Arithmetic special cycles and Jacobi forms

The impetus for this paper emerged from discussions during an AIM SQuaRE workshop; I’d like to thank the participants – Jan Bruinier, Stephan Ehlen, Stephen Kudla and Tonghai Yang – for the stimulating discussion and insightful remarks, and AIM for the hospitality. I’d also like to thank Craig Cowan for a helpful discussion on the theory of currents. This work was partially supported by an NSERC Discovery grant.

• •

  Throughout, we fix a totally real field $F$ with $[F:\mathbb{Q}]=d$. Let $\sigma_{1},\dots,\sigma_{d}$ denote the real embeddings. Using these embeddings, we identify $F\otimes_{\mathbb{Q}}\mathbb{R}$ with $\mathbb{R}^{d}$, and denote by $\sigma_{i}(\mathbf{t})$ the $i$’th component of $\mathbf{t}\in F\otimes_{\mathbb{Q}}\mathbb{R}$ under this identification.

• •

  For any matrix $A$, we denote the transpose by $A^{\prime}$.

• •

  If $A\in\mathrm{Mat}_{n}(F\otimes_{\mathbb{Q}}\mathbb{R})$, we write

  $$e(A)\ :=\ \prod_{i=1}^{d}\,\exp\big{(}2\pi i\,\mathrm{tr}\left(\sigma_{i}(A)\right)\big{)}$$

  (2.1)

• •

  If $(V,Q)$ is a quadratic space over $F$, let $\langle\mathbf{x},\mathbf{y}\rangle$ denote the corresponding bilinear form. Here we take the convention $Q(\mathbf{x})=\langle\mathbf{x},\mathbf{x}\rangle$. If $\mathbf{x}\in V$ and $\mathbf{y}=(\mathbf{y}_{1},\dots,\mathbf{y}_{n})\in V^{n}$ , we set $\langle\mathbf{x},\mathbf{y}\rangle=(\langle\mathbf{x},\mathbf{y}_{1}\rangle,\dots,\langle\mathbf{x},\mathbf{y}_{n}\rangle)\in\mathrm{Mat}_{1,n}(F)$.

• •

  For $i=1,\dots d$, we set $V_{i}=V\otimes_{F,\sigma_{i}}\mathbb{R}$.

• •

  Let

  $$\mathbb{H}_{n}^{d}=\{\bm{\tau}=\mathbf{u}+i\mathbf{v}\in\operatorname{Sym}_{n}(F\otimes_{\mathbb{Q}}\mathbb{R})\ |\ \mathbf{v}\gg 0\}$$

  (2.2)

  denote the Hilbert-Siegel upper half-space of genus $n$ attached to $F$. Via the fixed embeddings $\sigma_{1},\dots,\sigma_{d}$, we may identify $\operatorname{Sym}_{n}(F\otimes\mathbb{R})\simeq\operatorname{Sym}_{n}(\mathbb{R})^{d}$; we let $\sigma_{i}(\bm{\tau})=\sigma_{i}(\mathbf{u})+i\sigma_{i}(\mathbf{v})$ denote the corresponding component, so that, in particular, $\sigma_{i}(\mathbf{v})\in\operatorname{Sym}_{n}(\mathbb{R})_{>0}$ for $i=1,\dots,d$.

  If $\bm{\tau}\in\mathbb{H}_{n}^{d}$ and $T\in\operatorname{Sym}_{n}(F)$, we write

  $$q^{T}=e(\bm{\tau}T).$$

  (2.3)

In this section, we recall the theory of arithmetic Chow groups $\widehat{\mathrm{CH}}{}^{n}_{\mathbb{C}}(X)$, as conceived by Gillet and Soulé [GS90]; note here and throughout this paper, we work with complex coefficients. Recall that $X$ is defined over a number field $E$ endowed with a fixed complex embedding $\sigma:E\to\mathbb{C}$. We view $X$ as an arithmetic variety over the “arithmetic ring” $(E,\sigma,\text{complex conjugation})$ in the terminology of [GS90, §3.1.1].

An arithmetic cycle is a pair $(Z,g)$, where $Z$ is a formal $\mathbb{C}$-linear combination of codimension $n$ subvarieties of $X$, and $g$ is a Green current for $Z$; more precisely, $g$ is a current of degree $(p-1,p-1)$ on $X(\mathbb{C})$ such Green’s equation

$$\mathrm{dd^{c}}g\ +\ \delta_{Z(\mathbb{C})}=\omega$$

(2.4)

holds, where the right hand side is the current defined by integration¹¹1Here and throughout this paper, we will abuse notation and write $\omega$ both for the form and the current it defines. against some smooth form $\omega$. Given a codimension $n-1$ subvariety $Y$ and a rational function $f\in k(Y)^{\times}$ on $Y$, let

$$\widehat{\mathrm{div}}(f):=(\mathrm{div}(f),-\log|f|^{2}\,\delta_{Y})$$

(2.5)

denote the corresponding principal arithmetic divisor. The arithmetic Chow group $\widehat{\mathrm{CH}}{}^{n}_{\mathbb{C}}(X)$ is quotient of the space of arithmetic cycles by the subspace spanned by (a) the principal arithmetic divisors and (b) classes of the form $(0,\eta)$ with $\eta\in\mathrm{im}(\partial)+\mathrm{im}(\overline{\partial})$. For more details, see [GS90, Sou92]

In their paper [BGKK07], Burgos, Kramer and Kühn give an abstract reformulation and generalization of this theory: their main results describe the construction of an arithmetic Chow group $\widehat{\mathrm{CH}}{}^{*}(X,\mathcal{C})$ attached to a “Gillet complex” $\mathcal{C}$. One of the examples they describe is the group attached to the complex of currents $\mathcal{D}_{\mathrm{cur}}$; we will content ourselves with the superficial description of this group given below, which will suffice for our purposes, and the reader is invited to consult [BGKK07, §6.2] for a thorough treatment.

Unwinding the formal definitions in [BGKK07], one finds that classes in $\widehat{\mathrm{CH}}{}^{n}_{\mathbb{C}}(X,\mathcal{D}_{\mathrm{cur}})$ are represented by tuples $(Z,[T,g])$, with $Z$ as before, but now $T$ and $g$ are currents of degree $(n,n)$ and $(n-1,n-1)$ respectively such that²²2The reader is cautioned that in [BGKK07], the authors normalize delta currents and currents defined via integration by powers of $2\pi i$, resulting in formulas that look slightly different from those presented here; because we are working with $\mathbb{C}$-coefficients, the formulations are equivalent.

$$\mathrm{dd^{c}}g+\delta_{Z(\mathbb{C})}=T+\mathrm{dd^{c}}(\eta)$$

(2.6)

for some current $\eta$ with support contained in $Z(\mathbb{C})$; we can view this as a relaxation of the condition that the right hand side of (2.4) is smooth. A nice consequence of this description is that any codimension $n$ cycle $Z$ on $X$ gives rise to a canonical class (see [BGKK07, Definition 6.37])

$$\widehat{Z}^{\mathrm{can}}\ :=\ \left(Z,[\delta_{Z},0]\right).$$

(2.7)

By [BGKK07, Theorem 6.35], the natural map

$$\widehat{\mathrm{CH}}{}^{n}_{\mathbb{C}}(X)\to\widehat{\mathrm{CH}}{}^{n}_{\mathbb{C}}(X,\mathcal{D}_{\mathrm{cur}}),\qquad(Z,g)\mapsto(Z,[\omega,g])$$

(2.8)

is injective. Moreover while $\widehat{\mathrm{CH}}{}^{*}_{\mathbb{C}}(X,\mathcal{D}_{\mathrm{cur}})$ is not a ring in general, it is a module over $\widehat{\mathrm{CH}}{}^{*}_{\mathbb{C}}(X)$. As a special case of this product, let $(Z,g)\in\widehat{\mathrm{CH}}{}^{1}_{\mathbb{C}}(X)$ be an arithmetic divisor, where $g$ is a Green function with logarithmic singularities along the divisor $Z$. Suppose $\widehat{Y}^{\mathrm{can}}\in\widehat{\mathrm{CH}}{}^{m}(X,\mathcal{D}_{\mathrm{cur}})$ is the canonical class attached to a cycle $Y$ that intersects $Z$ properly; then by inspecting the proofs of [BGKK07, Theorem 6.23, Proposition 6.32] we find

$$(Z,g)\cdot\widehat{Y}^{\mathrm{can}}\ =\ \left(Z\cdot Y,[\omega\wedge\delta_{Y(\mathbb{C})},g\wedge\delta_{Y(\mathbb{C})}]\right)\in\widehat{\mathrm{CH}}{}^{m+1}_{\mathbb{C}}(X,\mathcal{D}_{\mathrm{cur}}).$$

(2.9)

Here we review Kudla’s construction of the family $\{Z(T)\}$ of special cycles on $X$, [Kud97]. First, recall that the symmetric space $\mathbb{D}$ has a concrete realization

$$\mathbb{D}=\{z\in\mathbb{P}^{1}(V\otimes_{\sigma_{1},F}\mathbb{C})\ |\ \langle z,z\rangle=0,\langle z,\overline{z}\rangle<0\}$$

(2.12)

where $\langle\cdot,\cdot\rangle$ is the $\mathbb{C}$-bilinear extension of the bilinear form on $V$; the two connected components of $\mathbb{D}$ are interchanged by conjugation.

Given a collection of vectors $\mathbf{x}=(\mathbf{x}_{1},\dots,\mathbf{x}_{n})\in V^{n}$, let

$$\mathbb{D}^{+}_{\mathbf{x}}\ :=\ \{z\in\mathbb{D}^{+}\ |\ z\perp\sigma_{1}(\mathbf{x}_{i})\ \text{ for }i=1,\dots,n\}.$$

(2.13)

where, abusing notation, we denote by $\sigma_{1}\colon V\to V_{1}=V\otimes_{F,\sigma_{1}}\mathbb{R}$ the map induced by inclusion in the first factor.

Let $\Gamma_{\mathbf{x}}$ denote the pointwise stabilizer of $\mathbf{x}$ in $\Gamma$; then the inclusion $\mathbb{D}^{+}_{\mathbf{x}}\subset\mathbb{D}^{+}$ induces a map

$$\Gamma_{\mathbf{x}}\big{\backslash}\mathbb{D}^{+}_{\mathbf{x}}\to\Gamma\big{\backslash}\mathbb{D}^{+}=X,$$

(2.14)

which defines a complex algebraic cycle that we denote $Z(\mathbf{x})$. If the span of $\{\mathbf{x}_{1},\dots,\mathbf{x}_{r}\}$ is not totally positive definite, then $\mathbb{D}^{+}_{\mathbf{x}}=\emptyset$ and $Z(\mathbf{x})=0$; otherwise, the codimension of $Z(\mathbf{x})$ is the dimension of this span.

Now suppose $T\in\operatorname{Sym}_{n}(F)$ and $\varphi\in S(L^{n})$, and set

$$Z(T,\varphi)^{\natural}\ :=\ \sum_{\begin{subarray}{c}\mathbf{x}\in\Omega(T)\\ \text{mod }\Gamma\end{subarray}}\varphi(\mathbf{x})\cdot Z(\mathbf{x}),$$

(2.15)

where

$$\Omega(T)\ :=\ \{\mathbf{x}=(\mathbf{x}_{1},\dots,\mathbf{x}_{n})\in V^{n}\ |\ \langle\mathbf{x}_{i},\mathbf{x}_{j}\rangle=T_{ij}\}.$$

(2.16)

This cycle is rational over $E$. If $Z(T,\varphi)^{\natural}\neq 0$, then $T$ is necessarily totally positive semidefinite, and in this case $Z(T,\varphi)^{\natural}$ has codimension equal to the rank of $T$.

Finally, we define a $S(L^{n})^{\vee}$-valued cycle $Z(T)^{\natural}$ by the rule

$$Z(T)^{\natural}\colon\varphi\mapsto Z(T,\varphi)^{\natural}.$$

(2.17)

for $\varphi\in S(L^{n})$.

Let $\mathcal{E}\to X$ denote the tautological bundle: over the complex points $X(\mathbb{C})=\Gamma\backslash\mathbb{D}^{+}$, the fibre $\mathcal{E}_{z}$ at a point $z\in\mathbb{D}^{+}$ is simply the line corresponding to $z$ in the model (2.12). There is a natural Hermitian metric $\|\cdot\|^{2}_{\mathcal{E}}$ on $\mathcal{E}(\mathbb{C})$, defined at a point $z\in\mathbb{D}^{+}$ by the formula $\|v_{z}\|^{2}_{\mathcal{E},z}\ =\ -\langle v_{z},v_{z}\rangle$ for $v_{z}\in z$.

Consider the arithmetic class

$$\widehat{\omega}=-\widehat{c}_{1}(\mathcal{E},\|\cdot\|_{\mathcal{E}})\ \in\ \widehat{\mathrm{CH}}{}^{1}_{\mathbb{C}}(X);$$

(2.18)

concretely, $\widehat{\omega}=-(\mathrm{div}s,-\log\|s\|_{\mathcal{E}}^{2})$, where $s$ is any meromorphic section of $\mathcal{E}$. Finally, for future use, we set

$$\Omega:=-c_{1}(\mathcal{E},\|\cdot\|_{\mathcal{E}})\in A^{1,1}(X(\mathbb{C}))$$

(2.19)

where $-\Omega=c_{1}(\mathcal{E},\|\cdot\|_{\mathcal{E}})$ is the first Chern form attached to $(\mathcal{E},\|\cdot\|_{\mathcal{E}})$; here the Chern form is normalized as in [Sou92, §4.2]. Note that $-\Omega$ is a Kähler form, cf. [GS19, §2.2].

In this section, we sketch the construction of a family of Green forms for the special cycles, following [GS19].

We begin by recalling that for any tuple $x=(x_{1},\dots x_{n})\in V_{1}^{n}=(V\otimes_{\sigma_{1},F}\mathbb{R})^{n}$, Kudla and Millson (see [KM90]) have defined a Schwartz form $\varphi_{\mathrm{KM}}(x)$, which is valued in the space of closed $(n,n)$ forms on $\mathbb{D}^{+}$, and is of exponential decay in $x$. Let $T(x)\in\operatorname{Sym}_{n}(\mathbb{R})$ denote the matrix of inner products, i.e. $T(x)_{ij}=\langle x_{i},x_{j}\rangle$, and consider the normalized form

$$\varphi^{o}_{\mathrm{KM}}(x)\ :=\ \varphi_{\mathrm{KM}}(x)\,e^{2\pi\mathrm{tr}T(x)}.$$

(2.20)

In [GS19, §2.2], another form $\nu^{o}(x)$, valued in the space of smooth $(n-1,n-1)$ forms on $\mathbb{D}^{+}$   is defined (this form is denoted by $\nu^{o}(x)_{[2n-2]}$ there). It satisfies the relation

$$\mathrm{dd^{c}}\nu^{o}(\sqrt{u}x)\ =\ -u\frac{\partial}{\partial u}\varphi_{\mathrm{KM}}^{o}(\sqrt{u}x),\qquad u\in\mathbb{R}_{>0}.$$

(2.21)

For a complex parameter $\rho\gg 0$, let

$$\mathfrak{g}^{o}(x;\rho)\ :=\ \int_{1}^{\infty}\nu^{o}(\sqrt{u}x)\frac{du}{u^{1+\rho}};$$

(2.22)

then $\mathfrak{g}^{o}(x,\rho)$ defines a smooth form for $Re(\rho)\gg 0$. The corresponding current admits a meromorphic continuation to a neighbourhood of $\rho=0$ and we set

$$\mathfrak{g}^{o}(x):=\mathop{CT}_{\rho=0}\,\mathfrak{g}^{o}(x;\rho).$$

(2.23)

Note that, for example,

$$\mathfrak{g}^{o}(0)\ =\ \nu^{o}(0)\,\mathop{CT}_{\rho=0}\int_{1}^{\infty}\frac{du}{u^{1+\rho}}=0.$$

(2.24)

In general, the current $\mathfrak{g}^{o}(x)$ satisfies the equation

$$\mathrm{dd^{c}}\mathfrak{g}^{o}(x)+\delta_{\mathbb{D}^{+}_{\mathbf{x}}}\wedge\Omega^{n-r(x)}=\varphi^{o}_{\mathrm{KM}}(x)$$

(2.25)

where $r(x)=\dim\mathrm{span}(x)=\dim\mathrm{span}(x_{1},\dots,x_{n})$; for details regarding all these facts, see [GS19, §2.6].

Now suppose $T\in\operatorname{Sym}_{n}(F)$. Following [GS19, §4], we define an $S(L^{n})^{\vee}$-valued current $\mathfrak{g}^{o}(T,\mathbf{v})$, depending on a parameter $\mathbf{v}\in\operatorname{Sym}_{n}(F\otimes_{\mathbb{Q}}\mathbb{R})_{\gg 0}$, as follows: let $v=\sigma_{1}(\mathbf{v})$ and choose any matrix $a\in\mathrm{GL}_{n}(\mathbb{R})$ such that $v=aa^{\prime}$. Then $\mathfrak{g}^{o}(T,\mathbf{v})$ is defined by the formula

$$\mathfrak{g}^{o}(T,\mathbf{v})(\varphi)\ :=\ \sum_{\mathbf{x}\in\Omega(T)}\varphi(\mathbf{x})\ \mathfrak{g}^{o}\left(\sigma_{1}(\mathbf{x})a\right),$$

(2.26)

where $\sigma_{1}(\mathbf{x})\in V_{1}^{n}$; by [GS19, Proposition 2.12], this is independent of the choice of $a\in\mathrm{GL}_{n}(\mathbb{R})$. Note that $\mathfrak{g}^{o}(T,\mathbf{v})$ is a $\Gamma$-invariant current on $\mathbb{D}^{+}$ and hence descends to $X(\mathbb{C})=\Gamma\backslash\mathbb{D}^{+}$.

Next, consider the $S(L^{n})^{\vee}$-valued differential form $\omega(T,\mathbf{v})$, defined by the formula

$$\omega(T,\mathbf{v})(\varphi)\ :=\ \sum_{\mathbf{x}\in\Omega(T)}\varphi(x)\,\varphi_{\mathrm{KM}}^{o}(\sigma_{1}(\mathbf{x})a),\qquad\sigma_{1}(\mathbf{v})=aa^{\prime},$$

(2.27)

and which is a $q$-coefficient of the Kudla-Millson theta series

$$\Theta_{\mathrm{KM}}(\bm{\tau})\ =\ \sum_{T\in\operatorname{Sym}_{n}(F)}\,\omega(T,\mathbf{v})\,q^{T},\qquad$$

(2.28)

where $\tau\in\mathbb{H}^{d}_{n}$, and $\mathbf{v}=\mathrm{Im}(\bm{\tau})$. We then have the equation of currents

$$\mathrm{dd^{c}}\mathfrak{g}^{o}(T,\mathbf{v})\ +\ \delta_{Z(T)(\mathbb{C})}\wedge\Omega^{n-\mathrm{rank}T}\ =\ \omega(T,\mathbf{v})$$

(2.29)

on $X$, see [GS19, Proposition 4.4]

In particular, if $T$ is non-degenerate, then $\mathrm{rank}(T)=n$ and $\mathfrak{g}^{o}(T,\mathbf{v})$ is a Green current for the cycle $Z(T)^{\natural}$; in this case, we obtain an arithmetic special cycle

$$\widehat{Z}(T,\mathbf{v}):=(Z(T)^{\natural},\mathfrak{g}^{o}(T,\mathbf{v}))\ \in\ \widehat{\mathrm{CH}}{}^{n}_{\mathbb{C}}(X)\otimes_{\mathbb{C}}S(L^{n})^{\vee}.$$

(2.30)

Now suppose $T\in\operatorname{Sym}_{n}(F)$ is arbitrary, let $r=\mathrm{rank}(T)$, and fix $\varphi\in S(L^{n})$. We may choose a pair $(Z_{0},g_{0})$ representing the class $\widehat{\omega}^{n-r}\in\widehat{\mathrm{CH}}{}^{n}_{\mathbb{C}}(X)$, such that $Z_{0}$ intersects $Z(T,\varphi)^{\natural}$ properly and $g_{0}$ has logarithmic type [Sou92, §II.2]. We then define

$$\widehat{Z}(T,\mathbf{v},\varphi):=\left(Z(T,\varphi)^{\natural}\cdot Z_{0},\ \mathfrak{g}^{o}(T,\mathbf{v},\varphi)+g_{0}\wedge\delta_{Z(T,\varphi)(\mathbb{C})}\right)\ \in\ \widehat{\mathrm{CH}}{}^{n}_{\mathbb{C}}(X).$$

(2.31)

The reader may consult [GS19, §5.4] for more detail on this construction, including the fact that it is independent of the choice of $(Z_{0},g_{0})$.

Finally, we define a class $\widehat{Z}(T,\mathbf{v})\in\widehat{\mathrm{CH}}{}^{n}_{\mathbb{C}}(X)\otimes S(L^{n})^{\vee}$ by the rule

$$\widehat{Z}(T,\mathbf{v})(\varphi)=\widehat{Z}(T,\mathbf{v},\varphi).$$

(2.32)

In this section, we briefly review the basic definitions of vector-valued (Hilbert) Jacobi modular forms, mainly to fix notions. For convenience, we work in “classical” coordinates and only with parallel scalar weight. Throughout, we fix an integer $n\geq 1.$

We begin by briefly recalling the theory of metaplectic groups and the Weil representation; a convenient summary for the facts mentioned here, in a form useful to us, is [JS07, §2]. For a place $v\leq\infty$, let $\operatorname{\widetilde{Sp}}_{n}(F_{v})$ denote the metaplectic group, a two-fold cover of $\operatorname{Sp}_{n}(F_{v})$; as a set, $\operatorname{\widetilde{Sp}}_{n}(F_{v})=\operatorname{Sp}_{n}(F_{v})\times\{\pm 1\}$. When $F_{v}=\mathbb{R}$, the group $\operatorname{\widetilde{Sp}}_{n}(\mathbb{R})$ is isomorphic to the group of pairs $(g,\phi)$, where $g=\left(\begin{smallmatrix}A&B\\ C&D\end{smallmatrix}\right)\in\operatorname{Sp}_{n}(\mathbb{R})$ and $\phi\colon\mathbb{H}_{n}\to\mathbb{C}$ is a function such that $\phi(\tau)^{2}=\det(C\tau+D)$; in this model, multiplication is given by

$$(g,\phi(\tau))\cdot(g^{\prime},\phi^{\prime}(\tau))=(gg^{\prime},\phi(g^{\prime}\tau)\phi^{\prime}(\tau)).$$

(2.34)

At a non-dyadic finite place, there exists a canonical embedding $\operatorname{Sp}_{n}(\mathcal{O}_{v})\to\operatorname{\widetilde{Sp}}_{n}(F_{v})$. Consider the restricted product $\prod^{\prime}_{v\leq\infty}\operatorname{\widetilde{Sp}}_{n}(F_{v})$ with respect to these embeddings; the global double cover $\operatorname{\widetilde{Sp}}_{n,\mathbb{A}}$ of $\operatorname{Sp}_{n}(\mathbb{A})$ is the quotient $\prod^{\prime}_{v\leq\infty}\operatorname{\widetilde{Sp}}_{n}(F_{v})/I$ of this restricted direct product by the subgroup

$$I:=\{(1,\epsilon_{v})_{v\leq\infty}\,|\,\prod_{v}\epsilon_{v}=1,\,\epsilon_{v}=1\text{ for almost all }v\}.$$

(2.35)

Moreover, there is a splitting

$$\iota_{F}\colon\operatorname{Sp}_{n}(F)\hookrightarrow\operatorname{\widetilde{Sp}}_{n,\mathbb{A}},\qquad\gamma\mapsto\prod_{v}(\gamma,1)_{v}\cdot I.$$

(2.36)

Let $\widetilde{\Gamma}^{\prime}$ denote the full inverse image of $\operatorname{Sp}_{n}(\mathcal{O}_{F})$ under the covering map $\prod_{v|\infty}\operatorname{\widetilde{Sp}}_{n}(F_{v})\to\operatorname{Sp}_{n}(F\otimes_{\mathbb{Q}}\mathbb{R})$. We obtain an action $\rho$ of $\widetilde{\Gamma}^{\prime}$ on the space $S(V(\mathbb{A}_{f})^{n})$ as follows. Let $\omega$ denote the³³3Here we take the Weil representation for the standard additive character $\psi_{F}\colon\mathbb{A}_{F}/F\to\mathbb{C}$, which we suppress from the notation. Weil representation of $\operatorname{\widetilde{Sp}}_{n,\mathbb{A}}$ on $S(V(\mathbb{A})^{n})$. Given $\widetilde{\gamma}\in\widetilde{\Gamma}^{\prime}$, choose $\widetilde{\gamma}_{f}\in\prod^{\prime}_{v<\infty}\operatorname{\widetilde{Sp}}_{n}(F_{v})$ such that $\widetilde{\gamma}\widetilde{\gamma}_{f}\in\mathrm{im}(\iota_{F})$ and set

$$\rho(\widetilde{\gamma})\ :=\ \omega(\widetilde{\gamma}_{f}).$$

(2.37)

Recall that we had fixed a lattice $L\subset V$. The subspace $S(L^{n})\subset S(V(\mathbb{A}_{f})^{n})$, as defined in (1.3), is stable under the action of $\widetilde{\Gamma}^{\prime}$; when we wish to emphasize this lattice, we denote the corresponding action by $\rho_{L}$.

For a half-integer $\kappa\in\frac{1}{2}\mathbb{Z}$, we define a (parallel, scalar) weight $\kappa$ slash operator, for the group $\widetilde{\Gamma}^{\prime}$ acting on the space of functions $f\colon\mathbb{H}^{d}_{n}\to S(L^{n})^{\vee}$, by the formula

$$f|_{\kappa}[\widetilde{\gamma}](\bm{\tau})\ =\ \prod_{v|\infty}\phi_{v}(\sigma_{v}(\bm{\tau}))^{-2\kappa}\rho_{L}^{\vee}(\widetilde{\gamma}^{-1})\cdot f(g\bm{\tau}),\qquad\widetilde{\gamma}=(g_{v},\phi_{v}(\tau))_{v|\infty}$$

(2.38)

where $g=(g_{v})_{v}$.

If $n>1$, consider the Jacobi group $G^{J}=G^{J}_{n,n-1}\subset\operatorname{Sp}_{n}$; for any ring $R$, its $R$-points are given by

$$G^{J}(R):=\left\{g=\left(\begin{array}[]{cc|cc}a&0&b&a\mu-b\lambda\\ \lambda^{t}&1_{n-1}&\mu^{t}&0\\ \hline\cr c&0&d&c\mu-d\lambda\\ 0&0&0&1_{n-1}\end{array}\right)\ |\ \begin{pmatrix}a&b\\ c&d\end{pmatrix}\in\mathrm{SL}_{2}(R),\ \mu,\tau\in M_{1,n-1}(R)\right\}.$$

(2.39)

Define $\widetilde{\Gamma}^{J}\subset\widetilde{\Gamma}^{\prime}$ to be the inverse image of $G^{J}(\mathcal{O}_{F})$ in $\widetilde{G}^{\prime}_{\mathbb{R}}$.

We now clarify what it should mean for generating series with coefficients in arithmetic Chow groups, such as those appearing in Theorem 1.1, to be modular.

First, let $D^{n-1}(X)$ denote the space of currents on $X(\mathbb{C})$ of complex bidegree $(n-1,n-1)$, and note that there is a map

$$a\colon D^{n-1}(X)\ \to\ \widehat{\mathrm{CH}}{}^{n}_{\mathbb{C}}(X,\mathcal{D}_{\mathrm{cur}}),\qquad a(g)=(0,\,[\mathrm{dd^{c}}g,g]).$$

(2.44)