2.1 The Weil representation

The aim of this section is to fix notation; see [Reference BruinierBru02, §1.1] for a reference for the facts mentioned here, and [Reference GelbartGel76, Reference StrömbergStr13, Reference WeilWei64] for a more general discussion of the Weil representation.

Let $(V,Q)$ be a quadratic space over $\mathbb{Q}$ of signature $(p,q)$ , and denote by $S(V(\mathbb{A}_{f}))$ the space of Schwartz–Bruhat functions on $V(\mathbb{A}_{f})$ , i.e., the space of complex valued functions on $V(\mathbb{A}_{f})$ that are locally constant and compactly supported. The Weil representation is an action

where $\operatorname{Mp}_{2}$ is the metaplectic extension of $\operatorname{SL}_{2}$ .

Elements of $\operatorname{Mp}_{2}(\mathbb{R})$ are represented by pairs $\tilde{\unicode[STIX]{x1D6FE}}=(\unicode[STIX]{x1D6FE},\unicode[STIX]{x1D719})$ where $\unicode[STIX]{x1D6FE}=\big(\!\begin{smallmatrix}a & b\\ c & d\end{smallmatrix}\!\big)\in \operatorname{SL}_{2}(\mathbb{R})$ and $\unicode[STIX]{x1D719}:\mathbb{H}\rightarrow \mathbb{C}$ is a holomorphic function satisfying $\unicode[STIX]{x1D719}(\unicode[STIX]{x1D70F})^{2}=c\unicode[STIX]{x1D70F}+d$ . The product of two elements is given by

The group $\widetilde{\unicode[STIX]{x1D6E4}}:=\operatorname{Mp}_{2}(\mathbb{Z})$ , defined as the inverse image of $\unicode[STIX]{x1D6E4}:=\operatorname{SL}_{2}(\mathbb{Z})$ under the covering map, is generated by the elements

Suppose $L\subset V$ is an even integral lattice (i.e., $L$ is a free $\mathbb{Z}$ -submodule for which $Q(x)\in \mathbb{Z}$ for all $x\in L$ ) and let $L^{\prime }$ be the dual lattice. We let $\widehat{\mathbb{Z}}=\prod _{p}\mathbb{Z}_{p}$ , where the product is over all rational primes, and define $\widehat{L}=L\otimes _{\mathbb{Z}}\widehat{\mathbb{Z}}$ . For a coset $\unicode[STIX]{x1D707}\in L^{\prime }/L\cong (\widehat{L})^{\prime }/\widehat{L}$ , let

the finite-dimensional space

is stable under $\widetilde{\unicode[STIX]{x1D6E4}}$ . The restriction is denoted

It can be seen, for example via the explicit formulas in [Reference BruinierBru02, §1.1], that the image of $\unicode[STIX]{x1D70C}_{L}$ is in fact a finite group.

Moreover, when $\dim (V)$ is even, $\unicode[STIX]{x1D70C}_{L}$ factors through the map $\widetilde{\unicode[STIX]{x1D6E4}}\rightarrow \operatorname{SL}_{2}(\mathbb{Z})$ , and we denote the resulting action of $\operatorname{SL}_{2}(\mathbb{Z})$ on $S(L)$ by the same symbol $\unicode[STIX]{x1D70C}_{L}$ .

For $v\in S(L)^{\vee }$ and $w\in S(L)$ , we frequently simply write $v\cdot w$ or $vw$ for the image of an element $v\otimes w\in S(L)^{\vee }\otimes S(L)$ under the canonical contraction map $S(L)^{\vee }\otimes _{\mathbb{C}}S(L)\rightarrow \mathbb{C}$ .

Finally, it will be useful to introduce the Hermitian pairing (conjugate-linear in the second argument)

It is easily checked, for example by using the explicit formulas in [Reference BruinierBru02, §1] that the induced $\mathbb{C}$ -linear isomorphism

identifies the conjugate representation $\overline{\unicode[STIX]{x1D70C}_{L}}$ with the dual representation $\unicode[STIX]{x1D70C}_{L}^{\vee }$ .

This discussion is intended to justify the following abuse of notation: if $\unicode[STIX]{x1D711}\in S(L)$ , let $\overline{\unicode[STIX]{x1D711}}\in S(L)^{\vee }$ denote the linear functional $\langle \cdot ,\unicode[STIX]{x1D711}\rangle _{S(L)}$ .

2.2 Some spaces of non-holomorphic modular forms

Suppose $(W,\unicode[STIX]{x1D70C})$ is a finite dimensional representation of $\widetilde{\unicode[STIX]{x1D6E4}}=\operatorname{Mp}_{2}(\mathbb{Z})$ . If $k\in \frac{1}{2}\mathbb{Z}$ and $\tilde{\unicode[STIX]{x1D6FE}}=(\unicode[STIX]{x1D6FE},\unicode[STIX]{x1D719})\in \widetilde{\unicode[STIX]{x1D6E4}}$ , define the slash operator Footnote ⁴ on functions $f:\mathbb{H}\rightarrow W$ by the formula

where $\unicode[STIX]{x1D6FE}=\big(\!\begin{smallmatrix}a & b\\ c & d\end{smallmatrix}\!\big)$ .

We say that a function $f:\mathbb{H}\rightarrow W$ transforms as a modular form of weight $k$ and representation $\unicode[STIX]{x1D70C}$ if

In this paper, we will primarily be interested in the case where $(W,\unicode[STIX]{x1D70C})$ is either $(S(L),\unicode[STIX]{x1D70C}_{L})$ or its dual.

Suppose that $(W,\unicode[STIX]{x1D70C})=(S(L),\unicode[STIX]{x1D70C}_{L})$ , and consider the element $\mathbf{Z}=(-\mathtt{Id},i)$ , so that $\mathbf{Z}^{2}=(\mathtt{Id},-1)$ . Then

for any function $f:\mathbb{H}\rightarrow S(L)$ , so there exist no non-zero functions satisfying (2.2) unless $2k\equiv q-p\hspace{0.2em}{\rm mod}\hspace{0.2em}2$ , in which case $\mid _{k}[\mathbf{Z}^{2}]$ acts trivially. Moreover, such functions then take values in the subspace of $S(L)$ spanned by the vectors

for $\unicode[STIX]{x1D707}\in L^{\prime }/L$ .

Note also that if $f:\mathbb{H}\rightarrow S(L)$ transforms as a modular form of weight $k$ , then

transforms as a modular form of weight $-k$ for the dual representation $\unicode[STIX]{x1D70C}_{L}^{\vee }$ .

Contained in $H_{k}(\unicode[STIX]{x1D70C}_{L})$ is the space

of weakly holomorphic forms; these are precisely the forms of weight $k$ that are holomorphic on $\mathbb{H}$ and meromorphic at the cusp $\infty$ . In turn, the space $M_{k}^{!}(\unicode[STIX]{x1D70C}_{L})$ contains the space of classical holomorphic modular forms $M_{k}(\unicode[STIX]{x1D70C}_{L})$ , namely those forms that are holomorphic at $\infty$ , and the space of cusp forms $S_{k}(\unicode[STIX]{x1D70C}_{L})$ .

We record some consequences of the definitions that will prove useful later, cf. [Reference Bruinier and FunkeBF04, §3]. First, the Fourier expansion of $f\in H_{k}(\unicode[STIX]{x1D70C}_{L})$ admits a decomposition

into its holomorphic and non-holomorphic parts, where the holomorphic part

has only finitely many negative terms. The non-holomorphic part

has non-zero Fourier coefficients only for negative indices; here

Moreover, a form $f$ is in $M_{k}^{!}(\unicode[STIX]{x1D70C}_{L})$ if and only if $f^{-}=0$ . Finally, we note that

for some constant $C>0$ .

2.3 A family of harmonic weak Maaß forms

In this section, we fix a special family $\{F_{m,\unicode[STIX]{x1D707}}\}\subset H_{k}(\unicode[STIX]{x1D70C}_{L})$ that will play an important role throughout.

Suppose for the moment that $k<0$ . Following [Reference BruinierBru02], if $m\in \mathbb{Q}_{{>}0}$ and $\unicode[STIX]{x1D707}\in L^{\prime }/L$ with $m\in Q(\unicode[STIX]{x1D707})+\mathbb{Z}$ , define

where $M(a,b,c)$ is Kummer’s confluent hypergeometric function (see [Reference Abramowitz and StegunAS64, 13.1.2, 13.1.32]); the observant reader will note that $F_{m,\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D70F})$ differs from its namesake in [Reference BruinierBru02] by a factor of $1/2$ . The forms $F_{m,\unicode[STIX]{x1D707}}$ span $H_{k}(\unicode[STIX]{x1D70C}_{L})$ if $k<0$ (see [Reference BruinierBru02, Proposition 1.12]).

The holomorphic part $F_{m,\unicode[STIX]{x1D707}}^{+}(\unicode[STIX]{x1D70F})$ is of the form

for some coefficients $c_{F_{m,\unicode[STIX]{x1D707}}}^{+}(n)$ and where $\tilde{\unicode[STIX]{x1D711}}_{\unicode[STIX]{x1D707}}=\frac{1}{2}(\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D707}}+(-1)^{k+(q-p)/2}\unicode[STIX]{x1D711}_{-\unicode[STIX]{x1D707}})$ ; in fact, $F_{m,\unicode[STIX]{x1D707}}$ is the unique harmonic weak Maaß form whose holomorphic part has this form [Reference Bruinier and FunkeBF04, Proposition 3.11] and moreover $\unicode[STIX]{x1D709}(F_{m,\unicode[STIX]{x1D707}})$ is a (holomorphic, cuspidal) Poincaré series (cf. [Reference Bruinier and FunkeBF04, Remark 3.10]).

This observation suggests the following approach to defining an analogous function $F_{m,\unicode[STIX]{x1D707}}$ in the case that $k\geqslant 0$ .

We first recall the following modularity criterion, due to Borcherds [Reference BorcherdsBor99]. Let $\mathsf{Sing}_{2-k}(\unicode[STIX]{x1D70C}_{L})$ be the space of formal Fourier polynomials with negative indices that are invariant under the action of the elements $\mathbf{Z}$ and $\mathbf{T}$ of $\widetilde{\unicode[STIX]{x1D6E4}}$ :

where $q=e^{2\unicode[STIX]{x1D70B}i\unicode[STIX]{x1D70F}}$ . We have a map

and also the principal part map

Note that this latter map factors through the quotient by the space of cusp forms $S_{2-k}(\unicode[STIX]{x1D70C}_{L}^{\vee })$ .

This sequence will be used to normalize the forms $F_{m,\unicode[STIX]{x1D707}}$ : for each $k$ , fix a splitting

The map $\unicode[STIX]{x1D713}^{\vee }:M_{k}(\unicode[STIX]{x1D70C}_{L})\rightarrow \mathsf{Sing}_{2-k}(\unicode[STIX]{x1D70C}_{L}^{\vee })^{\vee }$ may be extended to $H_{k}(\unicode[STIX]{x1D70C}_{L})$ , by setting

where $c_{F}^{+}(-m)$ are Fourier coefficients of the holomorphic part $F^{+}$ of $F$ .

When $m\notin Q(\unicode[STIX]{x1D707})+\mathbb{Z}$ , set $F_{m,\unicode[STIX]{x1D707}}=0$ . We may define an $S(L)^{\vee }\otimes _{\mathbb{C}}S(L)$ -valued version

by setting

whenever $\unicode[STIX]{x1D711}=\sum _{\unicode[STIX]{x1D707}}a_{\unicode[STIX]{x1D707}}\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D707}}\in S(L)$ .

2.4 Truncated Poincaré series

For a parameter $w\in \mathbb{R}_{{>}0}$ and $\unicode[STIX]{x1D70F}=u+iv\in \mathbb{H}$ , consider the cutoff function

Fix $w\in \mathbb{R}_{{>}0}$ and a half-integer $k\in \frac{1}{2}\mathbb{Z}$ such that $2k\equiv p\hspace{0.2em}{\rm mod}\hspace{0.2em}2$ . For a coset $\unicode[STIX]{x1D707}\in L^{\prime }/L$ and a rational number $m\in Q(\unicode[STIX]{x1D707})+\mathbb{Z}$ , define the truncated Poincaré series $P_{m,w,\unicode[STIX]{x1D707}}:\mathbb{H}\rightarrow S(L)$ by the formula

where $\widetilde{\unicode[STIX]{x1D6E4}}_{\infty }$ is the subgroup of $\widetilde{\unicode[STIX]{x1D6E4}}$ generated by $\mathbf{T}$ .

For a fixed $\unicode[STIX]{x1D70F}$ , there are only finitely many elements $\unicode[STIX]{x1D6FE}\in \unicode[STIX]{x1D6E4}_{\infty }\backslash \operatorname{SL}_{2}(\mathbb{Z})$ such that $\unicode[STIX]{x1D70E}_{w}(\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D70F})\neq 0$ ; thus the sum (2.9) is locally finite, and in particular, is absolutely convergent. By construction, $P_{m,w,\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D70F})$ is evidently invariant under the weight $k$ slash operator, i.e., it is a (discontinuous!) ‘modular form’ of weight $k$ .

2.5 A section of the Maaß lowering operator

In this section, we show that for a form $f$ satisfying certain mild analytic conditions, our Poincaré series can be used to generate the Fourier coefficients of a distinguished preimage $F\in \mathbf{L}^{-1}(f)$ of $f$ under the Maaß lowering operator.

We begin by fixing some notation. Suppose $L\subset V$ is an even integral lattice, and $\unicode[STIX]{x1D705}\in \frac{1}{2}\mathbb{Z}$ with $2\unicode[STIX]{x1D705}\equiv p-q\hspace{0.2em}{\rm mod}\hspace{0.2em}2$ .

The following Proposition generalizes [Reference Bruinier and FunkeBF04, Theorem 3.7].

Suppose $F\in A_{\unicode[STIX]{x1D705}}^{!}(\unicode[STIX]{x1D70C}_{L}^{\vee })$ and $f=\mathbf{L}(F)\in A_{\unicode[STIX]{x1D705}-2}^{\mathtt{mod}}(\unicode[STIX]{x1D70C}_{L}^{\vee })$ with Fourier expansions

where $\unicode[STIX]{x1D70F}=u+iv$ .

The relation $\mathbf{L}(F)=f$ implies that for each $m$ ,

Since $f(u+it)=O(t^{l})$ as $t\rightarrow \infty$ by assumption,

for all $t>1$ , where the implied constant is independent of $m$ . In particular, for all $m<0$ , substituting this bound in (2.14) gives

Moreover, it follows from the assumption that $F$ has exponential growth that

and

for any fixed $\unicode[STIX]{x1D716}>0$ , where the implied constant depends only on $\unicode[STIX]{x1D716}$ and $f$ .

Finally, we define $\unicode[STIX]{x1D705}_{F}(0)\in S(L)^{\vee }$ by noting that Definition 2.7(iii) implies that $c_{F}(0,v)$ can be written in the form

for some $\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D707}}\in \mathbb{C}$ . In other words, $\unicode[STIX]{x1D705}_{F}(0)$ is the constant part of the constant term.

Recall that we had defined the regularized pairing $\langle f,g\rangle ^{\text{reg}}$ for functions $f$ and $g$ transforming of weight $k$ and $-k$ respectively, as in Definition 2.1. Our next goal is to show this pairing exists in two particular situations.

Recall the fixed splitting

of the exact sequence in Theorem 2.3; note that we have relabelled the subscripts here. Define an extension of the map $P$ to $A_{\unicode[STIX]{x1D705}}^{!}(\unicode[STIX]{x1D70C}_{L}^{\vee })$ by setting

Note that when $\unicode[STIX]{x1D705}>2$ , the space $M_{2-\unicode[STIX]{x1D705}}(\unicode[STIX]{x1D70C}_{L})=\{0\}$ and so $\unicode[STIX]{x1D702}$ is identically zero. In this case, condition (ii) asserts that $P(\mathbf{L}^{\sharp }(f))=0$ , i.e., $\mathbf{L}^{\sharp }(f)$ has ‘trivial principal part’.

Our next aim to calculate regularized pairings against the harmonic Maaß forms of weight $k=2-\unicode[STIX]{x1D705}$ introduced previously. Fix $\unicode[STIX]{x1D707}\in L^{\prime }/L$ and $m\in Q(\unicode[STIX]{x1D707})+\mathbb{Z}$ . For convenience, write the Fourier expansion of the weight $k$ Poincaré series $F_{m,\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D70F})$ , defined for all $m$ and all weights as in Lemma 2.4, as

Recall from § 2.3 that $b_{m,\unicode[STIX]{x1D707}}(n,v)$ is independent of $v$ when $n\geqslant 0$ .

We arrive at the raison d’être of this section.

Extending the theorem to the $S(L)^{\vee }\otimes _{\mathbb{C}}S(L)$ -valued Poincaré series $P_{m,v}$ and $F_{m}$ yields the following corollary.

It remains to prove the following lemma.

3.1 Kudla’s Green function as a regularized theta lift

Let

and note that $\unicode[STIX]{x1D6FD}(r)+\log (r)=O(1)$ as $r\rightarrow 0$ .

For convenience, set

for any $\unicode[STIX]{x1D707}\in L^{\prime }/L$ . Our first aim is another construction of this Green function in terms of the Siegel theta function

which is defined by the formula

For a fixed $z$ , it is a standard fact that $\unicode[STIX]{x1D6E9}_{L}(\unicode[STIX]{x1D70F},z)$ transforms as a modular form of weight $p/2-1$ with respect to the Weil representation, and straightforward estimates imply that $\unicode[STIX]{x1D6E9}_{L}(\unicode[STIX]{x1D70F},z)$ is $O(v)$ as $v\rightarrow \infty$ and satisfies Definition 2.7(iii), which implies $\unicode[STIX]{x1D6E9}(\unicode[STIX]{x1D70F},z)\in A_{p/2-1}^{\mathtt{mod}}(\unicode[STIX]{x1D70C}_{L})$ .

For $\unicode[STIX]{x1D707}\in L^{\prime }/L$ , $m\in Q(\unicode[STIX]{x1D707})+\mathbb{Z}$ , and $w\in \mathbb{R}_{{>}0}$ , consider the regularized pairing

where $P_{m,w,\unicode[STIX]{x1D707}}$ is the truncated Poincaré series of weight $k=1-p/2$ and we view (3.2) as a function in $z\in \mathbb{D}^{o}(V)$ .

The natural $S(L)^{\vee }\otimes S(L)$ -valued version holds as well. Define

which defines a Green function for the $S(L)^{\vee }$ -valued cycle

Upon extending $S_{m,\unicode[STIX]{x1D707}}(z)$ to a functional $S_{m}(z)\in S(L)^{\vee }$ by linearity, the regularized theta lift

gives an extension of $\mathsf{Gr}_{o}^{\mathsf{K}}(m,w)$ to all $z\in \mathbb{D}^{o}(V)$ when $m\neq 0$ ; similarly,

gives an extension of $\mathsf{Gr}_{o}^{\mathsf{K}}(0,w)$ .

3.2 Bruinier’s Green functions and the archimedean generating series

The second family of Green functions arise as regularized theta lifts of the harmonic Maaß forms $F_{m,\unicode[STIX]{x1D711}}$ of weight $k=1-p/2$ : for $m\in \mathbb{Q}$ and $\unicode[STIX]{x1D711}\in S(L)$ , define Bruinier’s Green function

By analysing the singularities of this function, as in [Reference BruinierBru02, ch. 2], one finds that it defines a Green function for the cycle $Z(m,\unicode[STIX]{x1D711})$ ; this construction generalizes that of [Reference BorcherdsBor98], where weakly holomorphic forms were used. Note that we are extending the definition to the case that $m\leqslant 0$ , where $\mathsf{Gr}_{o}^{\mathsf{B}}(m,\unicode[STIX]{x1D711})$ is zero for all but finitely many $m\leqslant 0$ and is moreover a smooth function in this case, i.e., a Green function for the zero cycle.

As usual, there is an $S(L)^{\vee }$ -valued version

defining an $S(L)^{\vee }$ -valued Green function for $Z(m)$ .

The statement analogous to Lemma 3.2 holds for $\mathsf{Gr}_{o}^{\mathsf{B}}(m,\unicode[STIX]{x1D707})$ , and can be proved in much the same manner; see also [Reference Bruinier and YangBY09, Lemma 4.5] and [Reference SchoferSch09, Lemma 2.19] for the corresponding statement at a CM point.

Our next aim is to identify the differences of these Green functions as Fourier coefficients of a (non-holomorphic) modular form.

4.1 Unitary Shimura varieties

For any integer $n\geqslant 1$ , consider the functor ${\mathcal{M}}^{\text{Kr}\ddot{\text{a}}}(n-1,1)$ over $\operatorname{Spec}o_{\boldsymbol{k}}$ defined by the following moduli problem: for any scheme $S$ over $o_{\boldsymbol{k}}$ with structure map $\unicode[STIX]{x1D70F}_{S}:o_{\boldsymbol{k}}\rightarrow {\mathcal{O}}_{S}$ , the $S$ -points ${\mathcal{M}}^{\text{Kr}\ddot{\text{a}}}(n-1,1)(S)$ comprise the category

where:

1. (i) $A$ is an abelian scheme over $S$ of relative dimension $n$ ;

2. (ii) $i:o_{\boldsymbol{k}}\rightarrow \operatorname{End}(A)$ is an $o_{\boldsymbol{k}}$ -action;

3. (iii) $\unicode[STIX]{x1D706}$ is a principal polarization such that $\unicode[STIX]{x1D706}\circ i(a)=i(a^{\prime })^{\vee }\circ \unicode[STIX]{x1D706}$ for all $a\in o_{\boldsymbol{k}}$ ; and

4. (iv) (Krämer’s condition [Reference KrämerKrä03]) $\mathfrak{F}$ is a locally free subsheaf of $\mathsf{Lie}(A)$ of rank $n-1$ such that for all $a\in o_{\boldsymbol{k}}$ the induced map $\mathsf{Lie}(i(a))$ agrees with $\unicode[STIX]{x1D70F}_{S}(a)$ on $\mathfrak{F}$ , and with $\unicode[STIX]{x1D70F}_{S}(a^{\prime })$ on $\mathsf{Lie}(A)/\mathfrak{F}$ .

This moduli problem is represented by a Deligne–Mumford stack, which we also denote by ${\mathcal{M}}^{\text{Kr}\ddot{\text{a}}}(n-1,1)$ , which is regular, flat over $\operatorname{Spec}(o_{\boldsymbol{k}})$ , and smooth over $\operatorname{Spec}o_{\boldsymbol{k}}[1/d_{\boldsymbol{k}}]$ .

Similarly, we may consider the moduli space ${\mathcal{M}}(1,0)$ parametrizing principally polarized elliptic curves $E$ with complex multiplication by $o_{\boldsymbol{k}}$ , where the action is normalized to coincide with the structural morphism on $\mathsf{Lie}(E)$ . This stack is smooth and proper of relative dimension 0 over $\operatorname{Spec}(o_{\boldsymbol{k}})$ [Reference HowardHow15, Proposition 2.1.2]:

Finally, let

One of the main results of [Reference HowardHow15] is the construction of a canonical toroidal compactification of ${\mathcal{M}}$ , obtained by extending the moduli problem to generalized abelian varieties; its properties are summarized in the following proposition.

The stack ${\mathcal{M}}$ admits a decomposition

where $[{\mathcal{V}}]$ runs over the (finite) set of isomorphism classes of Hermitian vector spaces over $\boldsymbol{k}$ of signature $(n-1,1)$ that contain a self-dual lattice, see [Reference Kudla and RapoportKR14, Reference Bruinier, Howard and YangBHY15]. More precisely, suppose $z\in {\mathcal{M}}(\mathbb{C})$ corresponds to a pair of complex abelian varieties $(\text{}\underline{E},\text{}\underline{A})$ , equipped with $o_{\boldsymbol{k}}$ -actions and polarizations. These additional structures endow the homology groups $H_{1}(E(\mathbb{C}),\mathbb{Q})$ and $H_{1}(A(\mathbb{C}),\mathbb{Q})$ with $\boldsymbol{k}$ -Hermitian forms. The component ${\mathcal{M}}_{{\mathcal{V}}}$ is then characterized by the property that for any complex point $z\in {\mathcal{M}}_{{\mathcal{V}}}(\mathbb{C})$ corresponding to $(\text{}\underline{E},\text{}\underline{A})$ , there is an isomorphism

of Hermitian vector spaces.

As a consequence of (4.1), there is a decomposition

where ${\mathcal{M}}_{{\mathcal{V}}}^{\ast }$ is the Zariski closure of ${\mathcal{M}}_{{\mathcal{V}}}$ in ${\mathcal{M}}^{\ast }$ .

Next, we recall the complex uniformizations of ${\mathcal{M}}_{{\mathcal{V}}}$ and ${\mathcal{M}}_{{\mathcal{V}}}^{\ast }$ . For any Hermitian space ${\mathcal{V}}$ of signature $(n-1,1)$ , let

This is a complex manifold of dimension $n-1$ , and is a model for the locally symmetric space attached to $GU({\mathcal{V}})$ .

If ${\mathcal{L}}_{0}$ and ${\mathcal{L}}_{1}$ are self-dual Hermitian $o_{\boldsymbol{k}}$ -lattices of signature $(1,0)$ and $(n-1,1)$ respectively, and

then the group

acts on ${\mathcal{V}}_{{\mathcal{L}}_{0},{\mathcal{L}}_{1}}$ by unitary transformations, and hence on $\mathbb{D}({\mathcal{V}}_{{\mathcal{L}}_{0},{\mathcal{L}}_{1}})$ . We obtain a complex uniformization as in [Reference Bruinier, Howard and YangBHY15, §3.2], see also [Reference Kudla and RapoportKR14, §3]:

where the disjoint union is taken over isomorphism classes of pairs of lattices ${\mathcal{L}}_{0}$ and ${\mathcal{L}}_{1}$ such that ${\mathcal{V}}_{{\mathcal{L}}_{0},{\mathcal{L}}_{1}}\simeq {\mathcal{V}}$ . Implicitly in this statement, we have fixed a set of representatives $\{({\mathcal{L}}_{0},{\mathcal{L}}_{1})\}$ and for each such pair, we view

as a lattice in ${\mathcal{V}}$ via a fixed isomorphism ${\mathcal{V}}_{{\mathcal{L}}_{0},{\mathcal{L}}_{1}}\simeq {\mathcal{V}}$ ; in particular, the group $\unicode[STIX]{x1D6E4}_{{\mathcal{L}}_{0},{\mathcal{L}}_{1}}$ acts on $\mathbb{D}({\mathcal{V}})$ via this fixed isomorphism.

4.2 Kudla–Rapoport divisors

We now turn to the definition of the Kudla–Rapoport divisors, following [Reference Kudla and RapoportKR14, Reference Bruinier, Howard and YangBHY15]. Suppose $(\text{}\underline{E},\text{}\underline{A})\in {\mathcal{M}}(S)$ for some base scheme $S$ ; then the $o_{\boldsymbol{k}}$ -module

admits an $o_{\boldsymbol{k}}$ -Hermitian form, defined by the formula

For each $m\in \mathbb{Q}_{{>}0}$ and ideal $\mathfrak{a}\subset o_{\boldsymbol{k}}$ dividing $\unicode[STIX]{x2202}_{\boldsymbol{k}}$ , let ${\mathcal{Z}}(m,\mathfrak{a})$ be the Deligne–Mumford stack over $\operatorname{Spec}(o_{\boldsymbol{k}})$ representing the following moduli problem: for a scheme $S/\text{Spec}(o_{\boldsymbol{k}})$ , the points of ${\mathcal{Z}}(m,\mathfrak{a})$ comprise the category

where:

1. – $\text{}\underline{E}=(E,i_{E},\unicode[STIX]{x1D706}_{E})\in {\mathcal{M}}(1,0)(S)$ and $\text{}\underline{A}=(A,i_{A},\unicode[STIX]{x1D706}_{A},\mathfrak{F})\in {\mathcal{M}}^{\text{Kra}}(n-1,1)(S)$ ;

2. – $y\in \mathfrak{a}^{-1}L(E,A)$ with $(y,y)=m$ , and such that

   $$\begin{eqnarray}\operatorname{Lie}(i(\unicode[STIX]{x1D6FF}_{k})\circ y):\operatorname{Lie}(E)\rightarrow \operatorname{Lie}(A)\end{eqnarray}$$
   induces the trivial map $\operatorname{Lie}(E)\rightarrow \operatorname{Lie}(A)/\mathfrak{F}$ .

The forgetful map ${\mathcal{Z}}(m,\mathfrak{a})\rightarrow {\mathcal{M}}$ defines a divisor on ${\mathcal{M}}$ , cf. [Reference Bruinier, Howard and YangBHY15, §3.1], which we denote by the same symbol ${\mathcal{Z}}(m,\mathfrak{a})$ in a hopefully mild act of violence against notation.

Setting ${\mathcal{Z}}_{{\mathcal{V}}}(m,\mathfrak{a})={\mathcal{Z}}(m,\mathfrak{a})\times _{{\mathcal{M}}}{\mathcal{M}}_{{\mathcal{V}}}$ , there is a complex uniformization

here

This uniformization is compatible with (4.2), in the sense that the map ${\mathcal{Z}}_{{\mathcal{V}}}(m,\mathfrak{a})(\mathbb{C})\rightarrow {\mathcal{M}}_{{\mathcal{V}}}(\mathbb{C})$ is induced by the inclusions $\mathbb{D}_{y}{\hookrightarrow}\mathbb{D}({\mathcal{V}})$ .

It will be useful to take a ‘vector-valued’ approach, as follows. Suppose ${\mathcal{L}}\subset {\mathcal{V}}$ is a Hermitian self-dual lattice, and ${\mathcal{L}}_{[\mathbb{Z}]}$ the $\mathbb{Z}$ -lattice of signature $(2n-2,2)$ obtained by taking the trace of the Hermitian form. Then the $\mathbb{Z}$ -dual ${\mathcal{L}}_{[\mathbb{Z}]}^{\vee }$ satisfies

and in particular there is an action of $\operatorname{SL}_{2}(\mathbb{Z})$ on

via the Weil representation, as in § 2.1.

For each $\overline{m}\in \mathbb{Q}/\mathbb{Z}$ and $\mathfrak{a}|\unicode[STIX]{x2202}_{k}$ , define

note that $\unicode[STIX]{x1D711}_{\overline{m},\mathfrak{a}}=0$ if $m\notin d_{\boldsymbol{k}}^{-1}\mathbb{Z}/\mathbb{Z}$ .

By [Reference Bruinier, Howard and YangBHY15, Remark 3.9], the set $\{\unicode[STIX]{x1D711}_{\overline{m},\mathfrak{a}}\mid \unicode[STIX]{x1D6FF}_{k}\subset \mathfrak{a},\overline{m}\in N(\mathfrak{a})^{-1}\mathbb{Z}/\mathbb{Z}\}$ forms a basis for the space

of $\operatorname{Aut}(\unicode[STIX]{x2202}_{\boldsymbol{k}}^{-1}{\mathcal{L}}/{\mathcal{L}})$ invariant functions. The Weil representation commutes with $\operatorname{Aut}(\unicode[STIX]{x2202}_{\boldsymbol{k}}^{-1}{\mathcal{L}}/{\mathcal{L}})$ , and so ${\mathcal{S}}$ inherits an action of $\operatorname{SL}_{2}(\mathbb{Z})$ ; furthermore, this action depends only on ${\mathcal{V}}$ and not the choice of lattice ${\mathcal{L}}$ , since any two self-dual lattices in ${\mathcal{V}}$ are in the same genus.

We define vector-valued special cycles

by the formulas

For future use, we also define ${\mathcal{Z}}_{{\mathcal{V}}}(m)={\mathcal{Z}}_{{\mathcal{V}}}^{\ast }(m)=0$ whenever $m\leqslant 0$ .

4.3 Classes in arithmetic Chow groups

In this section, we recall two ways of equipping the divisors ${\mathcal{Z}}_{{\mathcal{V}}}^{\ast }(m,\mathfrak{a})$ with Green functions to obtain classes in the arithmetic Chow group $\widehat{\mathsf{CH}}\text{}_{\mathbb{C}}^{1}({\mathcal{M}}_{{\mathcal{V}}}^{\ast })$ . Here and throughout we work with the $\text{log}{-}\text{log}$ singular version due to Burgos Gil et al. [Reference Burgos Gil, Kramer and KühnBKK07], or more precisely, the ‘stacky’ extension described in [Reference HowardHow15, §3.1].

Roughly speaking, the extended Chow group $\widehat{\mathsf{CH}}\text{}_{\mathbb{C}}^{1}({\mathcal{M}}_{{\mathcal{V}}}^{\ast })$ is a complex vector space spanned by elements of the form

where $Z$ is a $\mathbb{C}$ -divisor on ${\mathcal{M}}_{{\mathcal{V}}}^{\ast }$ (i.e., a formal $\mathbb{C}$ -linear combination of closed substacks, each of which is étale locally cut out by a single non-zero equation), and $g_{Z}$ is a current on ${\mathcal{M}}_{{\mathcal{V}}}^{\ast }(\mathbb{C})$ that:

1. (i) is smooth outside the support of $Z(\mathbb{C})\cap {\mathcal{M}}_{{\mathcal{V}}}(\mathbb{C})$ with logarithmic singularities along this support;

2. (ii) has, along with its derivatives, at worst ‘ $\text{log}{-}\text{log}$ ’ singularities along the boundary $\unicode[STIX]{x2202}{\mathcal{M}}_{{\mathcal{V}}}^{\ast }(\mathbb{C})$ , cf. [Reference Burgos Gil, Kramer and KühnBKK07, Definition 7.1]; and

3. (iii) satisfies Green’s equation

   $$\begin{eqnarray}\operatorname{dd}^{\text{c}}[g_{Z}]+\unicode[STIX]{x1D6FF}_{Z(\mathbb{C})}=[\unicode[STIX]{x1D714}]\end{eqnarray}$$
   for some $(1,1)$ differential form $\unicode[STIX]{x1D714}$ that is smooth on ${\mathcal{M}}_{{\mathcal{V}}}(\mathbb{C})$ .

As with ordinary Chow groups, the principal divisors are deemed equivalent to zero: more precisely, if $f\in \mathbb{Q}({\mathcal{M}}_{{\mathcal{V}}}^{\ast })^{\times }$ is a global rational function, then the divisor

is called principal. The group $\widehat{\mathsf{CH}}\text{}_{\mathbb{C}}^{1}({\mathcal{M}}_{{\mathcal{V}}}^{\ast })$ is then defined to be the quotient of the space of arithmetic divisors by the subspace spanned by principal divisors.

We begin with the construction of Kudla’s Green functions $\mathsf{Gr}^{\mathsf{K}}(m,v)$ . In light of the complex uniformization (4.2), it suffices to specify $\mathsf{Gr}^{\mathsf{K}}(m,v)$ on each component $[\unicode[STIX]{x1D6E4}_{{\mathcal{L}}_{0},{\mathcal{L}}_{1}}\backslash \mathbb{D}({\mathcal{V}})]$ where, as before,

In § 3.1, we defined an $S({\mathcal{L}})^{\vee }$ -valued current $\mathsf{Gr}_{o}^{\mathsf{K}}(m,v)$ on $\mathbb{D}^{o}({\mathcal{V}}_{[\mathbb{Q}]})$ , with singularities along the cycle $Z(m)$ ; here ${\mathcal{V}}_{[\mathbb{Q}]}$ is the space ${\mathcal{V}}$ viewed as a quadratic space over $\mathbb{Q}$ of signature $(2n-2,2)$ . In parallel with Remark 4.4, we may restrict $\mathsf{Gr}_{o}^{\mathsf{K}}(m,v)$ to ${\mathcal{S}}^{\vee }$ , and pull back to the unitary Grassmannian $\mathbb{D}({\mathcal{V}})$ to obtain a $\unicode[STIX]{x1D6E4}_{{\mathcal{L}}_{0},{\mathcal{L}}_{1}}$ -invariant function on $\mathbb{D}({\mathcal{V}})$ ; descending to the quotient $[\unicode[STIX]{x1D6E4}_{{\mathcal{L}}_{0},{\mathcal{L}}_{1}}\backslash \mathbb{D}({\mathcal{V}})]$ gives a function $\mathsf{Gr}^{\mathsf{K}}(m,v)$ with singularities along the pullback of ${\mathcal{Z}}_{{\mathcal{V}}}(m)$ to $[\unicode[STIX]{x1D6E4}_{{\mathcal{L}}_{0},{\mathcal{L}}_{1}}\backslash \mathbb{D}({\mathcal{V}})]$ .

Putting the components together in the complex uniformization (4.2), we obtain a function $\mathsf{Gr}^{\mathsf{K}}(m,v)$ on ${\mathcal{M}}_{{\mathcal{V}}}(\mathbb{C})$ with logarithmic singularities along ${\mathcal{Z}}_{{\mathcal{V}}}(m)(\mathbb{C})$ .

Note that if $m\leqslant 0$ , then $\mathsf{Gr}^{\mathsf{K}}(m,v)$ is smooth.

As we wish to work with the toroidal compactification ${\mathcal{M}}_{{\mathcal{V}}}^{\ast }$ , we also need to understand the behaviour of $\mathsf{Gr}^{\mathsf{K}}(m,v)$ at the boundary. By the discussion in [Reference HowardHow15, §2.6], the components of the boundary $\unicode[STIX]{x2202}{\mathcal{M}}_{{\mathcal{V}}}^{\ast }={\mathcal{M}}_{{\mathcal{V}}}^{\ast }-{\mathcal{M}}_{{\mathcal{V}}}$ are parametrized by the set of isomorphism classes of triples

If $B=({\mathcal{L}}_{0},\mathfrak{m}\subset {\mathcal{L}}_{1})\in {\mathcal{B}}_{{\mathcal{V}}}$ , then

is an isotropic rank-1 submodule of ${\mathcal{L}}=\operatorname{Hom}_{o_{\boldsymbol{k}}}({\mathcal{L}}_{0},{\mathcal{L}}_{1})$ , and

is a self-dual Hermitian $o_{\boldsymbol{k}}$ -lattice of signature $(n-2,0)$ . There is a contraction map

defined by the formula

and is equivariant for the action of the Weil representation on $S({\mathcal{L}})$ and $S(\unicode[STIX]{x1D6EC}_{B})$ , cf. [Reference BorcherdsBor98, §5]. In the above sum, the equality $\unicode[STIX]{x1D707}|_{\mathfrak{n}^{\bot }}=\unicode[STIX]{x1D708}$ is to be interpreted as follows: a coset $\unicode[STIX]{x1D707}\in \unicode[STIX]{x2202}_{\boldsymbol{k}}^{-1}{\mathcal{L}}/{\mathcal{L}}$ determines a map

by choosing any representative $\tilde{\unicode[STIX]{x1D707}}\in \unicode[STIX]{x2202}_{\boldsymbol{k}}^{-1}{\mathcal{L}}$ and sending $x\in \mathfrak{n}^{\bot }$ to the image of $\langle x,\tilde{\unicode[STIX]{x1D707}}\rangle$ in $\unicode[STIX]{x2202}_{\boldsymbol{k}}^{-1}/o_{\boldsymbol{k}}$ . This is clearly independent of the choice of $\tilde{\unicode[STIX]{x1D707}}$ .

On the other hand, a coset $\unicode[STIX]{x1D708}\in \unicode[STIX]{x2202}_{\boldsymbol{k}}^{-1}\unicode[STIX]{x1D6EC}_{B}/\unicode[STIX]{x1D6EC}_{B}$ determines an element of $\operatorname{Hom}(\unicode[STIX]{x1D6EC}_{B},\unicode[STIX]{x2202}_{\boldsymbol{k}}^{-1}/o_{\boldsymbol{k}})$ by sending $\unicode[STIX]{x1D706}\in \unicode[STIX]{x1D6EC}_{B}$ to the image of $\langle \unicode[STIX]{x1D706},\tilde{\unicode[STIX]{x1D708}}\rangle$ in $\unicode[STIX]{x2202}_{\boldsymbol{k}}^{-1}/o_{\boldsymbol{k}}$ , for any representative $\tilde{\unicode[STIX]{x1D708}}$ of $\unicode[STIX]{x1D708}$ . Pulling back along the projection $\mathfrak{n}^{\bot }\rightarrow \unicode[STIX]{x1D6EC}_{B}$ defines an element of $\operatorname{Hom}(\mathfrak{n}^{\bot },\unicode[STIX]{x2202}_{\boldsymbol{k}}^{-1}/o_{\boldsymbol{k}})$ , and the equation $\unicode[STIX]{x1D707}|_{\mathfrak{n}^{\bot }}=\unicode[STIX]{x1D708}$ is interpreted as the equality of this element with the map (4.8).

Let $\unicode[STIX]{x1D717}_{\unicode[STIX]{x1D6EC}_{B}}\in M_{n-2}(S(\unicode[STIX]{x1D6EC}_{B})^{\vee })$ denote the theta function attached to $\unicode[STIX]{x1D6EC}_{B}$ ; it is a vector-valued modular form of weight $n-2$ defined by the formula

Pulling back along the map $\mathbf{c}_{B}:S({\mathcal{L}})\rightarrow S(\unicode[STIX]{x1D6EC}_{B})$ and restricting to ${\mathcal{S}}\subset S({\mathcal{L}})$ , yields a theta function

whose Fourier expansion we write as

with coefficients $\unicode[STIX]{x1D707}_{B}(m)\in {\mathcal{S}}^{\vee }$ . Concretely, a straightforward computation yields

As a consequence, for $v\in \mathbb{R}_{{>}0}$ and $m\in \mathbb{Q}_{\neq 0}$ we obtain classes

Here the superscript $\mathsf{K}$ serves as a reminder that Kudla’s Green functions are being used. Note if $m\notin d_{\boldsymbol{k}}^{-1}\mathbb{Z}$ then $\widehat{{\mathcal{Z}}}_{{\mathcal{V}}}^{\mathsf{K}}(m,v)=0$ . It remains to define the ‘constant term’ $\widehat{{\mathcal{Z}}}_{{\mathcal{V}}}^{\mathsf{K}}(0,v)$ , a task we will return to shortly.

Turning to Bruinier’s automorphic Green functions, suppose $m\in \mathbb{Q}_{\neq 0}$ . Recall that in § 3.2, we had considered the $S({\mathcal{L}})^{\vee }$ -valued current $\mathsf{Gr}_{o}^{\mathsf{B}}(m)$ on $\mathbb{D}^{o}({\mathcal{V}}_{[\mathbb{Q}]})$ , defined via the regularized theta lift against the Siegel theta function

here $F_{m}(\unicode[STIX]{x1D70F})$ is the unique weak harmonic Maaß form of weight $2-n$ specified in § 2.3.

Restricting $\mathsf{Gr}_{o}^{\mathsf{B}}(m)$ to ${\mathcal{S}}^{\vee }\subset S({\mathcal{L}})^{\vee }$ , and then to the unitary Grassmannian

yields a $\unicode[STIX]{x1D6E4}_{{\mathcal{L}}_{0},{\mathcal{L}}}$ -invariant current on $\mathbb{D}({\mathcal{V}})$ , which can be viewed as living on the component $[\unicode[STIX]{x1D6E4}_{{\mathcal{L}}_{0},{\mathcal{L}}}\backslash \mathbb{D}({\mathcal{V}})]$ ; repeating this construction for all the components appearing in (4.2) yields an ${\mathcal{S}}^{\vee }$ -valued current on ${\mathcal{M}}_{{\mathcal{V}}}(\mathbb{C})$ that we denote by $\mathsf{Gr}^{\mathsf{B}}(m)$ , with logarithmic singularities along ${\mathcal{Z}}_{{\mathcal{V}}}(m)(\mathbb{C})$ . Included in the following proposition is a description of the behaviour of $\mathsf{Gr}^{\mathsf{B}}(m)$ at the boundary.

Thus for each $m\in \mathbb{Q}$ , we may define an ${\mathcal{S}}^{\vee }$ -valued arithmetic cycle

where the superscript $\mathsf{B}$ reminds us that we are using Bruinier’s automorphic Green functions.

Finally, we turn to the constant terms. Let $\widehat{\unicode[STIX]{x1D714}}^{\mathsf{taut}}$ denote the tautological bundle on ${\mathcal{M}}_{{\mathcal{V}}}^{\ast }$ viewed as an element of $\widehat{\mathsf{CH}}\text{}_{\mathbb{C}}^{1}({\mathcal{M}}_{{\mathcal{V}}})$ . As this bundle plays only a marginal role in our present work, we refer the reader to [Reference Bruinier, Howard and YangBHY15, §6] for its construction, and content ourselves with the remark that when restricted to the open part ${\mathcal{M}}_{{\mathcal{V}}}(\mathbb{C})$ , the first Chern form $c_{1}(\widehat{\unicode[STIX]{x1D714}}^{\mathsf{taut}}|_{{\mathcal{M}}_{{\mathcal{V}}}(\mathbb{C})})$ is a Kähler form.

Set

and similarly, for $v\in \mathbb{R}_{{>}0}$ , let