over $k$ and $\pi\colon X\to C$ be an elliptic surface over $C$ with a zero section $\sigma\colon C\to X$ through which we also view $C$ as a curve on $X$ . The fundamental line bundle $L$ of $X/C$ is defined to be the dual of the normal bundle $N_{C/X}$ , or equivalently that of $R^{1}\pi_{*}\mathcal{O}_{X}$ . The degree of $L$ is defined to be the height of $X$ , which we denote by $\mathrm{ht}(X)$ . Set $V_{r}=\mathrm{H}^{0}(L^{r})$ . There exists a pair $(a_{4},a_{6})\in V_{4}\times V_{6}-\{0\}$ , which is unique up to the action of $\lambda\in k^{\times}$ by $\lambda\cdot(a_{4},a_{6})=(\lambda^{4}a_{4},\lambda^{6}a_{6})$ , such that $X$ is the minimal resolution of the hypersurface $X^{\prime}\subseteq\mathbb{P}(L^{2}\oplus L^{3}\oplus\mathcal{O}_{C})$ defined by the Weierstrass equation ([Kas, Thm 1])

y^{2}z=x^{3}-a_{4}xz^{2}-a_{6}z^{3},

(27)

where $x,y,z$ as homogenenous coordinates on $L^{2},L^{3},\mathcal{O}_{C}$ respectively. The hypersurface $X^{\prime}$ has at most rational double point singularities and is called the Weierstrass normal form of the original surface $X$ . If $X^{\prime}$ is smooth, then of course $X=X^{\prime}$ . In this paper, we only consider $X$ with $\mathrm{ht}(X)>0$ .

Next, we recall that Kodaira classified all the possible singular fibers in the elliptic fibration $\pi\colon X\to C$ when $k=\mathbb{C}$ in [Kodaira], and his classification is well known to hold verbatim in characteristic $\neq 2,3$ as well. We refer the reader to [SchSh, §4] for a summary. Set $\Delta\coloneqq 4a_{4}^{3}-27a_{6}^{2}$ . Let $c\in C$ be a point and denote by $\mathrm{val}_{c}$ the valuation defined by a uniformizer at $c$ . The only facts we shall need from loc. cit. are the following:

Proposition (7.1.1).

(a)

The fiber $X_{c}$ is singular if and only if $\Delta$ vanishes at $c$ , i.e., $\mathrm{val}_{c}(\Delta)\geq 1$ .
(b)

$X_{c}$ is of $\mathrm{I}_{n}$ type $(n>0)$ if and only if $\mathrm{val}_{c}(a_{4})=\mathrm{val}_{c}(a_{6})=0$ , and $n=\mathrm{val}_{c}(\Delta)$ .
(c)

$X_{c}$ is of $\mathrm{II}$ -type if and only if $\mathrm{val}_{c}(a_{4})\geq 1$ and $\mathrm{val}_{c}(a_{6})=1$ .
(d)

If $X_{c}$ is a singular fiber of any other type, $\mathrm{val}_{c}(\Delta)\geq 3$ .
(e)

If $X_{c}$ is of $\mathrm{I}_{1}$ -type or $\mathrm{II}$ -type, then $X_{c}=X^{\prime}_{c}$ . In other words, the singularity on $X^{\prime}_{c}$ is not a surface singularity.
(f)

If $X_{c}$ is of $\mathrm{I}_{2}$ -type, then $X^{\prime}_{c}$ has a unique ODP singularity given by contracting the irreducible component not meeting the zero section.

The degree of the discriminant $\Delta$ is $12\chi(\mathcal{O}_{X})=e(X)$ . Recall that the genus $g(C)$ is equal to the irregularity $q(X)$ and we have $p_{g}(X)=\chi(\mathcal{O}_{X})-1+g(C)$ . Therefore, elliptic surfaces with $p_{g}=1$ fall into two types:

•

$\chi(\mathcal{O}_{X})=2$ and $g(C)=0$ . These are elliptic K3 surfaces.
•

$\chi(\mathcal{O}_{X})=g(C)=1$ . These surfaces have Kodaira dimension $1$ .

We are interested in the latter class. Note that although these surfaces are elliptic fibrations over genus $1$ curves, one should not confuse them with bielliptic surfaces, which are of Kodaira dimension $0$ .

(7.1.2)

For future reference we introduce some notation. Let $B$ be a base scheme and $\mathcal{V}$ be a vector bundle over $B$ . We denote by $\mathbb{A}(\mathcal{V})$ the relative affine space over $B$ defined by $\mathcal{V}$ and $\mathbb{A}(\mathcal{V})^{*}$ the open part of $\mathbb{A}(\mathcal{V})$ minus the zero section. Given a sequence of numbers $q=(q_{0},\cdots,q_{m})$ such that $q_{i}$ ’s are invertible in $\mathcal{O}_{B}$ and vector bundles $\mathcal{V}_{0},\cdots,\mathcal{V}_{m}$ such that $\mathcal{V}=\oplus_{i=0}^{m}\mathcal{V}_{i}$ , we denote by $\mathcal{P}_{q}(\mathcal{V})$ the resulting weighted projective stack, i.e., the quotient stack of $\mathbb{G}_{m}$ -action on $\mathcal{V}$ given by

\lambda\colon(v_{0},\cdots,v_{m})\mapsto(\lambda^{q_{0}}v_{0},\cdots,\lambda^{q_{m}}v_{m})\text{ for }\lambda\in\mathbb{G}_{m},

and by $\mathbb{P}_{q}(\mathcal{V})$ the coarse moduli space of $\mathcal{P}_{q}(\mathcal{V})$ . It is well known that this coarse moduli space can be constructed explicitly by applying the relative Proj functor to a sheaf of graded algebras over $B$ . We omit the details. If $q$ is not specified, then it is assumed to be $(1,\cdots,1)$ .

Set-up (7.1.3).

Let $B$ be a $\mathbb{Z}[1/6]$ -scheme, $\varpi\colon\mathcal{C}\to B$ be a family of smooth projective curves over $B$ of genus $g$ and $\mathcal{L}$ be a relative line bundle on $\mathcal{C}$ of degree $h$ . Assume that $4h\geq 2g-1$ and $h\geq 1$ . Let $\mathcal{V}_{r}$ denote the vector bundle $\varpi_{*}\mathcal{L}^{r}$ for $r\geq 4$ . Let $\widetilde{\mathcal{X}}$ be the subscheme of $\mathbb{A}(\mathcal{V}_{4}\oplus\mathcal{V}_{6})^{*}\times_{B}\mathbb{P}(\mathcal{L}^{2}\oplus\mathcal{L}^{3}\oplus\mathcal{O}_{\mathcal{C}})$ defined by the Weierstrass equation (27) in the obvious way. Let $\mu$ be the $\mathbb{G}_{m}$ -action on $\mathbb{A}(\mathcal{V}_{4}\oplus\mathcal{V}_{6})^{*}\times_{B}\mathbb{P}(\mathcal{L}^{2}\oplus\mathcal{L}^{3}\oplus\mathcal{O}_{\mathcal{C}})$ defined by

\lambda\cdot((a_{4},a_{6}),[x:y:z])=((\lambda^{4}a_{4},\lambda^{6}a_{6}),[\lambda^{2}x:\lambda^{3}y:z]),\text{ for }\lambda\in\mathbb{G}_{m}.

(28)

Let $\mathcal{Q}(\mu)$ denote the quotient stack of the $\mathbb{G}_{m}$ -action $\mu$ . Then $\widetilde{\mathcal{X}}$ descends to an algebraic substack $\mathcal{X}$ of $\mathcal{Q}(\mu)$ . Note that $\mathcal{Q}(\mu)$ , and hence $\mathcal{X}$ , admit natural morphisms to $\mathcal{P}_{(4,6)}(\mathcal{V}_{4}\oplus\mathcal{V}_{6})$ . Set $\mathfrak{D}:=\mathrm{Disc}(\widetilde{\mathcal{X}}/\mathbb{A}(\mathcal{V}_{4}\oplus\mathcal{V}_{6})^{*})$ (see (6.1.3)). As a quasi-cone, it defines a closed substack $\overline{\mathfrak{D}}$ in $\mathcal{P}_{(4,6)}(\mathcal{V}_{4}\oplus\mathcal{V}_{6})$ . Let $\mathbb{U}$ denote the open complement of $\mathfrak{D}$ in $\mathbb{A}(\mathcal{V}_{4}\oplus\mathcal{V}_{6})^{*}$ .

Remark (7.1.4).

Note that with fixed $(\mathcal{C},\mathcal{L})$ , the formation of $\mathcal{V}_{4},\mathcal{V}_{6},\widetilde{\mathcal{X}},\mathcal{X}$ and $\mathbb{U}$ naturally commutes with base change among $B$ -schemes. We will implicitly use this for the rest of Sec. 7. However, a priori $\mathfrak{D}$ and $\overline{\mathfrak{D}}$ might not commute with non-flat base change as they can become non-reduced (cf. (6.1.4)). Much of Sec. 7 is devouted to giving conditions to exclude this possibility. The key intermediate result is (7.3.4), which will play an important role in the proof of Theorem A.

We remark that $\mathcal{X}$ is only “stacky” because of the base.

Lemma (7.1.5).

Let $T$ be an $B$ -scheme and $\nu\colon T\to\mathcal{P}_{(4,6)}(\mathcal{V}_{4}\oplus\mathcal{V}_{6})$ be a morphism. Then the pullback $\nu^{*}\mathcal{Q}(\mu)$ , and hence $\nu^{*}\mathcal{X}$ , are flat projective schemes over $T$ .

Proof.

The reader can check that $\mathcal{Q}(\mu)$ is in fact a $\mathbb{P}^{2}$ -bundle over $\mathcal{P}_{(4,6)}(\mathcal{V}_{4}\oplus\mathcal{V}_{6})\times_{B}\mathcal{C}$ . Therefore, the pullback $\nu^{*}\mathcal{Q}(\mu)$ is a $\mathbb{P}^{2}$ -bundle over the scheme $T\times_{B}\mathcal{C}$ . Being a closed substack of the scheme $\nu^{*}\mathcal{Q}(\mu)$ , $\nu^{*}\mathcal{X}$ has to be a projective scheme. The flatness of $\nu^{*}\mathcal{Q}(\mu)$ is clear, and one deduces the flatness of $\nu^{*}\mathcal{X}$ using that it is locally cut out by a single equation, and its geometric fibers are of codimension $1$ (cf. [stacks-project, 00MF]). ∎

Proposition (7.1.6).

The morphism $\widetilde{\mathcal{X}}\to B$ is smooth.

Proof.

As $\widetilde{\mathcal{X}}$ is flat over $B$ , it suffices to check smoothness of geometric fibers. Hence we may assume that $B=\mathrm{Spec}(k)$ , where $k$ is an algebraically closed field of characteristic $\neq 2,3$ . Let us simply write $\mathbb{A}^{*}$ for $\mathbb{A}(\mathcal{V}_{4}\oplus\mathcal{V}_{6})^{*}$ . Choose a point $u\in\mathbb{A}^{*}(k)$ and $c\in\mathcal{C}(k)$ . Choose a uniformizer $t$ of $C$ at $c$ and bases $\{\sigma_{i}\},\{\theta_{j}\}$ for $\mathcal{V}_{4},\mathcal{V}_{6}$ respectively. Then the formal complection of $\mathbb{A}^{*}\times C$ at $(u,c)$ can be identified with $\mathrm{Spf}(R)$ , where $R=k[\![t,\alpha_{0},\cdots,\alpha_{4h-g},\beta_{0},\cdots,\beta_{6h-g}]\!]$ .

By choosing a local $\mathcal{O}_{C}$ -generator of $L$ at $c$ , we turn $\sigma_{i}$ ’s and $\theta_{j}$ ’s into elements in $k[\![t]\!]$ . Let $(\{a_{i}\},\{b_{j}\})\in k^{10h-2g+2}$ be the affine coordinates of $u$ in $\mathbb{A}(\mathcal{V}_{4}\oplus\mathcal{V}_{6})$ . Then the restriction of $\widetilde{\mathcal{X}}$ to $\mathrm{Spf}(R)$ can be identified with the subscheme of $\mathrm{Proj\,}R[x,y,z]$ defined by the equation

W\coloneqq y^{2}z-x^{3}+(\sum_{i=0}^{4h-g}(a_{i}+\alpha_{i})\sigma_{i}(t))xz^{2}+(\sum_{j=0}^{6h-g}(b_{j}+\beta_{j})\theta_{j}(t))z^{3}=0.

(29)

Let $r$ be the special point of $\mathrm{Spf}(R)$ . The singularity of the (generalized) elliptic curve $\widetilde{\mathcal{X}}_{t}$ defined by the above equation when $t$ , $\alpha_{i}$ ’s and $\beta_{j}$ ’s all vanish cannot appear on the $z=0$ chart. So we may set $z=1$ in the above equation and consider the resulting scheme in $\mathrm{Spec}R[x,y]$ . Since $\theta_{j}(0)\neq 0$ for some $j$ , we deduce that the partial derivative $\partial W/\partial\beta_{j}$ remains nonzero on the special fiber. This implies that the total space of the restriction of $\widetilde{\mathcal{X}}$ to $\mathrm{Spf}(R)$ is smooth. But the choice of $(u,c)$ is arbitrary, so $\widetilde{\mathcal{X}}$ is smooth. ∎

7.2 Nonlinear Bertini Theorems for Families of Elliptic Surfaces

In this section, $k$ remains an algebraically closed field of characteristic $\neq 2,3$ .

Proposition (7.2.1).

Let $C$ be a smooth projective curve of genus $g$ over $k$ and let $L$ be a line bundle on $C$ with degree $h$ . Set $V_{r}\coloneqq\mathrm{H}^{0}(L^{r})$ for every $r\in\mathbb{N}$ . For every $d\in\mathbb{N}$ , consider the closed subset of $\mathbb{A}(V_{4}\oplus V_{6})\times C$ defined by ( $\Delta\in V_{12}$ is defined by $4a_{4}^{3}-27a_{6}^{2}$ as before)

\mathcal{K}_{d}\coloneqq\{(a_{4},a_{6},c)\in V_{4}\times V_{6}\times C\mid\mathrm{val}_{c}(\Delta)\geq d\}

and endow it with the reduced subscheme structure. Likewise, let $D\subseteq C\times C$ be the diagonal and define a closed subscheme in $\mathbb{A}(V_{4}\oplus V_{6})\times(C\times C-D)$ by

\mathcal{K}_{2}^{+}\coloneqq\{(a_{4},a_{6},c,c^{\prime})\in V_{4}\times V_{6}\times(C\times C-D)\mid\mathrm{val}_{c}(\Delta)\text{ and }\mathrm{val}_{c^{\prime}}(\Delta)\text{ are both }\geq 2\}.

If $2h\geq g+1$ , then we have the following:

(a)

$\mathcal{K}_{d}$ has codimension $d$ for $d\leq 3$ .
(b)

$\mathcal{K}_{2}$ has two irreducible components $\mathcal{K}_{2}(\mathrm{I}_{2})$ and $\mathcal{K}_{2}(\mathrm{II})$ characterized by conditions $\mathrm{val}_{c}(a_{6})=0$ and $\mathrm{val}_{c}(a_{6})\geq 1$ respectively.
(c)

$\mathcal{K}_{2}^{+}$ has codimension $4$ .

Proof.

Recall that by the Riemann-Roch theorem, for any line bundle $M$ on $C$ , if $\mathrm{deg}(M)\geq 2g-1$ , then $h^{0}(M)=\mathrm{deg}(M)-g+1$ ; if $\mathrm{deg}(M)=2g-2$ , then $h^{0}(M)=\mathrm{deg}(M)-g+1$ unless $M\,{\cong}\,\omega_{C}$ .

Fix any point $c\in C$ and consider the projection $\mathcal{K}_{d}\to C$ . It suffices to show that the fiber $\mathcal{K}_{d,c}$ over $c$ , viewed naturally as a closed subscheme of $\mathbb{A}(V_{4}\oplus V_{6})$ , has codimension $d$ . We identify the completion of $C$ along $c$ with $\mathrm{Spf}(k[\![t]\!])$ by choosing a uniformizer $t$ . After choosing a local generator of $L$ , we may consider the Taylor series of any $\sigma\in\mathrm{H}^{0}(L^{r})$ , which is a power series $\sigma(t)\in k[\![t]\!]$ . By the first paragraph, for $r\geq 4$ and $d\leq 3$ , we have

h^{0}(L^{r}((1-d)c))=h^{0}(L^{r}(-dc))+1.

Therefore, we may choose a basis $\sigma_{0},\cdots,\sigma_{4h-g}$ for $V_{4}$ such that $\mathrm{val}_{c}(\sigma_{i})=0$ for $i=0,1,2$ and $\{\sigma_{i}\}_{3\leq i\leq 4h-g}$ forms a basis for $\mathrm{H}^{0}(L^{4}(-3c))$ . We may assume that $\sigma_{0}(t)\equiv 1,\sigma_{1}(t)\equiv t$ and $\sigma_{2}(t)\equiv t^{2}$ modulo $t^{3}$ . We choose a basis $\{\theta_{0},\cdots,\theta_{6h-g}\}$ in an entirely similar way.

With the given choices of bases, we use $\{\alpha_{i}\}$ and $\{\beta_{j}\}$ for the coordinates of $V_{4}$ and $V_{6}$ respectively, so that $\Delta$ can be expressed as

\Delta=4\left(\sum_{i=0}^{4h-g}\alpha_{i}\sigma_{i}\right)^{3}-27\left(\sum_{j=0}^{6h-g}\beta_{j}\theta_{j}\right)^{2}.

(30)

Then the fiber $\mathcal{K}_{d,c}$ ( $d\leq 3$ ) is cut out in $\mathbb{A}(V_{4}\oplus V_{6})$ by the first $d$ equations from below:

\displaystyle\begin{cases}\Delta(0)&=4\alpha_{0}^{3}-27\beta_{0}^{2}\\ \Delta^{\prime}(0)&=3(4\alpha_{0}^{2}\alpha_{1}-18\beta_{0}\beta_{1})\\ \Delta^{\prime\prime}(0)&=24(\alpha_{0}\alpha_{1}^{2}+\alpha_{0}^{2}\alpha_{2})-54(\beta_{1}^{2}+2\beta_{0}\beta_{2})\end{cases}

(31)

The statement (a) is clear for $d=0,1$ . For $d=2$ , it is clear that $\mathcal{K}_{2}$ contains the following subscheme

\mathcal{K}_{2}(\mathrm{II})\coloneqq\{(a_{4},a_{6},c)\in V_{4}\times V_{6}\times C\mid\mathrm{val}_{c}(a_{4})\geq 1,\mathrm{val}_{c}(a_{6})\geq 1\},

such that the fiber of $\mathcal{K}_{2}(\mathrm{II})$ over $c$ is simply cut out by $\alpha_{0}=\beta_{0}=0$ . Let $\mathbb{A}^{2}\coloneqq\mathbb{A}(k\langle\sigma_{0},\theta_{0}\rangle)$ be the affine space with coordinates $(\alpha_{0},\beta_{0})$ and $C^{\prime}\subseteq\mathbb{A}^{2}$ be the cuspidal curve defined by $\Delta(0)=0$ . Then the fiber of $\mathcal{K}_{2}-\mathcal{K}_{2}(\mathrm{II})$ over a point in $C^{\prime}-\{(0,0)\}$ is given by a codimension $1$ hyperplane in $\mathbb{A}(k\langle\sigma_{i},\beta_{j}\rangle_{i,j\geq 1})$ . This implies that $\mathcal{K}_{2}-\mathcal{K}_{2}(\mathrm{II})$ is irreducible of codimension $2$ in $\mathbb{A}(V_{4}\oplus V_{6})$ , and we denote this component by $\mathcal{K}_{2}(\mathrm{I}_{2})$ . Note that this implies (b). To see the $d=3$ case for (a), just note that $\Delta^{\prime\prime}(0)$ does not vanish identically on both $\mathcal{K}_{2}(\mathrm{I}_{2})$ and $\mathcal{K}_{2}(\mathrm{II})$ .

Finally we treat (c). We consider the projection $\Phi\colon\mathcal{K}_{2}^{+}\to(C\times C-D)$ and take a point $(c,c^{\prime})\in(C\times C-D)$ . Denote the fiber of $\Phi$ over $(c,c^{\prime})$ by $\Phi_{(c,c^{\prime})}$ . We assume first that $L^{4}\not\cong\omega_{C}(2c+2c^{\prime})$ . This condition is automatically satisfied when $2h>g+1$ and ensures that $h^{0}(L^{r}(-2c-2c^{\prime}))=rh+g-5$ for $r\geq 4$ . Then we may choose $\sigma_{0},\cdots,\sigma_{3}\in V_{4}$ with the following vanishing orders:

	$\sigma_{0}$	$\sigma_{1}$	$\sigma_{2}$	$\sigma_{3}$
$\mathrm{val}_{c}$	$0$	$1$	$\geq 2$	$\geq 2$
$\mathrm{val}_{c^{\prime}}$	$\geq 2$	$\geq 2$	$0$	$1$

(32)

Then we complete $\{\sigma_{0},\cdots,\sigma_{3}\}$ to a basis $\{\sigma_{i}\}$ of $V_{4}$ by adjoining a basis for $\mathrm{H}^{0}(L^{4}(-2c-2c^{\prime}))$ . Let $t,s$ be uniformizers of the completions of $C$ along $c$ and $c^{\prime}$ respectively. After choosing local generators of $L$ , we may consider Taylor series $\sigma_{i}(t)\in k[\![t]\!]$ and $\sigma_{i}(s)\in k[\![s]\!]$ , and assume that $\sigma_{0}(t)\equiv 1,\sigma_{1}(t)\equiv t\mod t^{2}$ and $\sigma_{2}(s)\equiv 1,\sigma_{3}(s)\equiv s\mod s^{2}$ . Choose an entirely similar basis $\{\theta_{j}\}$ for $V_{6}$ and express $\Delta$ again as in (30). Then the defining equations for $\Phi_{(c,c^{\prime})}$ in $\mathbb{A}(V_{4}\oplus V_{6})$ are

\displaystyle\begin{cases}&4\alpha_{0}^{3}-27\beta_{0}^{2}=4\alpha_{2}^{3}-27\beta_{2}^{2}=0\\ &3(4\alpha_{0}^{2}\alpha_{1}-18\beta_{0}\beta_{1})=3(4\alpha_{2}^{2}\alpha_{3}-18\beta_{2}\beta_{3})=0\end{cases}

(33)

By the same argument for the $d=2$ case in (a), the above equations define a codimension $4$ subscheme. The point is that the variables with indices $0,1$ do not interfere with those with $2,3$ .

It remains to deal with the case when $2h=g+1$ and $L^{4}\,{\cong}\,\omega_{C}(2c+2c^{\prime})$ . Note that in this case $g\geq 1$ , so the condition $L^{4}\,{\cong}\,\omega_{C}(2c+2c^{\prime})$ defines a closed subscheme of $(C\times C-D)$ of codimension at least $1$ . Therefore, it is enough to show that the codimension of $\Phi_{(c,c^{\prime})}$ is at least $3$ . Note that we are able to choose a basis $\{\theta_{j}\}$ just as before, but this time choose $\{\sigma_{0},\sigma_{1},\sigma_{2}\}$ with the following vanishing orders:

	$\sigma_{0}$	$\sigma_{1}$	$\sigma_{2}$
$\mathrm{val}_{c}$	$0$	$1$	$2$
$\mathrm{val}_{c^{\prime}}$	$2$	$1$	$0$

(34)

and complete it to a basis of $V_{4}$ by adjoining a basis of $H^{0}(L^{4}(-2c-2c^{\prime}))$ . Assume that $\sigma_{0}(t)=1,\sigma_{1}(t)=t\mod t^{3}$ and $\sigma_{2}(s)=1\mod s$ . Then the conditions $\mathrm{val}_{c}(\Delta)\geq 2$ and $\mathrm{val}_{c^{\prime}}(\Delta)\geq 1$ give us $3$ equations which are necessarily satisfied by $\Phi_{(c,c^{\prime})}$ :

\displaystyle\begin{cases}&4\alpha_{0}^{3}-27\beta_{0}^{2}=4\alpha_{2}^{3}-27\beta_{2}^{2}=0\\ &3(4\alpha_{0}^{2}\alpha_{1}-18\beta_{0}\beta_{1})=0\end{cases}

(35)

It is clear that these indeed cut out a subscheme of codimension $3$ . ∎

Proposition (7.2.2).

Assume $2h\geq g+1$ . Apply set-up(7.1.3) to $B\coloneqq\mathrm{Spec\,}(k)$ . The resulting discriminant $\mathfrak{D}$ is a proper subvariety of $\mathbb{A}(\mathcal{V}_{4}\oplus\mathcal{V}_{6})^{*}$ . If $\mathrm{codim\,}\mathfrak{D}=1$ , then $\mathfrak{D}$ has a unique irreducible component $\mathfrak{D}_{0}$ of maximal dimension; moreover, for a general point $a$ on $\mathfrak{D}_{0}$ , $\widetilde{\mathcal{X}}_{a}$ is smooth away from a single ODP.

Proof.

As $B=\mathrm{Spec\,}(k)$ , $(\mathcal{V}_{4},\mathcal{V}_{6},\mathcal{C})$ above is the same as $(V_{4},V_{6},C)$ in (7.2.1), and we use the notations from (7.2.1) and the results in (7.1.1) throughout the proof below.

Note that for $a=(a_{4},a_{6})\in\mathbb{A}(V_{4}\oplus V_{6})^{*}$ such that $\Delta_{a}=4a_{4}^{3}-27a_{6}^{3}\in\mathrm{H}^{0}(L^{12})$ does not vanish identically on $C$ , $\widetilde{\mathcal{X}}_{a}$ is singular if and only if its elliptic fibration has a reducible singular fiber. It is clear that $\mathfrak{D}$ is contained in the image of $\mathcal{K}_{2}$ in $\mathbb{A}(V_{4}\oplus V_{6})$ , and hence has codimension at least $1$ . If $\mathrm{codim\,}\mathfrak{D}=1$ , then by (7.2.1)(c), there exists an open dense subset $U\subseteq\mathfrak{D}$ such that if $a\in U$ , $\widetilde{\mathcal{X}}_{a}$ has at most one singular fiber not of $\mathrm{I}_{1}$ -type. If moreover this singular fiber is of $\mathrm{II}$ -type, then $\widetilde{\mathcal{X}}_{a}$ is smooth and $a\not\in\mathfrak{D}$ . Therefore, the only possible irreducible component of maximal dimension in $\mathfrak{D}_{0}$ is the Zariski closure of the image of $\mathcal{K}_{2}(\mathrm{I}_{2})$ . This implies the second statement in the proposition. ∎

7.3 Mod $p$ Behavior of Discriminants

Set-up (7.3.1).

Suppose that in (7.1.3) $\mathcal{O}_{B}$ is a local ring, so that the vector bundles $\mathcal{V}_{4}$ and $\mathcal{V}_{6}$ are trivial $\mathcal{O}_{B}$ -modules. By choosing $\mathcal{O}_{B}$ -generators for $\mathcal{V}_{4}$ and $\mathcal{V}_{6}$ , we identify $\mathbb{A}(\mathcal{V}_{4}\oplus\mathcal{V}_{6})$ with $\mathbb{A}^{d_{1}}\oplus\mathbb{A}^{d_{2}}$ , where $d_{1}=4h-g+1$ and $d_{2}=6h-g+1$ . Assume that $2h\geq g+1$ as in the results in §7.2. Consider $\mathbb{P}^{1}=\mathrm{Proj\,}\mathcal{O}_{B}[u,v]$ . Let $\mathbb{A}^{1}=\mathrm{Spec\,}(\mathcal{O}_{B}[u])$ be the $v=1$ chart on $\mathbb{P}^{1}$ , and let $\infty:B\to\mathbb{P}^{1}$ denote the section defined by $v=0$ . Let $\mathcal{W}_{r}$ be the $\mathcal{O}_{B}$ -module of degree $r$ homogenous polynomials in $\mathcal{O}_{B}[u,v]$ or equivalently the module of degree $\leq r$ polynomials in $\mathcal{O}_{B}[u]$ . Consider the open subscheme $T\subseteq\mathbb{A}(\mathcal{W}_{4}^{d_{1}}\oplus\mathcal{W}_{6}^{d_{2}})$ consisting of the points of the form

\{(f_{1},\cdots,f_{d_{1}},g_{1},\cdots,g_{d_{2}})\mid\text{the common vanishing locus\,}V(\{f_{i},g_{j}\})=\emptyset\}.

Then it is clear that there is a natural morphism $\mathbb{P}^{1}_{B}\times_{B}T=\mathbb{P}^{1}_{T}\to\mathcal{P}_{(4,6)}(\mathcal{V}_{4}\oplus\mathcal{V}_{6})$ . By setting $v=1$ in the polynomials $f_{i}$ ’s and $g_{j}$ ’s, we also obtain an $B$ -morphism $\mathbb{A}^{1}_{B}\times_{B}T=\mathbb{A}^{1}_{T}\to\mathbb{A}(\mathcal{V}_{4}\oplus\mathcal{V}_{6})^{*}$ . Recall that $\mathbb{A}(\mathcal{V}_{4}\oplus\mathcal{V}_{6})^{*}$ denotes the affine space $\mathbb{A}(\mathcal{V}_{4}\oplus\mathcal{V}_{6})$ minus the zero section. This morphism fits into a commutative diagram

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(36)

For the content below, recall (6.1.1) and definition of $\widetilde{\mathcal{X}},\mathcal{X},\mathfrak{D},\overline{\mathfrak{D}}$ and $\mathbb{U}$ in (7.1.3). Assume that $k$ is an algebraically closed field of characteristic $\neq 2,3$ .

Proposition (7.3.2).

Suppose that $B=\mathrm{Spec\,}(k)$ . Then the family $\mathbb{A}^{1}_{T}/T$ strongly dominates $\mathbb{A}(\mathcal{V}_{4}\oplus\mathcal{V}_{6})^{*}$ via $\varphi$ . Moreover, for a general point $t\in T$ , $\overline{\varphi}_{t}(\infty)\not\in\overline{\mathfrak{D}}$ .

Proof.

The first statement is an exercise of dimension counting, so we omit the details. For the second statement, it suffices to exhibit a single such $t$ as the condition is open. Note that the automorphism group of $\mathbb{P}^{1}$ naturally acts on $T$ . We start with any point $t^{\prime}\in T$ be such that for some point $w\in\mathbb{A}^{1}$ , $\varphi_{t}(w)\not\in\mathfrak{D}$ . Then we can always apply an automorphism of $\mathbb{P}^{1}$ to switch $w$ and $\infty$ . This gives us a point $t\in T$ and by construction $\overline{\varphi}_{t}(\infty)\not\in\overline{\mathfrak{D}}$ . ∎

Remark (7.3.3).

We expect that $\mathfrak{D}$ to always be of codimension exactly $1$ in $\mathbb{A}(\mathcal{V}_{4}\oplus\mathcal{V}_{6})^{*}$ , but not smaller. We make a simple observation for a case when this is easily guaranteed: Suppose that $k=\mathbb{C}$ in (7.3.2). If for a general point $t$ on $T$ , the $H^{2}$ of the restriction of $\widetilde{\mathcal{X}}$ to $\varphi^{*}_{t}(\mathbb{U})$ defines a non-isotrivial VHS, then $\mathrm{codim\,}\mathfrak{D}=1$ . Indeed, otherwise for a general $t$ , the family $\overline{\varphi}_{t}^{*}(\mathcal{X})$ is a smooth family over $\mathbb{P}^{1}_{\mathbb{C}}$ , which cannot support a non-isotrivial VHS.

Lemma (7.3.4).

Suppose that $\mathrm{char\,}k=p\geq 5$ and in set-up(7.3.1) $B$ is taken to be $\mathrm{Spec\,}(W)$ for $W:=W(k)$ . Assume that $\mathfrak{D}_{K}$ has codimension $1$ in $\mathbb{A}(\mathcal{V}_{4}\oplus\mathcal{V}_{6})^{*}_{K}$ over $K:=W[1/p]$ . Then for a general $k$ -point $t\in T_{k}$ , $\overline{\varphi}_{t}$ has the following properties:

(a)

$\overline{\varphi}_{t}(\infty)\not\in\overline{\mathfrak{D}}_{k}$ , the total space of the family $\overline{\varphi}^{*}_{t}(\mathcal{X})\to\mathbb{P}^{1}_{k}$ is smooth and every fiber has at most a single ODP singularity.
(b)

$\overline{\varphi}^{*}_{t}(\overline{\mathfrak{D}}_{k})$ , or equivalently $\varphi_{t}^{*}(\mathfrak{D}_{k})$ , is reduced.
(c)

For every $\widetilde{t}\in T(W)$ lifting $t$ , $\varphi_{\widetilde{t}}^{*}(\mathfrak{D})$ is étale over $W$ , so that the open subcurve $\varphi_{\widetilde{t}}^{*}(\mathbb{U})\subseteq\mathbb{A}^{1}_{\widetilde{t}}$ has a good compactification relative to $W$ .

Proof.

Since $\overline{\mathfrak{D}}$ is proper over $W$ , it is not hard to see that $\mathfrak{D}_{k}$ is also of codimension $1$ in $\mathbb{A}(\mathcal{V}_{4}\oplus\mathcal{V}_{6})^{*}_{k}$ . We break into 3 steps.
Step 1: By (7.2.2) and (6.1.4), the irreducible components of maximal dimension of $\mathfrak{D}_{K}$ and $(\mathfrak{D}_{k})_{\mathrm{red}}$ are unique. Let us denote them by $\mathfrak{D}_{K}^{\circ}$ and $\mathfrak{D}_{k}^{\circ}$ respectively. Moreover, by (7.2.2), there exists an open dense subscheme $V\subseteq\mathfrak{D}^{\circ}_{k}$ , such that for every $v\in V(k)$ , $\mathcal{X}_{v}$ has at most a single ODP singularity. In particular, by (7.3.2), as well as (6.1.5) and its proof, for a general point $t\in T(k)$ , $\overline{\varphi}_{t}(\infty)\not\in\overline{\mathfrak{D}}_{k}$ , the total space of the pullback family $\varphi_{t}^{*}(\mathcal{X})$ is smooth, and the image of $\varphi_{t}$ only intersects $\mathfrak{D}^{\circ}_{k}$ on $V$ , so that every singular fiber of $\varphi_{t}^{*}(\mathcal{X})$ over $\mathbb{A}^{1}_{t}$ has a single ODP; moreover, as $\overline{\varphi}_{t}(\infty)\not\in\overline{\mathfrak{D}}_{k}$ , the total space of $\overline{\varphi}_{t}^{*}(\mathcal{X})$ is smooth. Hence we may conclude (a).
Step 2: Next, we show the following claim $(\star)$ : If $t$ is a general point, for any $\widetilde{t}\in T(W)$ lifting $t$ , $Z:=\varphi_{\widetilde{t}}^{*}(\mathfrak{D})$ is flat over $W$ . Let $\mathfrak{D}_{0},\cdots,\mathfrak{D}_{r}$ be the irreducible components of $\mathfrak{D}$ such that $\mathfrak{D}_{0}$ is the component which contains $\mathfrak{D}_{K}^{\circ}$ . We claim that $\mathfrak{D}_{0}$ contains $\mathfrak{D}^{\circ}_{k}$ as well. To simplify notation, let us write $\mathbb{A}(\mathcal{V}_{4}\oplus\mathcal{V}_{6})$ as $\mathscr{A}$ . Then $\mathfrak{D}_{0,K}\subseteq\mathscr{A}_{K}^{*}$ is cut out by a single polynomial $F$ in the coordinates of the affine space $\mathscr{A}_{K}$ . By minimally clearing denominators, we may assume that the coefficients of $F$ are defined in $W$ and generate $W$ . Using the fact that $\mathscr{A}$ is affine and $\mathcal{O}_{\mathscr{A}}$ is a UFD, one checks that the Zariski closure of $\mathfrak{D}_{0,K}$ in $\mathscr{A}$ contains the vanishing locus $V(F)$ of $F\in\mathcal{O}_{\mathscr{A}}$ . Note that $F$ is weighted-homogeneous, so $V(F)_{k}\subseteq\mathscr{A}_{k}$ at least contains the origin. In paricular, $V(F)\to B$ is surjective. By [stacks-project, Tag 0B2J], $\dim V(F)_{k}=\dim V(F)_{K}$ . This implies that $\dim\mathfrak{D}_{0,k}=\dim\mathfrak{D}_{0,K}$ . By the uniqueness of $\mathfrak{D}_{k}^{\circ}$ as an irreducible component of maximal dimension, we conclude that $\mathfrak{D}_{k}^{\circ}\subseteq\mathfrak{D}_{0}$ . By applying [stacks-project, Tag 0B2J] again, we also conclude that for any $i>0$ , $\mathfrak{D}_{i,k}$ has codimension $\geq 2$ in $\mathscr{A}^{*}_{k}$ .

Set $U\subseteq\mathfrak{D}_{0}$ be the complement of the closed subscheme $\cup_{i>1}(\mathfrak{D}_{0}\cap\mathfrak{D}_{i})$ . Then $(U_{k})_{\mathrm{red}}$ is dense in $\mathfrak{D}_{k}^{\circ}$ . As $t$ is general and the family $\mathbb{A}^{1}_{T_{k}}$ strongly dominates $\mathbb{A}(\mathcal{V}_{4}\oplus\mathcal{V}_{6})_{k}$ , we may assume that the intersection $\mathrm{im}(\varphi_{t})\cap(\mathfrak{D}_{k})_{\mathrm{red}}$ is transverse and lies in $U_{k}$ . Now we can prove the claim $(\star)$ . Indeed, note that $\mathfrak{D}_{0}$ is a Weil divisor, and hence also a Cartier divisor of $\mathscr{A}^{*}$ , as $\mathscr{A}^{*}$ is regular. This implies that $Z=\varphi^{*}_{\widetilde{t}}(\mathfrak{D}_{0})$ is everywhere locally cut out in $\mathbb{P}^{1}_{W}$ by a single equation. Since $Z_{k}\subseteq\mathbb{P}^{1}_{k}$ is of codimension $1$ , $Z$ is flat over $W$ by [stacks-project, Tag 00MF].
Step 3: Finally, we show (b) and (c) simultaneously. Note that if we show (c) for some $\widetilde{t}$ , then we can already conclude (b), which conversely implies (c) for all $\widetilde{t}$ . Indeed, $\mathcal{O}_{Z}$ is isomorphic to $W[x]/(f(x))$ for some $f(x)\not\in pW[x]$ . If $Z_{k}=\varphi^{*}(\mathfrak{D}_{k})$ is reduced, then by Hensel’s lemma $Z$ has to be ad disjoint union of several copies of $W$ . Hence it suffices to show (c) for some $\widetilde{t}$ , for which we may assume that the generic fiber $\varphi_{\widetilde{t}}{\otimes}K$ intersects $\mathfrak{D}_{K}$ transversely. What we are using here is that $\mathbb{A}^{1}_{T}{\otimes}\bar{K}$ strongly dominates $\mathbb{A}(\mathcal{V}_{4}\oplus\mathcal{V}_{6})^{*}_{\bar{K}}$ and the points on $T(W)$ which lift $t$ are Zariski dense on $T_{K}$ .

Since $Z$ is finite and flat, $\mathrm{deg}_{K}\,Z_{K}=\mathrm{deg}_{k}\,Z_{k}$ . As $Z_{K}$ is reduced, $\mathrm{deg}Z_{K}$ is the number of closed points on $Z_{\bar{K}}$ . Therefore, to show that $Z_{k}$ is reduced, it suffices to show that $Z_{k}$ has the same number of closed points, i.e., $|Z_{k}(k)|=|Z_{\bar{K}}(\bar{K})|$ . Now we compute $|Z_{k}(k)|$ by topology. Choose a prime $\ell\neq p$ . By combining the Leray spectral sequence and the Grothendieck-Ogg-Shafarevich formula, we have

\chi(\overline{\varphi}_{t}^{*}(\mathcal{X}))=\chi(\mathcal{X}_{\overline{\eta}_{t}})\chi(\mathbb{P}^{1}_{k})+\sum_{z\in Z_{k}(k)}\left(\chi(\mathcal{X}_{z})-\chi(\mathcal{X}_{\overline{\eta}_{t}})-\mathrm{sw}_{z}(\mathrm{H}_{\mathrm{{\acute{e}}t}}^{*}(\mathcal{X}_{\overline{\eta}_{t}},\mathbb{Q}_{\ell}))\right)

where $\overline{\eta}_{t}$ is a geometric generic point over $\mathbb{P}^{1}_{t}$ , and $\mathrm{sw}_{z}(\mathrm{H}_{\mathrm{{\acute{e}}t}}^{*}(\mathcal{X}_{\overline{\eta}_{t}},\mathbb{Q}_{\ell}))$ denotes the alternating sum of Swan conductors

\mathrm{sw}_{z}(\mathrm{H}_{\mathrm{{\acute{e}}t}}^{*}(\mathcal{X}_{\overline{\eta}_{t}},\mathbb{Q}_{\ell}))\coloneqq\sum_{i=0}^{4}(-1)^{i}\mathrm{sw}_{z}(\mathrm{H}_{\mathrm{{\acute{e}}t}}^{i}(\mathcal{X}_{\overline{\eta}_{t}},\mathbb{Q}_{\ell})).

Since for every $z\in Z_{k}(k)$ , $\mathcal{X}_{z}$ is smooth away from an ODP, [WeilI, §4.2] tells us that the local monodromy action on $\mathrm{H}_{\mathrm{{\acute{e}}t}}^{*}(\mathcal{X}_{\overline{\eta}_{t}},\mathbb{Q}_{\ell})$ factors through $\mathbb{Z}/2\mathbb{Z}$ and hence is tamely ramified as $p\neq 2$ . This implies that the Swan conductors all vanish. Moreover, by loc. cit. we also know that $\chi(\mathcal{X}_{z})-\chi(\mathcal{X}_{\overline{\eta}_{t}})=-1$ , so

|Z_{k}(k)|=\chi(\mathcal{X}_{\overline{\eta}_{t}})\chi(\mathbb{P}^{1}_{k})-\chi(\overline{\varphi}_{t}^{*}(\mathcal{X})).

(37)

By considering how singularities might degenerate, we easily see that the fiber of $\mathcal{X}$ over each point in $Z_{\bar{K}}(\bar{K})$ has at most an ODP singularity. Therefore, by the same computation as above, for $u:=\widetilde{t}_{\bar{K}}$ and a geometric generic point $\overline{\eta}_{u}$ of $\mathbb{P}^{1}_{u}$ , we have

|Z_{\bar{K}}(\bar{K})|=\chi(\mathcal{X}_{\overline{\eta}_{u}})\chi(\mathbb{P}^{1}_{\bar{K}})-\chi(\overline{\varphi}_{u}^{*}(\mathcal{X})).

(38)

Using the smooth and proper base change theorem for étale cohomology, it is not hard to see that $\chi(\overline{\varphi}_{u}^{*}(\mathcal{X}))=\chi(\overline{\varphi}_{t}^{*}(\mathcal{X}))$ and $\chi(\mathcal{X}_{\overline{\eta}_{u}})=\chi(\mathcal{X}_{\overline{\eta}_{t}})$ . Hence we conclude that $|Z_{k}(k)|=|Z_{\bar{K}}(\bar{K})|$ as desired. ∎

Note that the discussion of Swan conductors fundamentally uses the $p\neq 2$ assumption. Of course it is irrelevant here because we are working with the $p\geq 5$ , but we remark that in [Saitodisc, Thm 4.2] the non-reducedness of the relevant discriminant scheme modulo $2$ can indeed be explained by the fact that ordinary quadratic singularities behave differently in characteristic $2$ .¹²¹²12This was explained in the appendix in the earlier arXiv version of the paper, which will be published separately. We also remark that for results in §7.2 crucially uses the $2h\geq g+1$ assumption, which is indeed satisfied by the case we care about ( $g=h=1$ ).

7.4 Proof of Theorem A

In this section we work with the following set-up:

Set-up (7.4.1).

Let $\mathscr{M}_{1,1}$ be the moduli stack of the pair of a genus $1$ curve together with a degree $1$ line bundle (i.e., an elliptic curve). Let $B$ be the $\mathbb{Z}[1/6]$ -scheme defined by

\{(a,b)\in\mathrm{Spec}(\mathbb{Z}[\frac{1}{6}][a,b])\mid 4a^{3}-27b^{2}\neq 0\}.

Then the Weierstrass equation equips $B$ with a surjective morphism $B\to\mathscr{M}_{1,1}$ . Let $(\varpi:\mathcal{C}\to B,\mathcal{L})$ be the restriction of the universal family over $\mathscr{M}_{1,1}$ to $B$ . Apply the constructions in (7.1.3) with this triple $(B,\mathcal{C},\mathcal{L})$ and define the objects $\mathcal{V}_{r}=\varpi_{*}\mathcal{L}^{r}$ ( $r=4,6$ ), $\widetilde{\mathcal{X}}\to\mathbb{A}(\mathcal{V}_{4}\oplus\mathcal{V}_{6})^{*}$ and $\mathfrak{D}=\mathrm{Disc}(\widetilde{\mathcal{X}}/\mathbb{A}(\mathcal{V}_{4}\oplus\mathcal{V}_{6})^{*})$ accordingly. Below we write $\mathbb{A}(\mathcal{V}_{4}\oplus\mathcal{V}_{6})^{*}$ as $\widetilde{\mathsf{M}}$ , the open subscheme $\widetilde{\mathsf{M}}-\mathfrak{D}$ as $\mathsf{M}$ , and the restriction of $\widetilde{\mathcal{X}}$ to $\mathsf{M}$ as $\mathcal{X}^{\circ}$ .

Remark (7.4.2).

For any algebraically closed field $k$ of characteristic $p\neq 2,3$ and elliptic surface $X$ over $k$ with $p_{g}=q=1$ , there exists a point $z\in\widetilde{\mathsf{M}}(k)$ such that $X$ is the minimal model of $\widetilde{\mathcal{X}}_{z}$ . Moreover, there are no reducible fibers in the elliptic fibration of $X$ if and only if $z\in\mathsf{M}(k)$ . The choice of $z$ is unique up to the $\mathbb{G}_{m}$ -action on $\mathsf{M}$ given by $\lambda\cdot(a_{4},a_{6})=(\lambda^{4}a_{4},\lambda^{6}a_{6})$ where $(a_{4},a_{6})$ is the relative coordinate on $\mathbb{A}(\mathcal{V}_{4}\oplus\mathcal{V}_{6})$ .

We need a lower bound on the rank of the Kodaira-Spencer map, or equivalently the image of the period morphism over $\mathbb{C}$ .¹³¹³13This bound is now obsolete due to a more recent preprint [EGW], but we keep it here to illustrate that the proof of (7.4.4) below works under much weaker inputs, in case the reader wishes to study other varieties (cf. (1.0.4)). Note also that the period map over $\mathbb{C}$ in [EGW] is not flat, despite being generically étale (Cor. 5.1 ibid), and its mod $p$ properties are unclear, so computations in §7.2, 7.3 remain necessary for the current method.

Lemma (7.4.3).

For a general $z\in\mathsf{M}_{\mathbb{C}}$ and $X\coloneqq\mathcal{X}_{z}$ , the Kodaira-Spencer map

T_{z}\mathsf{M}_{\mathbb{C}}\to\mathrm{Hom}(\mathrm{H}^{1}(\Omega^{1}_{X}),\mathrm{H}^{2}(\mathcal{O}_{X}))

has rank at least $3$ .

Proof.

This follows a construction of Ikeda and the Artin-Brieskorn resolution. In [Ikeda], Ikeda constructed a subfamily of elliptic surfaces over $\mathbb{C}$ with the given invariants $p_{g}=q=1$ using bielliptic curves of genus $3$ . Let $\widetilde{C}$ be a bielliptic curve of genus $3$ , equipped with an involution $\sigma$ such that $\widetilde{C}/\sigma$ is a smooth genus $1$ curve $C$ . On the symmetric square $\widetilde{C}^{(2)}$ of $\widetilde{C}$ , $\sigma$ also lifts to an involution $\sigma^{(2)}$ . Consider the surface $Y^{\prime}={\widetilde{C}}^{(2)}/\sigma^{(2)}$ , which is shown to be a projective surface of Kodaira dimension $1$ with $6$ ODPs. Its minimal resolution $Y$ is an elliptic surface with $p_{g}=q=1$ . By Prop. 2.9 in loc. cit., the morphism $\widetilde{C}\to C$ can be recovered from $Y$ . Note however the Weierstrass model of $Y$ is singular, so $Y$ is not given by a point on $\mathsf{M}$ .

Note that for every $\mathbb{C}$ -point $s\in\widetilde{\mathsf{M}}_{\mathbb{C}}$ with image $t$ in $B_{\mathbb{C}}$ , $s$ is given by a pair $(a_{4},a_{6})\in\mathrm{H}^{0}(\mathcal{L}_{t}^{4})\times\mathrm{H}^{0}(\mathcal{L}_{t}^{6})$ . Let $\mathsf{M}^{\prime}_{\mathbb{C}}\subseteq\widetilde{\mathsf{M}}_{\mathbb{C}}$ be the open subscheme which consists of those $s$ such that $\Delta_{s}=4a_{4}^{3}-27a_{6}^{2}$ does not vanish identically on the base curve $\mathcal{C}_{t}$ . Then $\widetilde{\mathcal{X}}_{s}$ is the Weierstrass normal form of an elliptic surface and by [Kas, Thm 1] has at most rational double point singularities. By applying the Artin-Brieskorn resolution [Artin-Res] to $\widetilde{\mathcal{X}}|_{\mathsf{M}^{\prime}_{\mathbb{C}}}$ , we obtain a smooth and proper algebraic space $\mathcal{X}_{\mathbb{C}}^{\sharp}\to\mathsf{M}_{\mathbb{C}}^{\sharp}$ , where $\mathsf{M}_{\mathbb{C}}^{\sharp}$ is an algebraic space which admits a morphism to $\mathsf{M}^{\prime}_{\mathbb{C}}$ bijective on geometric points. Moreover, [Artin-Res, Thm 2] tells us that for any $\mathbb{C}$ -point $z$ of $\mathsf{M}^{\sharp}_{\mathbb{C}}$ which maps to a point $s$ of $\mathsf{M}^{\prime}_{\mathbb{C}}$ , the Henselianization of $\mathsf{M}^{\sharp}_{\mathbb{C}}$ at $z$ maps surjectively to that of $\mathsf{M}^{\prime}_{\mathbb{C}}$ at $s$ . Let $\mathsf{M}^{+}_{\mathbb{C}}$ be a resolution of singularities of $\mathsf{M}^{\sharp}_{\mathbb{C}}$ and pullback the family $\mathcal{X}^{\sharp}|_{\mathsf{M}^{\prime}_{\mathbb{C}}}$ to $\mathsf{M}^{+}_{\mathbb{C}}$ .

Note that all elliptic surfaces which can be constructed as in the first paragraph can be found as fibers of this family over $\mathsf{M}^{+}_{\mathbb{C}}$ . Let $\Omega$ be the period domain parametrizing Hodge structures of K3-type on the integral lattice $\Lambda$ given by the Betti cohomology of any complex elliptic surface with $p_{g}=q=1$ . Let $\widetilde{\mathsf{M}}^{+}_{\mathbb{C}}$ be the universal cover of $\mathsf{M}^{+}_{\mathbb{C}}$ . Then up to an action of $\mathrm{O}(\Lambda)$ there is a well defined period map $\widetilde{\mathsf{M}}^{+}_{\mathbb{C}}\to\Omega$ . The moduli space of bielliptic curves of genus $3$ over $\mathbb{C}$ is $4$ -dimensional, so [Ikeda, Thm 1.1(1)] implies that the period image of $\widetilde{\mathsf{M}}^{+}_{\mathbb{C}}$ is of dimension $\geq 3$ . Since $\mathsf{M}_{\mathbb{C}}\subseteq\mathsf{M}^{\prime}_{\mathbb{C}}$ is open and dense, the discussion on the Henselization of $\mathsf{M}^{\sharp}_{\mathbb{C}}$ at $\mathbb{C}$ -points in the preceeding paragraph implies that the preimage of $\mathsf{M}_{\mathbb{C}}\subseteq\mathsf{M}^{\prime}_{\mathbb{C}}$ in $\mathsf{M}^{+}_{\mathbb{C}}$ is also open and dense. This implies that the preimage of $\mathsf{M}_{\mathbb{C}}$ in $\widetilde{\mathsf{M}}^{+}_{\mathbb{C}}$ also has period image of $\dim\geq 3$ by a continuity argument. ∎

We are now ready to prove (a more general form of) Theorem A:

Theorem (7.4.4).

Assume that $k$ is a field finitely generated over $\mathbb{F}_{p}$ for $p\geq 5$ and let $\bar{k}$ be a separable closure. Let $X$ be an elliptic surface over $k$ with $p_{g}=q=1$ . If all fibers in the elliptic fibration of $X_{\bar{k}}$ are irreducible, then the Tate conjecture holds for $X$ .

Proof.

Let $\eta$ be the generic point of $\mathsf{M}$ . Let $\mathfrak{f}$ and $\mathfrak{s}$ be the classes in $\mathrm{NS}(\mathcal{X}^{\circ}_{\eta})$ such that $\mathfrak{f}$ (resp. $\mathfrak{s}$ ) is given by a smooth fiber (resp. zero section) in the elliptic fibration of $\mathcal{X}^{\circ}_{\eta}$ . We may polarize the family $\mathcal{X}^{\circ}/\mathsf{M}$ with the subspace $\mathbb{Q}\langle\mathfrak{f},\mathfrak{s}\rangle$ , as it contains the class of $\boldsymbol{\xi}_{\eta}$ for some relatively ample line bundle $\boldsymbol{\xi}$ . Since $\langle\mathfrak{f},\mathfrak{f}\rangle=0,\langle\mathfrak{f},\mathfrak{s}\rangle=1$ and $\langle\mathfrak{s},\mathfrak{s}\rangle=-1$ , one deduces that for any $s\in\mathsf{M}(\mathbb{C})$ , the lattice $\mathrm{PH}^{2}(\mathcal{X}^{\circ}_{s},\mathbb{Z}_{(p)})_{\mathrm{tf}}$ is self-dual for every $p\geq 5$ . Note that $\mathsf{M}_{\mathbb{C}}$ is clearly connected, and by (7.4.3) and (2.2.5), the family $(\mathcal{X}^{\circ}/\mathsf{M},\boldsymbol{\xi})|_{\mathsf{M}_{\mathbb{C}}}$ has maximal monodromy, as defined in (2.2.3). By (7.4.2), it suffices to apply (5.3.3)(a) to the family $f:\mathcal{X}^{\circ}\to\mathsf{M}$ , for which we only need to prove that $\mathbb{L}:=R^{2}f_{*}\mathbb{Q}_{2}$ has constant $\lambda^{\mathrm{geo}}$ over $\mathrm{Spec\,}(\mathbb{Z}[1/6])$ . We do so in two steps:

Step 1: Fix a prime $p\geq 5$ , set $W:=W(\bar{\mathbb{F}}_{p})$ , $K:=W[1/p]$ and choose an isomorphism $\bar{K}\,{\cong}\,\mathbb{C}$ . It suffices to show that there exists a smooth connected $W$ -curve $C$ with a morphism to $\mathsf{M}$ such that (recall (5.1.1) and (5.1.2))

(i)

$\lambda(\mathbb{L}|_{C_{\mathbb{C}}})=\lambda(\mathbb{L}|_{\mathsf{M}_{\mathbb{C}}})$ , and
(ii)

$C$ has a good compactification over $W$ .

Assume the existence of such curves for the moment, we first show how the theorem follows. Note that $\mathsf{M}_{\bar{\mathbb{F}}_{p}}$ and $\mathsf{M}_{\mathbb{C}}$ are both irreducible. Choose a base point $s\in\mathsf{M}(\mathbb{C})$ which lies over the generic point $\eta$ of $\mathsf{M}$ . Let $s_{p}$ be a geometric point over the generic point $\eta_{p}$ of $\mathsf{M}_{\bar{\mathbb{F}}_{p}}$ . Since $s$ specializes to $s_{p}$ , $\dim\mathrm{NS}(\mathcal{X}^{\circ}_{s_{p}})_{\mathbb{Q}}\geq\dim\mathrm{NS}(\mathcal{X}^{\circ}_{s})_{\mathbb{Q}}$ ; moreover, as every element in $\mathrm{NS}(\mathcal{X}^{\circ}_{s_{p}})_{\mathbb{Q}}$ is stabilized by an open subgroup of $\pi_{1}^{\mathrm{{\acute{e}}t}}(\mathsf{M}_{\bar{\mathbb{F}}_{p}},s_{p})$ , $\lambda(\mathbb{L}|_{\mathsf{M}_{\bar{\mathbb{F}}_{p}}})\geq\dim\mathrm{NS}(\mathcal{X}_{s_{p}})_{\mathbb{Q}}$ . On the other hand, as $\pi_{1}$ is a functor, we have $\lambda(\mathbb{L}|_{\mathsf{M}_{\bar{\mathbb{F}}_{p}}})\leq\lambda(\mathbb{L}|_{C_{\bar{\mathbb{F}}_{p}}})$ by default. But the curve $C$ has constant $\lambda^{\mathrm{geo}}$ because it has a good relative compactification (5.1.3), so we have

\dim\mathrm{NS}(\mathcal{X}^{\circ}_{s})_{\mathbb{Q}}\stackrel{{\scriptstyle\ref{prop: thm of fixed part}}}{{=}}\lambda(\mathbb{L}|_{\mathsf{M}_{\mathbb{C}}})=\lambda(\mathbb{L}|_{C_{\mathbb{C}}})=\lambda(\mathbb{L}|_{C_{\bar{\mathbb{F}}_{p}}})\geq\lambda(\mathbb{L}|_{\mathsf{M}_{\bar{\mathbb{F}}_{p}}}).

This implies that $\lambda(\mathbb{L}|_{\mathsf{M}_{\mathbb{C}}})=\lambda(\mathbb{L}|_{\mathsf{M}_{\bar{\mathbb{F}}_{p}}})$ , i.e., $\mathsf{M}_{W}$ has constant $\lambda^{\mathrm{geo}}$ .

Step 2: It remains to construct the curve $C$ which satisfies (i) and (ii). Let $\mathsf{V}$ be the VHS on $\mathsf{M}_{\mathbb{C}}$ given by $R^{2}f_{\mathbb{C}*}\mathbb{Q}(1)$ . We say that an irreducible smooth $\mathbb{C}$ -variety $Z$ admitting an understood morphism to $\mathsf{M}_{\mathbb{C}}$ admissible if its image is not contained in the Noether-Lefschetz loci of $\mathsf{V}$ and the restriction $\mathsf{V}|_{Z}$ is non-isotrivial. This implies, by (5.1.4), that $\lambda(\mathbb{L}|_{Z})=\lambda(\mathbb{L}|_{\mathsf{M}_{\mathbb{C}}})$ .

We construct $C$ in two steps. First, let $b\in B(k)$ be any point and $\widehat{B}_{b}$ be the formal completion of $B_{W}$ at $b$ . We claim that for some $\widetilde{b}\in\widehat{B}_{b}(W)$ , $\mathsf{M}_{\widetilde{b}}{\otimes}\mathbb{C}$ is admissible. Indeed, we first observe that since $\dim B_{\mathbb{C}}=2$ , (7.4.3) implies that for a general point $z\in B_{\mathbb{C}}$ , $\mathsf{V}|_{\mathsf{M}_{z}}$ is non-isotrivial. Note that the $K$ -points given by an analytically dense subset of $\widehat{B}_{b}(W)$ are Zariski dense on $B_{K}$ (and hence also on $B_{\bar{K}}$ ). As the Noether-Lefschetz (NL) loci of $\mathsf{V}$ is a countable union of proper closed subvarieties, we may now apply (6.2.1) to the case $S$ being $\mathsf{M}$ and $N$ being the NL loci to conclude that the desired lifting $\widetilde{b}$ exists.

Next, we take the base $B$ in the context of (7.3.1) to be $\widetilde{b}$ above (and hence $\mathbb{A}(\mathcal{V}_{4}\oplus\mathcal{V}_{6})^{*}$ and $\mathbb{U}$ in (7.3.1) are $\widetilde{\mathsf{M}}_{\widetilde{b}}$ and $\mathsf{M}_{\widetilde{b}}$ respectively in the current context). Then we obtain an irreducible smooth $W$ -scheme $T$ and a morphism $\varphi:\mathbb{A}^{1}_{T}\to\widetilde{\mathsf{M}}_{\widetilde{b}}$ such that $\varphi_{\mathbb{C}}$ defines a strongly dominating families of curves on $\widetilde{\mathsf{M}}_{\widetilde{b}}{\otimes}\mathbb{C}$ by (7.3.2). Let $t$ be a general $k$ -point on $T_{k}$ such that the conclusion of (7.3.4) holds. Since $\mathsf{V}|_{\mathsf{M}_{\widetilde{b}}{\otimes}\mathbb{C}}$ is non-isotrivial, for a general $z\in T_{\mathbb{C}}$ , the restriction of $\mathsf{V}$ to $\varphi_{z}^{*}(\mathsf{M}_{\widetilde{b}}{\otimes}\mathbb{C})$ is also non-isotrivial. Similarly, applying (6.2.2) to the NL loci on $\mathsf{M}_{\widetilde{b}}{\otimes}\mathbb{C}$ again, we conclude that for some $\widetilde{t}\in T(W)$ lifting $t$ , $\varphi_{\widetilde{t}{\otimes}\mathbb{C}}^{*}(\mathsf{M}_{\widetilde{b}}{\otimes}\mathbb{C})$ is admissible. Therefore, if we set $C$ to be $\varphi^{*}_{\widetilde{t}}(\mathsf{M}_{\widetilde{b}})$ , then it indeed satisfies condition (i), and (7.3.4)(c) guarantees that this $C$ also satisfies (ii), as desired. ∎

Remark (7.4.5).

Let $Z_{\mathbb{C}}$ be an irreducible component of the Noether-Lefschetz (NL) loci on $\mathsf{M}_{\mathbb{C}}$ and take $Z$ to be the Zariski closure of $Z_{\mathbb{C}}$ in $\mathsf{M}$ . Then by specialization of line bundles we know that every geometric fiber of $\mathcal{X}^{\circ}$ over $Z$ has Picard rank $\geq 3$ . By the Shioda-Tate formula¹⁴¹⁴14See Landesman’s note people.math.harvard.edu/~landesman/assets/shioda-tate.pdf for a proof of the formula over finite fields., this implies that, if $k$ is a finite field, $s\in Z(k)$ and $\mathcal{E}$ is the generic fiber of the elliptic fibration on $\mathcal{X}^{\circ}_{s}$ , then up to replacing $k$ by a finite extension $\mathrm{rank\,}\mathrm{MW}(\mathcal{E})>0$ . Moreover, as $\mathcal{X}^{\circ}_{s}$ has even second Betti number, the Weil conjecture implies that $\mathrm{rank\,}\mathrm{NS}(\mathcal{X}^{\circ}_{s}{\otimes}\bar{k})$ is also even, so up to extending $k$ again we must in fact have $\mathrm{rank\,}\mathrm{MW}(\mathcal{E})\geq 2$ .

Note that $Z$ is of codimension $1$ in $\mathsf{M}$ , so that $Z$ has relative dimension $10$ over $\mathbb{Z}[1/6]$ . The $\mathbb{G}_{m}$ -action on $\mathsf{M}$ stabilizes $Z$ and the quotient $Z/\mathbb{G}_{m}$ gives rise to a $9$ -dimensional subfamily in the coarse moduli of elliptic surfaces with $p_{g}=q=1$ over $\mathbb{Z}[1/6]$ (cf. (7.4.2)). By [Moonen, Prop. 6.4], the NL loci has infinitely many components, so there are lots of these examples.

8 Surfaces with $p_{g}=K^{2}=1$ and $q=0$

Recall our notations for weighted projective spaces in (7.1.2) and set $Q:=(1,2,2,3,3)$ . Throughout this section, $k$ denotes an arbitrary algebraically closed field of characteristic $\neq 2,3$ unless otherwise specified.

Theorem (8.0.1).

Let $X$ be a minimal surface over $k$ with $p_{g}=K_{X}^{2}=1$ and $q=0$ . Then the canonical model $X^{\prime}$ of $X$ only has rational double point singularities. Moreover, if we let $(x_{0},x_{1},x_{2},x_{3},x_{4})$ be the coordinates of $\mathbb{P}_{Q}(k^{\oplus 5})$ , then for $j=1,2$

Finiteness of for some Elliptic Curves of Analytic Rank >1>1

Abstract

1 Introduction

Theorem A.

Theorem B.

Theorem C.

Sketch of Proofs

(1.0.1)

(1.0.2)

(1.0.3)

Remark (1.0.4).

Organization of Paper

(1.0.5)

2 Preliminaries

2.1 Motives and Norm Functors

(2.1.1)

Theorem (2.1.2).

(2.1.3)

(2.1.4)

Notation (2.1.5).

Lemma (2.1.6).

Proof.

2.2 Motives of Varieties with h2,0=1h^{2,0}=1

Definition (2.2.1).

Theorem (2.2.2) ([Zarhin, §2]).

Definition (2.2.3).

(2.2.4)

Proposition (2.2.5).

Proof.

(2.2.6)

Theorem (2.2.7).

Proof.

Proposition (2.2.8).

Proof.

Lemma (2.2.9).

Proof.

3 Moduli Interpretations of Shimura Varieties

3.1 Systems of realizations

Definition (3.1.1).

Definition (3.1.2).

Example (3.1.3).

Notation (3.1.4).

Definition (3.1.5).

Lemma (3.1.6).

Lemma (3.1.7).

Proof.

3.2 Shimura varieties

(3.2.1)

Theorem (3.2.2).

Assumption (3.2.3).

(3.2.4)

Theorem (3.2.5).

3.3 Orthogonal Shimura Varieties over Totally Real Fields

(3.3.1)

Lemma (3.3.2).

Proof.

(3.3.3)

Remark (3.3.4).

(3.3.5)

Lemma (3.3.6).

Proof.

(3.3.7)

3.4 Integral Model

Definition (3.4.1).

(3.4.2)

(3.4.3)

Theorem (3.4.4).

Proof.

4 Period Morphisms

4.1 The Basic Set-up

Definition (4.1.1).

Lemma (4.1.2).

Proof.

Set-up (4.1.3).

Lemma (4.1.4).

Proof.

(4.1.5)

(4.1.6)

(4.1.7)

(4.1.8)

Finiteness of for some Elliptic Curves of Analytic Rank $>1$

2.2 Motives of Varieties with $h^{2,0}=1$

7 Elliptic Surfaces with $p_{g}=q=1$