The Shafarevich conjecture for hypersurfaces in abelian varieties

The argument of [47] derives bounds on $\mathcal{X}(\mathcal{O}_{K,S})$ from a family $f\colon Y\rightarrow X$, through a study of various cohomology objects $R^{i}f_{*}(-)$ on $X$. The étale local system $R^{i}f_{*}(\mathbb{Q}_{p})$ gives rise to the global Galois representations to which Faltings’s lemma is applied; a filtered $F$-isocrystal coming from crystalline cohomology is used to study the $p$-adic variation of these Galois representations; and a variation of Hodge structure allows one to relate a $p$-adic period map to topological monodromy. The method allows one to conclude that $\mathcal{X}(\mathcal{O}_{K,S})$ is not Zariski dense in $X$.

In the present setting, we will apply these results to $R^{i}f_{*}(\mathsf{L})$, where $\mathsf{L}$ is a nontrivial local system on $Y$. We now outline the main construction; details are in the proof of Theorem 9.2. For technical reasons, ¹¹1Most importantly, Lemma 8.1 holds over $\mathbb{Q}$ but not over an arbitrary number field. Additionally, $p$-adic Hodge theory plays well with tensor categories over $\mathbb{Q}_{p}$. Since passing to extensions (even unramified extensions) gives rise to semilinear operators, we need to set up the Tannakian formalism over $\mathbb{Q}_{p}$. we prefer to work with $X$ a variety over $\mathbb{Q}$, so we take $X$ to be a Zariski-closed subset of the Weil restriction from $K$ to $\mathbb{Q}$ of the moduli space of smooth hypersurfaces in $A$. In fact, we simply take $X$ to be an irreducible component the Zariski closure of the set of integral points. We take $Y$ the universal family of smooth hypersurfaces over $X$; that is, points $x\in X(\mathbb{Q})$ are in bijection with smooth hypersurfaces $Y_{x}\subseteq A$ over $K$. To conclude Theorem 9.2, we need to show that $Y$ is a translate of a constant family over $X$.

Let $n=\dim A$. We will study cohomology objects $R^{i}f_{*}(\mathsf{L})$, where $\mathsf{L}$ is a local system on $Y$, given as a direct sum of characters. Any finite-order character $\chi$ on $\pi_{1}(A)$ defines a local system on $Y$, by pullback via $Y\hookrightarrow A$. If $\chi$ is defined over some field $L$, the same is true of the corresponding local system $\mathsf{L}_{\chi}$. We construct a local system $\mathsf{L}$ defined over $\mathbb{Q}$ by descent, as a sum of Galois conjugates of $\mathsf{L}_{\chi}$; and we take $\mathsf{V}=R^{n-1}f_{*}(\mathsf{L})$. The construction is given in Lemma 5.29 in §5.5.

We think of $\mathsf{V}$ and $\mathsf{L}$ as “motives” with various realizations satisfying some compatibilities, as in Deligne [16]. The precise realizations and compatibilities we need are formalized in the notion of a Hodge–Deligne system (Definition 5.2).

The étale realization of $\mathsf{V}=R^{n-1}f_{*}(\mathsf{L})$ gives, for every $x\in\mathcal{X}(\mathbb{Z}[1/S])$, a Galois representation

By Faltings’s lemma (Lemma 5.49), there are only finitely many possibilities for the semisimplification $\rho_{x}^{ss}$ of $\rho_{x}$, as $x$ varies over $\mathcal{X}(\mathbb{Z}[1/S])$.

As in [47], we want to show that the fibers of the map

are not Zariski dense. To do this, we consider the map that takes a $p$-adic point $x\in X(\mathbb{Q}_{p})$ to a local Galois representation. By $p$-adic Hodge theory, the local Galois representation

determines the filtered $\phi$-module

We recall from [47, §3] some facts about the variation of $(V_{x},\phi_{x},F_{x})$ with $x$. For $x$ in a fixed mod-$p$ residue disk $\Omega$, the pair $(V_{x},\phi_{x})$ is constant: these spaces are canonically identified with the crystalline cohomology of the mod-$p$ reduction of $Y_{x}$. The filtration $F_{x}$ varies with $x$. The assignment $x\mapsto F_{x}$ defines a $p$-adic period map

to a certain flag variety. The $p$-adic period map is analogous to the classical complex-analytic period map of Hodge theory, and indeed the two maps are closely related, a fact we will exploit in Section 6 (see for example the proof of Lemma 6.1).

The global semisimplification $\rho_{x}\mapsto\rho_{x}^{ss}$ causes substantial technical difficulties. Before our main argument in Section 8, we give (Section 7) an alternative, simpler proof under the additional assumption that every relevant representation $\rho_{x}$ is semisimple. For this sketch, to illustrate ideas, let us make the same assumption; that is, let us imagine that every global representation $\rho_{x}$ is semisimple. Then there are literally only finitely many possibilities for the isomorphism class $\rho_{x}$, so (restricting to the local Galois representation and applying the crystalline Dieudonné functor) there are only finitely many possibilities (up to isomorphism) for the filtered $\phi$-module $(V_{x},\phi_{x},F_{x})$, as $x$ ranges over all integral points. In this simplified setting, we need only show that

is contained in a positive-codimension algebraic subset of $X$.

Isomorphism classes of triples $(V_{x},\phi_{x},F_{x})$ correspond to orbits of the Frobenius centralizer $Z(\phi)$ on $\mathcal{H}$, so we want to control $\Phi_{p}^{-1}(Z)$, where $Z$ is an orbit of the Frobenius centralizer. We’ll have the result we want if we can prove precise versions of the following two conditions.

1. (a)

   The Frobenius centralizer is small.

2. (b)

   The image of $\Phi_{p}$ is not contained in a small algebraic set.

In fact, since we don’t know that the global Galois representations are semisimple, we need a stronger form of a.

1. (a’)

   (See Lemma 8.16.) Fix a $\phi$-module $(V,\phi)$ and a semisimple global Galois representation $\rho^{ss}$. Consider all global Galois representations $\rho$ whose semisimplification is $\rho^{ss}$, and such that $D_{cris}(\rho|_{\mathbb{Q}_{p}})\cong(V,\phi,F)$, for some filtration $F$ on $V$.

   The $F$ that arise in this way all lie in a subvariety $Z\subseteq\mathcal{H}$ of low dimension.

Once we have items a and b, we know that $X(\mathbb{Z}[1/S])$ is contained in $\Phi_{p}^{-1}(Z)$. A $p$-adic version of the Bakker–Tsimerman transcendence theorem (Theorem 6.4) will imply that $X(\mathbb{Z}[1/S])$ is not Zariski dense.

Condition a comes from two ingredients. First, the semilinearity of Frobenius gives an upper bound on its centralizer (Lemma 5.33). Second, the possible subrepresentations of a global Galois representation are constrained by purity (Lemma 8.1), which restricts the structure of local Galois representations coming from global $\rho$ having a given semisimplification. It is this latter result that requires us to work over $\mathbb{Q}$ (or at least a number field that has no CM subfield). As mentioned above, we can always pass to this situation by restriction of scalars.

We introduce “$H^{0}$-algebras” (§5.3) to package the cohomological data that arise in this situation. The Hodge–Deligne system $\mathsf{V}$ will have the structure of module over a certain algebra object $\mathsf{E}$ in the category of Hodge–Deligne systems; this module structure allows us to keep track of the Galois actions on embeddings of the field $K$, isomorphism classes of local systems $\mathsf{L}$, and on the coefficient field of the local system, in a uniform and convenient way.

Condition b is a question about the monodromy of the variation of Hodge structure given by $\mathsf{V}$. As mentioned above, we only need a very weak lower bound on the Zariski closure of the monodromy group. We call the relevant condition “strongly $c$-balanced” (Definition 6.6; see Corollary 4.10 and Lemma 6.8 for precise statements). It depends on a parameter $c$ which must be taken sufficiently large. The technical difficulty in Corollary 4.10 is that it applies uniformly to any family of hypersurfaces in an abelian variety, as is required to prove finiteness.

It is now crucial that, in our case, $Y$ is a subvariety of $A\times X$, with the map $f$ the restriction of the projection map $A\times X$ to $X$. The abelian variety $A$ has many rank-one local systems $\mathsf{L}$, each of which we can pull back to $Y$, push forward to $X$, and apply this machinery to. These local systems are associated to characters of the fundamental group $\pi_{1}(A)$.

To apply the $p$-adic Hodge theory argument described above, it suffices to have a local system $\mathsf{L}$ on $Y$ such that $R^{n-1}f_{*}(\mathsf{L})$ has big monodromy in our sense. (There are some additional technical conditions that we suppress here to focus on the main difficulty.) In fact we will show big monodromy for almost all rank one local systems $\mathsf{L}$, in a precise sense (Theorem 3.5 and Corollary 4.10). To do this, it is necessary to have a framework in which the vector spaces $R^{n-1}f_{*}(\mathsf{L})_{x}$ for different local systems $\mathsf{L}$ can be studied all at once. This is accomplished by the Tannakian theory of sheaf convolution [44].

The fundamental objects of the Tannakian theory of sheaf convolution are perverse sheaves. The fundamental perverse sheaf for us is the constant sheaf on $Y_{x}$, pushed forward to $A$, and placed in degree $1-n$. The vector space $R^{n-1}f_{*}(\mathsf{L})_{x}$ can be recovered from this by applying the functor $K\mapsto H^{0}(A,K\otimes\mathsf{L})$. The theory of [44] views (a slight modification of) the category of perverse sheaves on $A$ as the category of representations of a certain group; the functors $K\mapsto H^{0}(A,K\otimes\mathsf{L})$ are almost all isomorphic to the functor taking a representation to the underlying vector space. The image of this group on the representation associated to a perverse sheaf $K$ is the convolution monodromy group.

We show that, if the convolution monodromy group of the constant sheaf on $Y_{x}$ contains a classical group for some $x\in X$, and if the family $Y$ over $X$ is not equal to a translate of the constant family, then for almost all $\mathsf{L}$, the monodromy groups of $R^{n-1}f_{*}(\mathsf{L})_{x}$ contain a classical group (Theorem 4.7, Corollary 4.10). To check the condition that the convolution monodromy group of the smooth hypersurface $Y_{x}$ contains a classical group, we use recent results of Krämer [42, 43], to reduce to a small number of cases—essentially, the simple algebraic groups acting by their minuscule representations—and then some intricate but elementary combinatorics involving Hodge numbers to eliminate the non-classical cases.

We now conclude the argument by taking $\mathrm{Hilb}$ to be the moduli space of hypersurfaces in a given Néron-Severi class in $A$ and $X$ to be an irreducible component of the Zariski closure of $\mathrm{Hilb}(\mathcal{O}_{K,S})$. Assuming that the universal family over $X$ is not equal to a translate of the constant family, we find a sheaf $R^{n-1}f_{*}(\mathsf{L})_{x}$ with big monodromy and, using $p$-adic Hodge theory, show the integral points of $X$ are not Zariski dense. This contradicts the definition of $X$ as an irreducible component, so we conclude the universal family over each component is equal to a translate of the constant family. It follows that all the fibers $Y_{x}$ contained in each irreducible component are identical up to translation, so because there are finitely many irreducible components, there are finitely many hypersurfaces up to translation – our desired conclusion.

Given an abelian variety $A$ over an algebraically closed field, Krämer and Weissauer [44] construct a Tannakian category as a quotient of the category of perverse sheaves on $A$. A perverse sheaf $N$ on $A$ is said to be negligible if its Euler characteristic is zero; the negligible sheaves form a thick subcategory, and the sheaf convolution category is defined as the quotient of the category of all perverse sheaves by the negligible sheaves. The convolution of two perverse sheaves has negligible perverse homology in nonzero degrees; in other words, it is “perverse up to negligible sheaves,” and convolution defines a tensor structure on this quotient category.

One original motivation for this construction was the Schottky problem [45]. Given a principally polarized abelian variety $A$ (say of dimension $g$) with theta divisor $\Theta$, one wants to know whether $A$ is isomorphic to a Jacobian, say $\operatorname{Jac}C$. In this case, $\Theta$ would be the $(g-1)$-st convolution power of $C$. Informally, the role of the Tannakian formalism here is to determine whether $\Theta$ is “a $(g-1)$-st convolution power of something.”

An alternate motivation for the sheaf convolution theory comes from work of Katz. This time, one works with an abelian variety $A$ over a finite field $\mathbb{F}_{q}$. A perverse sheaf $K$ on $A$ has a trace function $f_{K}$ on $A(\mathbb{F}_{q})$. Associated to a character $\chi$ of $A(\mathbb{F}_{q})$ is the character sum $\sum_{x\in A(\mathbb{F}_{q})}f_{K}(x)\chi(x)$. Katz showed (in unpublished work analogous to [37]) that the distribution of $\sum_{x\in A(\mathbb{F}_{q})}f_{K}(x)\chi(x)$, viewed as a random variable for uniformly random $\chi$, converges to a distribution determined by the convolution monodromy group, in the limit as $q$ goes to $\infty$. More precisely, the distribution is like the trace of a random element in the maximal compact subgroup of the convolution monodromy group. To gain some intuition for this, note that given representations $V_{1}$, $V_{2}$, we have $\operatorname{tr}(g,V_{1}\otimes V_{2})=\operatorname{tr}(g,V_{1})\operatorname{tr}(g,V_{2})$; that is, taking the tensor product of representations has the effect of multiplying the traces. For the character sums $\sum_{x\in A(\mathbb{F}_{q})}f_{K}(x)\chi(x)$, convolution has the same effect:

In other words, convolution of these functions $f_{K}$ has a similar effect on this sum as tensor product of the representations $V$ has on the trace. It stands to reason that a framework where perverse sheaves correspond to representations, and convolution of sheaves correspond to tensor product of representations, would have relevance to the distribution of the trace. In particular, this should be plausible if one is familiar with Deligne’s equidistribution theorem [15, Theorem 3.5.3], whose proof is similar to the argument Katz uses to establish the relationship between the distribution and the convolution monodromy group [37, Corollary 7.4].

For non-algebraically closed fields, such as finite fields, we can construct a Tannakian category in almost the same way as Krämer and Weissauer did—again defining negligible sheaves as those with zero Euler characteristic. The key facts (for example, that the convolution of two perverse sheaves has negligible perverse cohomology in nonzero degrees) hold over the base field once checked over its algebraic closure.

To relate these two categories, it is convenient to restrict attention to geometrically semisimple perverse sheaves on $A_{k}$, and to perverse sheaves on $A_{\overline{k}}$ which are summands of the pullback from $A_{k}$ to $A_{\overline{k}}$ of geometrically semisimple perverse sheaves. Having done this, we obtain (in Lemma 2.8) an exact sequence of pro-algebraic groups

where $G_{k}$ is the Tannakian group of a suitable category of perverse sheaves on $A_{k}$, $G_{\overline{k}}$ is the Tannakian group of a suitable category of perverse sheaves on $A_{\overline{k}}$, and $\operatorname{Gal}_{k}$ is the Tannakian group of the category of $\ell$-adic $\operatorname{Gal}(\overline{k}/k)$-representations – in other words, the Zariski closure of $\operatorname{Gal}(\overline{k}/k)$ in the product of the general linear groups of all its finite-dimensional $\ell$-adic representations. We think of this as a close analogue of the usual exact sequence

for a variety $X$ over a field $k$.

Just like this usual exact sequence, (1) often has splittings. In our case, splittings arise from certain local systems $\mathsf{L}$ on $A$ defined over $k$, as the cohomology of a perverse sheaf twisted by a local system is a Galois representation, on which $\operatorname{Gal}_{k}$ acts, and we can check that this action factors through the Tannakian group $G_{k}$, giving the splitting.

Fix now a subvariety $X$ of the moduli space of smooth hypersurfaces in an abelian variety $A$. Let $k$ be the field of functions on the generic point of $X$. Let $H$ be the universal hypersurface in $A$, defined over $k$. Let $K$ be the constant sheaf on $H$, pushed forward to $A$, placed in degree $1-n$; this is our perverse sheaf of interest. Associated to $k$ is a representation of $G_{k}$. The action of $G_{\overline{k}}$ on this representation is a purely geometric object. By geometric methods, we will show in Theorem 3.5 that the image of $G_{\overline{k}}$ acting on this representation contains $SL_{N},SO_{N}$, or $Sp_{N}$ as a normal subgroup. So the image of $G_{k}$ on the representation associated to $k$ contains the same classical group as a normal subgroup. Because the action of $\operatorname{Gal}_{k}$ in this setting matches the action of the fundamental group, it will suffice for our big monodromy theorem to show that the action of $\operatorname{Gal}_{k}$ also contains (as a normal subgroup) the same classical group.

To do this, we construct in Lemma 4.5 a battery of tests, each consisting of a representation of the normalizer of the classical group, such that any subgroup of the normalizer contains the classical group if and only if it has no invariants on any of these representations. Associated to each of these representations is a perverse sheaf on $A_{k}$. We prove Lemma 4.4 showing that the action of $\operatorname{Gal}_{k}$ on the cohomology of a perverse sheaf, defined using a generic local system $\mathsf{L}$, has invariants if and only if the perverse sheaf has a very special form. Using Lemma 4.1, we check that the relevant perverse sheaves do not have this special form unless the family of hypersurfaces over $X$ is constant, up to translation by a section of $A$, completing the proof of Theorem 4.7.

Next we describe how we check in Theorem 3.5 that the image of the $G_{\overline{k}}$-action on the representation associated to a smooth hypersurface in $A$ contains a classical group acting by the standard representation as a normal subgroup. This proceeds in two steps. The first step shows (in Lemmas 3.8 and 3.9) that the commutator of the identity component of this image group is a simple algebraic group acting by a minuscule representation. (Recall that a minuscule representation is one where the eigenvalues of the maximal torus action are conjugate under the Weyl group.) The second step eliminates (in Lemmas 3.12 and 3.14 and Proposition 3.15) all such pairs of a group and a representation except the standard representations of the classical groups. The first step is a conceptual proof using sophisticated machinery from [42, 43], while the second uses no additional machinery (except a bit of Hodge theory) but involves an intricate combinatorial argument.

For the first step, we apply results of Krämer that study the characteristic cycle of a perverse sheaf. This is a fundamental invariant of any perverse sheaf on a smooth variety, defined as an algebraic cycle on the cotangent bundle of that variety. (For abelian varieties, the cotangent bundle is a trivial vector bundle.) By examining how the characteristic cycle of a perverse sheaf changes when it is convolved with another perverse sheaf, Krämer was able to relate the convolution monodromy group to the characteristic cycle. In particular, he gave criteria for the commutator subgroup of the identity component to be a simple group, and for the representation of it to be minuscule. The fact that our hypersurface is smooth makes its characteristic cycle relatively simple—it is simply the conormal bundle to the hypersurface. This makes Krämer’s minisculeness criterion straightforward to check, but to check simplicity we must make a small modification to Krämer’s argument. The reason for this is that Krämer, motivated by the theta divisor and the Schottky problem, assumed that a hypersurface in $A$ was invariant under the inversion map, while we do not wish to assume this.

For the second step, the exceptional groups and spin groups are not too hard to eliminate, as they only occur for representations of very specific dimensions. The Tannakian dimension in our setting is the topological Euler characteristic of the hypersurface, which we have an explicit formula for. Comparing these, we can see in Lemma 3.12 that the problematic cases only occur for curves in an abelian surface, which are excluded by the assumption $\dim(A)\geq 3$. The only remaining case, except for the good classical cases, is the case of a special linear group acting by a wedge power representation. For this representation, the Euler characteristic formula is not sufficient, but we are eventually able to rule this case out using a more sophisticated numerical invariant, the Hodge numbers. If the convolution monodromy group acts on the representation associated to $H$ by the $k$-th wedge power of an $m$-dimensional representation, we might expect that the Hodge structure on the cohomology of $H$, or the cohomology of $H$ twisted by a rank one local system, is the $k$-th wedge power of an $m$-dimensional Hodge structure. This would place some restrictions on the Hodge numbers. We don’t prove this, but instead prove in Lemma 3.14 a $p$-adic Hodge-theoretic analogue, using the $\operatorname{Gal}_{k}$-action discussed earlier. On the other hand, we can calculate the Hodge numbers of the cohomology of $H$ twisted by a rank one local system using the Hirzebruch-Riemann-Roch formula. Working this out gives a complicated set of combinatorial relations between the Hodge numbers of the original $m$-dimensional Hodge structure. By a lengthy combinatorial argument in Appendix B, we find all solutions of these relations, noting in particular that they occur only for abelian varieties of dimension less than four. This completes the proof.

The argument proceeds in three parts.

First, we use the sheaf convolution formalism to prove a big monodromy result for families of hypersurfaces. In Section 2 we introduce the sheaf convolution category, a Tannakian category of perverse sheaves on an abelian variety. In Section 3 we investigate the convolution monodromy group of a hypersurface; we show in many cases that this group must be as big as possible. In Section 4 we relate the convolution monodromy group to the geometric monodromy group, which gives the big monodromy statement we need.

Sections 5–8 explain how to deduce non-density of integral points, following the strategy in [47]. Section 5 contains some technical preliminaries. We introduce the notion of Hodge–Deligne system, which is closely related to Deligne’s “system of realizations” of a motive, although we include only the realizations that are relevant for our argument. We discuss “$H^{0}$-algebras”, roughly, algebra objects in the category of Artin motives with rational coefficients, which we need to express the semilinearity of Frobenius. We also recall some facts from the theory of not-necessarily-connected reductive groups. Section 6 relates the big monodromy statement from Section 4 to the $p$-adic period map, via the theorem of Bakker and Tsimerman ([3]). In Section 7, we deduce the non-density of integral points, under the simplifying assumption that all the global representations that arise are semisimple. In Section 8, we prove the theorem in full generality. The argument used to handle the global semisimplification involves combinatorics on reductive groups, analogous to [47, §11]. We conclude with Theorem 8.17, which is analogous to Lemma 4.2, Prop. 5.3, and Thm. 10.1 in [47].

Finally, we wrap up the proof of our main theorem in Section 9.

Appendices A, B, and C contain some purely combinatorial calculations involving Eulerian numbers. Appendix A verifies the two numerical conditions in the hypotheses of Theorem 8.17. Appendix B is devoted to the proof of Prop. 3.15, which is used to show that the representation of the Tannakian group associated to a smooth hypersurface is not the wedge power of a smaller-dimensional representation—the last remaining case where the Tannakian group could be too small, and Appendix C contains inequalities that are used in Appendix B.

We would like to thank Johan de Jong, Matthew Emerton, Sergey Gorchinskiy, Ariyan Javanpeykar, Caleb Ji, Shizhang Li, Benjamin Martin, Bjorn Poonen, Akshay Venkatesh, Thomas Krämer, and Marco Maculan for interesting discussions related to this project. We would like to thank the three anonymous referees for numerous helpful comments.

This work was conducted while Will Sawin served as a Clay Research Fellow, and, later, was supported by NSF grant DMS-2101491. Brian Lawrence would like to acknowledge support from the National Science Foundation. We met to work on this project at the Oberwolfach Research Institute for Mathematics, Columbia University, and the University of Chicago; we would like to thank these institutions for their hospitality.