2.1 The Shimura variety

As in the introduction, we denote by $G=\mathrm{GSpin}(V)$ the group of spinor similitudes of V. By construction, G is an algebraic subgroup of the group of units of the Clifford algebra C(V). The bilinear form associated to the quadratic form Q is denoted by

(2.1) \begin{equation} [x,y]=Q(x+y)-Q(x)-Q(y). \end{equation}

If we define a Hermitian symmetric domain

$$\mathcal{D} = \{ z\in V_\mathbb C : [z,z]=0, [z,\overline{z}] \lt 0 \} / \mathbb C^\times \subset \mathbb{P}(V_\mathbb C),$$

then the pair $(G,\mathcal{D})$ is a Hodge-type Shimura datum with reflex field $\mathbb Q$ .

Any choice of compact open subgroup in $G(\mathbb A_f)$ determines a Shimura variety, but we shall only consider subgroups of a particular type. Fix a $\mathbb Z$ -lattice $L\subset V$ satisfying $Q(L) \subset \mathbb Z$ , and let $L^\vee$ denote the dual lattice relative to the bilinear form (2.1). For every prime p, abbreviate $L_p=L\otimes \mathbb Z_p$ , and let $C(L_p) \subset C(V_p)$ be the $\mathbb Z_p$ -subalgebra generated by $L_p \subset C(V_p)$ . The compact open subgroup

(2.2) \begin{equation}K_p = G(\mathbb Q_p) \cap C(L_p)^\times\end{equation}

of $G(\mathbb Q_p)$ is the largest one that stabilizes the lattice $L_p$ and acts trivially on the discriminant group $L_p^\vee / L_p$ . The compact open subgroup

(2.3) \begin{equation}K = \prod_p K_p \subset G (\mathbb A_f)\end{equation}

determines a complex orbifold

$$M (\mathbb C) = G (\mathbb Q) \backslash \mathcal{D} \times G (\mathbb A_f) / K,$$

whose canonical model $M \to \mathrm{Spec}(\mathbb Q)$ is a smooth Deligne–Mumford stack of dimension n.

Remark 2.1.2. As the derived subgroup $\mathrm{Spin}(V)\subset G$ is simply connected, we find by [Reference DeligneDel71, (2.7.1)] that the space of connected components of $M(\mathbb C)$ is a torsor under

$$\pi_0(\mathbb A^\times/\mathbb Q^\times\nu(K)\mathbb R_{\gt 0})\cong \mathbb A_f^\times/\mathbb Q_{\gt 0}^\times\nu(K),$$

where $\nu:G\to \mathbb{G}_m$ is the spinor norm. It follows that if $\nu(K) = \widehat{\mathbb Z}^\times$ , then M is geometrically connected. This holds in particular if L contains isotropic vectors $\ell,\ell_*\in L$ with $[\ell,\ell_*] = 1$ .

Suppose $V^\sharp$ is a quadratic space of signature $(n^\sharp,2)$ , and let $(G^\sharp, \mathcal{D}^\sharp)$ be the associated Shimura datum. A $\mathbb Z$ -lattice $L^\sharp \subset V^\sharp$ on which the quadratic form is $\mathbb Z$ -valued determines a compact open subgroup $K^\sharp \subset G^\sharp(\mathbb A_f)$ as in (2.3), and hence a Shimura variety $M^\sharp$ over $\mathbb Q$ .

An isometric embedding $L\hookrightarrow L^\sharp$ determines an injection of Clifford algebras $C(V) \to C(V^\sharp)$ , which then induces a closed immersion of algebraic groups $G \hookrightarrow G^\sharp$ exhibiting G as the pointwise stabilizer of the orthogonal complement of $V \subset V^\sharp$ . This embedding of groups induces an embedding of Shimura data

$$(G , \mathcal{D})\to (G^\sharp ,\mathcal{D}^\sharp).$$

As $K \subset K^\sharp \cap G(\mathbb A_f)$ , the theory of canonical models implies the existence of a finite and unramified morphism

(2.4) \begin{equation}M \to M^\sharp\end{equation}

of Deligne–Mumford stacks, given on $\mathbb C$ -points by

$$G(\mathbb Q) \backslash \mathcal{D} \times G (\mathbb A_f) / K \xrightarrow{ ( z,g) \mapsto (z,g) } G^\sharp (\mathbb Q) \backslash \mathcal{D} ^\sharp \times G^\sharp (\mathbb A_f) / K^\sharp.$$

More generally, for any $g\in G^\sharp (\mathbb A_f)$ we may replace L by the quadratic lattice $L_g = V \cap g L^\sharp_{\widehat{\mathbb Z}}$ throughout the previous discussion. The compact open subgroup associated to this lattice is

$$K_g = gK^\sharp g^{-1}\cap G(\mathbb A_f),$$

and the associated Shimura variety $M_g$ admits a finite unramified morphism

(2.5) \begin{equation}M_g \to M^\sharp\end{equation}

given on $\mathbb C$ -points by

$$G(\mathbb Q) \backslash \mathcal{D} \times G (\mathbb A_f) / K_g\xrightarrow{ ( z,h) \mapsto (z,hg) }G^\sharp(\mathbb Q) \backslash \mathcal{D} ^\sharp \times G^\sharp (\mathbb A_f) / K^\sharp.$$

2.2 Integral models and special cycles

We use [Reference Howard and Madapusi PeraHMP20, 6] as our primary reference for the theory of integral models of M. See also [Reference Madapusi PeraKMP16, Reference Andreatta, Goren, Howard and Madapusi PeraAGH+17, Reference Andreatta, Goren, Howard and Madapusi PeraAGH+18].

• • $L_p$ is self-dual; or

• • $p=2$ , $\dim_\mathbb Q(V)$ is odd, and $[ L^\vee_2 : L_2 ]$ is not divisible by 4.

We call L maximal or hyperspecial if $L_p$ has this property for every p.

Remark 2.2.2. Note that

$$ L_p \mbox{ self-dual} \implies L_p \mbox{ hyperspecial} \implies L_p \mbox{ maximal}, $$

and that $ L_p \mbox{ not maximal} \implies p^2 \mbox{ divides } [L^\vee : L].$

Remark 2.2.3. A hyperspecial lattice $L_p$ was called an almost self-dual lattice in [Reference Howard and Madapusi PeraHMP20, Definition 6.1.1]. If $L_p$ is hyperspecial, then (2.2) is a hyperspecial subgroup in the usual sense, justifying the terminology. See [Reference Howard and Madapusi PeraHMP20, 6.3].

As in the introduction, $\Sigma$ will always denote a finite set of primes containing all primes p for which $L_p$ is not maximal. The constructions of [Reference Howard and Madapusi PeraHMP20, 6] provide us with a normal and flat Deligne–Mumford stack

$$\mathcal{M} \to \mathrm{Spec}(\mathbb Z[\Sigma^{-1}])$$

with generic fiber M. Strictly speaking, [Reference Howard and Madapusi PeraHMP20, 6] constructs an integral model over the localization $\mathbb Z_{(p)}$ for any p at which $L_p$ is maximal; these can be collated into a model over $\mathbb Z[\Sigma^{-1}]$ as in [Reference Howard and Madapusi PeraHMP20, 9.1].

• • $p\in \Sigma$ for all primes p such that $p^2$ divides $[ L^\vee : L]$ ;

• • if $L_2$ is not hyperspecial, then $2\in \Sigma$ .

The stack $\mathcal{M}$ is regular, and for any $p \not\in \Sigma$ the localization $\mathcal{M}_{\mathbb Z_{(p)}}$ is the canonical integral model of M in the sense of [Reference Madapusi PeraMP16, Definition 4.3].

Remark 2.2.6. If $p \not\in \Sigma$ is an odd prime with $\mathrm{ord}_p([L^\vee:L]) = 2$ , the model $\mathcal{M}_{\mathbb Z_{(p)}}$ is no longer regular. It does admit a regular resolution constructed by Pappas and Zachos [Reference Pappas and ZachosPZ22], which has a certain canonicity property formulated by Pappas [Reference PappasPap23], and now proven by Daniels and Youcis [Reference Daniels and YoucisDY24]. It would be interesting to extend the results of this paper to these regular integral models as well.

The integral model $\mathcal{M}$ comes with a tautological line bundle

(2.6) \begin{equation}\omega \in \mathrm{Pic}( \mathcal{M} ),\end{equation}

called the line bundle of weight one modular forms in [Reference Howard and Madapusi PeraHMP20, 6.3], whose fiber at a complex point

$$(z,g) \in G (\mathbb Q) \backslash \mathcal{D} \times G(\mathbb A_f) / K \cong \mathcal{M} (\mathbb C)$$

is identified with the isotropic line $\mathbb C z \subset V_\mathbb C$ .

Remark 2.2.7. Recall from (2.4) the morphism $M \to M^\sharp$ of canonical models induced by an isometric embedding $L \hookrightarrow L^\sharp$ . If $\Sigma$ also contains all primes p for which $L_p^\sharp$ is not maximal, then $M^\sharp$ has its own flat and normal integral model

$$\mathcal{M}^\sharp \to \mathrm{Spec}(\mathbb Z[\Sigma^{-1}])$$

and the morphism above extends uniquely to a finite morphism

$$\mathcal{M} \to \mathcal{M}^\sharp.$$

The tautological bundle on $\omega^\sharp$ on the target pulls back to the tautological bundle $\omega$ on the source. See [Reference Howard and Madapusi PeraHMP20, Proposition 6.6.1].

The integral model $\mathcal{M}$ also comes with a Kuga–Satake abelian scheme $\mathcal{A} \to \mathcal{M}.$ For every scheme $S\to \mathcal{M}$ there is a canonical (e.g. functorial in S) subspace

$$ V(\mathcal{A}_S)_\mathbb Q\subset \mathrm{End}(\mathcal{A}_S)_{\mathbb Q} $$

of special quasi-endomorphisms, carrying a positive-definite quadratic form defined by $ Q(x) = x\circ x $ as elements of $\mathbb Q\subset \mathrm{End}(\mathcal{A}_S)_\mathbb Q$ . More generally, for every coset $\mu\in L^\vee/L$ there is a subset

$$ V_\mu(\mathcal{A}_S)\subset V(\mathcal{A}_S)_\mathbb Q $$

of special quasi-endomorphisms with denominator $\mu$ . When $\mu=0$ this agrees with

$$ V(\mathcal{A}_S) \stackrel{\mathrm{def}}{=} V(\mathcal{A}_S)_\mathbb Q\cap \mathrm{End}(\mathcal{A}_S) , $$

and the subsets indexed by distinct cosets are disjoint. Again, we refer the reader to [Reference Howard and Madapusi PeraHMP20, 6] for details.

Remark 2.2.8. For any $x\in V_\mu(\mathcal{A}_S)$ , we have

$$Q(x) \equiv Q(\tilde{\mu})\pmod{\mathbb Z},$$

where $\tilde{\mu}\in L^\vee$ is any lift of $\mu$ . See [Reference Andreatta, Goren, Howard and Madapusi PeraAGH+18, Proposition 4.5.4].

Remark 2.2.9. Suppose $s\in \mathcal{M}(\mathbb C)$ is a complex point corresponding to a pair

$$(z,g) \in G (\mathbb Q) \backslash \mathcal{D} \times G(\mathbb A_f) / K \cong \mathcal{M} (\mathbb C).$$

Recalling that $z\in V_\mathbb C$ is a nonzero isotropic vector, there is a canonical identification

$$V(\mathcal{A}_s)_\mathbb Q = \{x\in V : \, [x,z] = 0\} ,$$

respecting quadratic forms and satisfying

$$V_\mu(\mathcal{A}_s) = \{x\in V :\, x\in g \cdot ( \mu + L_{\widehat{\mathbb Z}} ) \} .$$

Here we are regarding $\mu + L_{\widehat{\mathbb Z}} \subset V_{\mathbb A_f}$ .

Definition 2.2.10. For $t\in \mathbb Q$ and $\mu \in L^\vee/L$ , the special divisor

(2.7) \begin{equation}\mathcal{Z} (t,\mu) \to \mathcal{M}\end{equation}

is the finite, unramified, and relatively representable $\mathcal{M}$ -stack whose functor of points assigns to any scheme $S\to \mathcal{M}$ the set

$$\mathcal{Z}(t,\mu)(S) = \{ x \in V_\mu(\mathcal{A}_S) : Q(x) = t \}.$$

The definition of special divisors can be generalized as follows.

Definition 2.2.11. Given an integer $d \ge 1$ , a matrix $T\in \mathrm{Sym}_d(\mathbb Q)$ , and a tuple of cosets $\mu = (\mu_1 ,\ldots, \mu_d) \in (L^\vee / L)^d,$ the special cycle

(2.8) \begin{equation}\mathcal{Z}(T,\mu) \to \mathcal{M}\end{equation}

is the finite, unramified, and relatively representable $\mathcal{M}$ -stack whose functor of points assigns to a scheme $S \to \mathcal{M}$ the set of all tuples

$$x=(x_1,\ldots, x_d) \in V_{\mu_1}(\mathcal{A}_S) \times \cdots \times V_{\mu_d}(\mathcal{A}_S),$$

whose moment matrix

(2.9) \begin{equation}Q(x) \stackrel{\mathrm{def}}{=} \Big( \frac{[x_i,x_j]}{2} \Big) \in \mathrm{Sym}_d(\mathbb Q)\end{equation}

satisfies $Q(x)=T$ .

2.3 Special cycles as Shimura varieties

Given a special cycle (2.8), we explain how to write its generic fiber

$$Z(T,\mu) = \mathcal{Z}(T,\mu)_\mathbb Q$$

as a disjoint union of Shimura varieties. We may assume that $T\in \mathrm{Sym}_d(\mathbb Q)$ is positive semi-definite, as otherwise $Z(T,\mu)=\emptyset$ by Remark 2.2.12.

Endow the space of column vectors $\mathbb Q^{d}$ with the (possibly degenerate) quadratic form $Q ( w ) = {}^tw T w ,$ let $\mathrm{rad}(Q) \subset \mathbb Q^d$ be its radical, and define a positive definite quadratic space

(2.10) \begin{equation}W = \mathbb Q^d / \mathrm{rad}(Q)\end{equation}

of dimension $\mathrm{rank}(T)$ . Let $e_1,\ldots, e_d \in W$ be the images of the standard basis vectors in $\mathbb Q^d$ . Using the notation (2.9), the tuple $e=(e_1,\ldots, e_d)$ has moment matrix $Q(e)=T$ .

If S is any scheme, a morphism $S\to \mathcal{Z}(T,\mu)$ determines a tuple

$$x = (x_1,\ldots, x_d) \in V(\mathcal{A}_S)_\mathbb Q^d$$

with $Q(x) = T$ , and hence an isometric embedding

(2.11) \begin{equation}W\xrightarrow{e_i \mapsto x_i} V(\mathcal{A}_S )_\mathbb Q.\end{equation}

Proof. Using Remark 2.2.9, a complex point $s\in Z(T,\mu)(\mathbb C)$ determines an isometric embedding

By Lemma 2.3.1 we may assume there exists an isometric embedding $W \hookrightarrow V$ , which we now fix. Any two embeddings lie in the same $G(\mathbb Q)$ -orbit, so the particular choice is unimportant. Let $V^\flat \subset V$ be the orthogonal complement of W, so that $V^\flat$ has signature $(n^\flat,2)$ with $n^\flat = n -\mathrm{rank}(T)$ , and

$$V = V^\flat \oplus W.$$

Applying the constructions of § 2.1 to $V^\flat$ , we obtain a reductive group $G^\flat = \mathrm{GSpin}(V^\flat)$ and an embedding of Shimura data

$$(G^\flat,\mathcal{D}^\flat )\hookrightarrow (G,\mathcal{D}).$$

The vectors $e_1,\ldots, e_d \in W \subset V$ determine a subset

$$\Xi( T, \mu ) \stackrel{\mathrm{def}}{=} \{ g\in G(\mathbb A_f) : e_i\in g \cdot ( \mu_i + L_{\widehat{\mathbb Z}} ) \mbox{ for all } 1\le i \leq d \} ,$$

in which we regard $\mu + L_{\widehat{\mathbb Z}} \subset V_{\mathbb A_f}$ , and each $g \in \Xi( T, \mu )$ determines $\mathbb Z$ -lattices

(2.12) \begin{equation} L_g^\flat = V^\flat \cap g L_{\widehat{\mathbb Z}} ,\quad \Lambda_g = W \cap g L_{\widehat{\mathbb Z}}. \end{equation}

As the quadratic form on $V^\flat$ is $\mathbb Z$ -valued on $L^\flat_g$ , the constructions of § 2.1 associate to it a Shimura datum $(G^\flat, \mathcal{D}^\flat)$ , Shimura variety $M^\flat_g$ over $\mathbb Q$ , and a finite unramified morphism $M^\flat_g \to M$ as in (2.5).

Proposition 2.3.2. The set $\Xi(T,\mu)$ is stable under left multiplication by $G^\flat(\mathbb A_f)\subset G(\mathbb A_f)$ and right multiplication by the compact open subgroup $K \subset G(\mathbb A_f)$ of (2.3). The Shimura variety $M_g^\flat$ depends only on the double coset $G^\flat(\mathbb Q) g K$ , and there is an isomorphism of M-stacks

(2.13) \begin{equation}\bigsqcup_{ g \in G^\flat(\mathbb Q)\backslash \Xi(T,\mu)/K } M^\flat_g \cong Z(T, \mu).\end{equation}

Proof. Only the decomposition (2.13) is nontrivial. For that we use Remark 2.2.9 to identify points of $Z(T,\mu)(\mathbb C)$ with

$$G(\mathbb Q) \big\backslash \left\{ (z,x,g) \in \mathcal{D} \times V^d \times G(\mathbb A_f) :\begin{array}{c} Q(x) = T \\ { [z,x_i] =0, \ \forall\, 1 \le i \le d } \\ { x_i \in g \cdot( \mu_i + L_{\widehat{\mathbb Z}}) , \ \forall\, 1 \le i \le d } \end{array} \right\} \big/ K. $$

The key point is that the group $G(\mathbb Q)$ acts transitively on the set of tuples $x\in V^d$ satisfying $Q(x)=T$ . Thus, any element of the double quotient above is represented by a triple of the form (z,e,g), where $e=(e_1,\ldots, e_d) \in W^d \subset V^d$ . As the stabilizer of e is precisely $G^\flat(\mathbb Q) \subset G(\mathbb Q)$ , and the condition $[z,e_i]=0$ for $1\le i \le d$ is equivalent to $z\in \mathcal{D}^\flat \subset \mathcal{D}$ , we may rewrite the double quotient above as

$$ Z(T,\mu)(\mathbb C) \cong G^\flat(\mathbb Q) \backslash \mathcal{D}^\flat \times \Xi(T,\mu) / K. $$

Over the complex fiber, the decomposition (2.13) follows easily from this. In particular, for every g, we have maps $M_{g,\mathbb C}\to Z(T,\mu)_{\mathbb C}$ of finite unramified stacks over $M_{\mathbb C}$ . To finish, it is enough to know that these maps descend over $\mathbb Q$ : this will give the map underlying the isomorphism (2.13), and that it is an isomorphism can be checked over $\mathbb C$ . The desired descent to $\mathbb Q$ is, in fact, a consequence of the theory of canonical models for Shimura varieties and uses the moduli interpretation of the cycle $Z(T,\mu)$ ; see the argument in [Reference Madapusi PeraMP16, Proposition 6.5].

2.4 Basic properties of special cycles

First we explain in what sense the morphisms (2.7), which are not closed immersions, deserve to be called special divisors.

Definition 2.4.1. Suppose $D\to X$ is any finite, unramified, and relatively representable morphism of Deligne–Mumford stacks. By [Sta22, Tag 04HG] there is an étale cover $U \to X$ by a scheme such that the pullback $D_U\to U$ is a finite disjoint union

$$D_U = \bigsqcup_i D_U^i$$

with each map $D_U^i \to U$ a closed immersion. If each of these closed immersions is an effective Cartier divisor on U in the usual sense (the corresponding ideal sheaves are invertible), then we call $D\to X$ a generalized Cartier divisor.

1. (1) If $t \gt 0$ , then $\mathcal{Z}(t,\mu) \to \mathcal{M} $ is a generalized Cartier divisor.

2. (2) If $t \lt 0$ , then $\mathcal{Z}(t,\mu) =\emptyset$ .

3. (3) If $t=0$ , then

   $$\mathcal{Z}(0,\mu) = \begin{cases}\mathcal{M} & \mbox{if }\mu=0 ,\\\emptyset& \mbox{if }\mu\neq 0.\end{cases}$$

For future reference, we recall the main ingredient of the proof of condition (1). Suppose the following

is a commutative diagram of stacks in which $S \to \widetilde{S}$ is a closed immersion of schemes defined by an ideal sheaf $J \subset \mathcal O_{\widetilde{S}}$ with $J^2=0$ . The top horizontal arrow corresponds to a special quasi-endomorphism $x \in V_\mu(\mathcal{A}_S),$ and we want to know when x lies in the image of the (injective) restriction map

(2.14) \begin{equation}V_\mu(\mathcal{A}_{\widetilde{S}}) \to V_\mu(\mathcal{A}_S).\end{equation}

Equivalently, when there is a (necessarily unique) dotted arrow

making the diagram commute. In this situation, [Reference Howard and Madapusi PeraHMP20, Proposition 6.5.1] provides us with a canonical section

\begin{equation*}\mathrm{obst}_x \in H^0 ( \widetilde{S} , \omega|^{-1}_{\widetilde{S}} ),\end{equation*}

called the obstruction to deforming x, with the property that x lies in the image of (2.14) if and only if $\mathrm{obst}_x=0$ .

Using this and Nakayama’s lemma, one shows that at any geometric point $z \to \mathcal{Z}(t,\mu)$ , the kernel of the natural surjection

$$ \mathcal O^\mathrm{et}_{\mathcal{M} , z} \to \mathcal O^\mathrm{et}_{\mathcal{Z}(t,\mu) , z}$$

is a principal ideal, and condition (1) follows from this. Again, see [Reference Howard and Madapusi PeraHMP20, Proposition 6.5.2] for details.

Proposition 2.4.4. Fix a special cycle (2.8). Every irreducible component $\mathcal{Z} \subset \mathcal{Z}(T,\mu)$ satisfies

$$ \mathrm{dim}( \mathcal{Z} ) \ge \mathrm{dim}(\mathcal{M})-\mathrm{rank}(T). $$

If equality holds, then $\mathcal{Z}(T,\mu)$ is a local complete intersection over $\mathbb Z[\Sigma^{-1}]$ at every point of $\mathcal{Z}$ .

Proof. For any geometric point $z \to \mathcal{Z}(T,\mu)$ , the kernel of the natural surjection

$$ \mathcal O^\mathrm{et}_{\mathcal{M},z} \to \mathcal O^\mathrm{et}_{\mathcal{Z}(T,\mu),z}$$

is generated by $d = \mathrm{rank}(T)$ elements. This follows from Nakayama’s lemma and the deformation theory used in the proof of Proposition 2.4.3; see also [Reference Madapusi PeraMP16, Corollary 5.17]. From this, the asserted inequality is immediate. Moreover, it is clear that $\mathcal{Z}(T,\mu)$ is a local complete intersection at z whenever

$$\dim( \mathcal O^\mathrm{et}_{\mathcal{Z}(T,\mu),z} ) = \dim( \mathcal O^\mathrm{et}_{\mathcal{M},z} )- d = \dim(\mathcal{M}) - \mathrm{rank}(T).$$

Proposition 2.4.5. For any special cycle (2.8) and any $A\in \mathrm{GL}_d(\mathbb Z)$ , there is an isomorphism of $\mathcal{M}$ -stacks

$$\mathcal{Z}(T ,\mu ) \cong \mathcal{Z}({}^tA T A , \mu A).$$

Proof. Given a scheme $S \to \mathcal{M}$ , the isomorphism sends a tuple

$$(x_1,\ldots, x_d) \in V(\mathcal{A}_S)_\mathbb Q^d$$

to the tuple $(x_1,\ldots, x_d) \cdot A \in V(\mathcal{A}_S)_\mathbb Q^d$ .

Proposition 2.4.6. Given positive integers d’ and d”, symmetric matrices

$$T' \in \mathrm{Sym}_{d'}(\mathbb Q) \quad \mbox{and}\quad T'' \in \mathrm{Sym}_{d''}(\mathbb Q),$$

and tuples $\mu' \in (L^\vee/L)^{d'}$ and $\mu'' \in (L^\vee / L)^{d''}$ , there is a canonical isomorphism of $\mathcal{M}$ -stacks

$$\mathcal{Z}(T',\mu') \times_{\mathcal{M}} \mathcal{Z}(T'',\mu'') \cong \bigsqcup_{ T = (\begin{smallmatrix} T' & * \\ * & T'' \end{smallmatrix} ) } \mathcal{Z}(T,\mu)$$

where theisjoint union is over all $T\in \mathrm{Sym}_{d'+d''}(\mathbb Q)$ of the indicated form, and $\mu = ( \mu',\mu'') \in (L^\vee/L)^{d'+d''}$ is the concatenation of the tuples $\mu'$ and $\mu''$ .

Proof. For any $\mathcal{M}$ -scheme S, the S-valued points on both sides can be identified with the set of tuples

$$(x',x'')\in \prod_{i=1}^{d'} V_{\mu'_i}(\mathcal{A}_S)\times\prod_{j=1}^{d''} V_{\mu''_{j}}(\mathcal{A}_S)$$

such that $Q(x')=T'$ and $Q(x'') = T''$ .

Suppose we have an isometric embedding $L\hookrightarrow L^\sharp$ as in the discussion leading to (2.4). As in Remark 2.2.7, we assume that $\Sigma$ contains all primes p for which $L_p^\sharp$ is not maximal, so that there is a morphism of integral models

$$ \mathcal{M} \to \mathcal{M} ^\sharp$$

over $\mathbb Z[\Sigma^{-1}]$ . The target of this morphism has its own special cycles

$$\mathcal{Z}^\sharp(T^\sharp,\mu^\sharp) \to \mathcal{M}^\sharp$$

indexed by $T^\sharp \in \mathrm{Sym}_d(\mathbb Q)$ and $\mu^\sharp \in (L^{\sharp,\vee}/ L^\sharp)^d$ , and we wish to describe their pullbacks to $\mathcal{M}$ .

Denoting by $\Lambda \subset L^\sharp$ the set of vectors orthogonal to L, there are inclusions of lattices

$$L \oplus \Lambda \subset L^\sharp \subset L^{\sharp,\vee} \subset L^\vee \oplus \Lambda^\vee .$$

Given cosets

$$\mu \in L^\vee/L ,\quad \nu \in \Lambda^\vee /\Lambda, \quad \mu^\sharp \in L^{\sharp,\vee}/L^\sharp,$$

we write $\mu+ \nu= \mu^\sharp$ to indicate that the natural map

$$(L^\vee \oplus \Lambda^\vee) / (L \oplus \Lambda) \to (L^\vee \oplus \Lambda^\vee ) / L^\sharp$$

sends

$$\mu + \nu \mapsto \mu^\sharp \in L^{\sharp,\vee} / L^\sharp \subset ( L^\vee \oplus \Lambda^\vee ) / L^\sharp .$$

Proposition 2.4.7. There is an isomorphism of $\mathcal{M}$ -stacks

$$ \mathcal{Z}^\sharp (T^\sharp, \mu^\sharp) \times_{ \mathcal{M}^\sharp} \mathcal{M}\cong \bigsqcup_{ \substack{ T \in \mathrm{Sym}_d(\mathbb Q) \\ \mu\in (L^\vee / L)^d } } \bigsqcup_{ \substack{ \nu \in ( \Lambda^\vee/\Lambda)^d \\ \mu+\nu = \mu^\sharp } } \bigsqcup_{ \substack{ y\in \nu + \Lambda^d \\ T+ Q(y) = T^\sharp } } \mathcal{Z} ( T , \mu ),$$

where $\mu+\nu = \mu^\sharp$ is understood as above, but componentwise (that is, $\mu_i +\nu_i = \mu^\sharp_i$ for every $1\le i \le d$ ).

Proof. By [Reference Howard and Madapusi PeraHMP20, Proposition 6.6.2], for any scheme $S \to \mathcal{M}$ there is a canonical isometric embedding

$$V(\mathcal{A}_S)\oplus \Lambda \hookrightarrow V(\mathcal{A}^\sharp_S).$$

Here $\mathcal{A} \to \mathcal{M}$ and $\mathcal{A}^\sharp \to \mathcal{M}^\sharp$ are the Kuga–Satake abelian schemes, and the $\oplus$ on the left is the orthogonal direct sum. This embedding determines a $\mathbb Q$ -linear isometry

$$V(\mathcal{A}^\sharp_S)_\mathbb Q \cong V(\mathcal{A}_S)_\mathbb Q \oplus \Lambda_\mathbb Q ,$$

which restricts to a bijection

$$V_{\mu^\sharp}(\mathcal{A}^\sharp_S) \cong \bigsqcup_{ \substack{ \mu \in L^\vee /L \\ \nu \in \Lambda^\vee/ \Lambda \\ \mu+\nu = \mu^\sharp } }V_{\mu}(\mathcal{A}_S) \times (\nu+\Lambda)$$

for every $\mu^\sharp\in L^{\sharp, \vee} /L^\sharp$ . The proposition follows easily from this and the definition of special cycles.

3.1 Connectedness in low codimension

Our notion of smallness of $\mathrm{rank}(T)$ is always relative to the fixed lattice L, and depends on the following integer r(L) associated to it.

Remark 3.1.2. For any $g\in G(\mathbb A_f)$ we have

$$r(L) = r(L_g),$$

where $L_g = V \cap g L_{\widehat{\mathbb Z}}$ . This is immediate from the fact that if L embeds isometrically as a $\mathbb Z$ -module direct summand of $L^\sharp$ , then $L_g$ embeds isometrically as a $\mathbb Z$ -module direct summand of $L^\sharp_g = V^\sharp \cap g{L^\sharp_{\widehat{\mathbb Z}}}$ .

Proposition 3.1.3. Suppose $T\in \mathrm{Sym}_d(\mathbb Q)$ and $\mu \in (L^\vee/L)^d$ . If

$$ \mathrm{rank}(T) \leq \frac{ n - 2 r(L) -4 }{3} ,$$

then every connected component of the generic fiber $ Z(T,\mu) = \mathcal{Z}(T,\mu)_\mathbb Q $ is geometrically connected.

In general, if N is a quadratic $\mathbb Z$ -module with $N_\mathbb Q$ nondegenerate, let $\gamma(N)$ the minimal number of elements needed to generate the finite abelian group $N^\vee / N$ . This quantity only depends on the $\widehat{\mathbb Z}$ -quadratic space $N_{ \widehat{\mathbb Z}}$ . Moreover, if we realize $N \subset N^\sharp$ as a $\mathbb Z$ -module direct summand of a self-dual quadratic $\mathbb Z$ -module as in Definition 3.1.1, there is a canonical surjection

$$N^\sharp\cong N^{\sharp,\vee}\to N^\vee$$

whose restriction to N is just the inclusion $N\to N^\vee$ . The induced surjection $N^\sharp / N \to N^\vee / N$ shows that

$$\gamma(N) \le \mathrm{rank}_\mathbb Z(N^\sharp) - \mathrm{rank}_\mathbb Z(N).$$

As in Remark 3.1.2, set $L_g = V \cap g L_{\widehat{\mathbb Z}}$ and abbreviate

$$r=r(L)=r(L_g).$$

Fix an embedding $L_g \to L^\sharp$ as a $\mathbb Z$ -module direct summand of a self-dual quadratic lattice of signature $(n+r,2)$ . As the submodule $L^\flat_g \subset L_g$ of (2.12) is a $\mathbb Z$ -module direct summand, the paragraph above implies

$$\gamma(L_g^\flat ) \le \mathrm{rank}_\mathbb Z(L^\sharp) - \mathrm{rank}_\mathbb Z(L_g^\flat)= \mathrm{rank}(T) +r .$$

This implies the first inequality in

$$2 \cdot \gamma(L_g^\flat) +6 \le 2 \cdot \mathrm{rank}(T) + 2r + 6 \le n - \mathrm{rank}(T) + 2 = \mathrm{rank}_\mathbb Z (L_g^\flat)$$

(the second is by the hypotheses of the proposition), and so Proposition B.1.2 implies the existence of the desired isotropic vectors $\ell,\ell_* \in L_g^\flat$ .

3.2 Geometric properties in low codimension: the self-dual case

In this subsection, we assume that L is self-dual. In particular, L is hyperspecial (Definition 2.2.1), and the integral model $\mathcal{M}$ is a smooth $\mathbb Z[\Sigma^{-1}]$ -stack by the proof of Proposition 2.2.4.

Let $\Lambda$ be a positive-definite quadratic $\mathbb Z$ -module. Set

$$\mathsf{L}(\Lambda) = \{ \mathbb Z\mbox{-lattices }\Lambda'\subset \Lambda_\mathbb Q : \;\Lambda\subset \Lambda'\subset(\Lambda')^\vee\subset\Lambda^\vee\},$$

and for each $\Lambda'\in \mathsf{L}(\Lambda)$ , write $\mathcal{Z}(\Lambda')$ for the finite unramified stack over $\mathcal{M}$ with functor of points

$$\mathcal{Z}(\Lambda')(S) = \{\mbox{isometric embeddings }\iota:\Lambda'\hookrightarrow V(\mathcal{A}_S)\}$$

for any scheme $S \to \mathcal{M}$ .

Remark 3.2.1. The above stacks are actually special cycles under a different name. In what follows, we fix a basis $e_1,\ldots, e_d \in \Lambda$ , and let $T=Q(e)\in \mathrm{Sym}_d(\mathbb Q)$ be the moment matrix of $e=(e_1,\ldots, e_d)$ . There is a canonical isometry $\Lambda_\mathbb Q \cong W$ , where the right-hand side is the quadratic space (2.10) determined by T, and a canonical isomorphism of $\mathcal{M}$ -stacks

$$\mathcal{Z}(\Lambda) \cong \mathcal{Z}(T,0)$$

where $0=(0,\ldots,0) \in (L^\vee/L)^d$ . Indeed, an S-valued point of the left-hand side is an isometric embedding $\iota: \Lambda \to V(\mathcal{A}_S)$ , and the tuple $x= (\iota(e_1) , \ldots, \iota(e_d)) \in V(\mathcal{A}_S),$ defines an S-point of the right-hand side.

For each $\Lambda'\in \mathsf{L}(\Lambda)$ , the natural map

$$\mathcal{Z}(\Lambda') \xrightarrow{\iota\mapsto \iota\vert_{\Lambda}}\mathcal{Z}(\Lambda)$$

is a closed immersion. Henceforth, we regard $\mathcal{Z}(\Lambda')$ as a closed substack of $\mathcal{Z}(\Lambda)$ , so that $\mathcal{Z}(\Lambda')\subset \mathcal{Z}(\Lambda'')$ whenever $\Lambda''\subset \Lambda'$ is an inclusion of lattices in $\mathsf{L}(\Lambda)$ . The open substack of $\mathcal{Z}(\Lambda')$ defined by

$${}^{\circ}\mathcal{Z}(\Lambda') = \mathcal{Z}(\Lambda')\smallsetminus \bigcup_{\Lambda'\subsetneq \Lambda''}\mathcal{Z}(\Lambda'')$$

is then a locally closed substack of $\mathcal{Z}(\Lambda)$ .

By construction, we have the equality of sets

(3.1) \begin{equation} \mathcal{Z}(\Lambda)(k) = \bigsqcup_{\Lambda'\in \mathsf{L}(\Lambda)}{}^{\circ}\mathcal{Z}(\Lambda')(k)\end{equation}

for any algebraically closed field k with $\mathrm{char}(k)\not\in \Sigma$ . In fact, given a point $s\in \mathcal{Z}(\Lambda)(k)$ corresponding to an isometric embedding $\iota: \Lambda\to V(\mathcal{A}_s)$ , we have $s\in {}^{\circ}\mathcal{Z}(\Lambda')(k)$ if and only if

$$\Lambda' = V(\mathcal{A}_s)\cap \iota(\Lambda)_\mathbb Q .$$

This last equality says simply that $\Lambda' \subset \Lambda_\mathbb Q$ is the largest lattice such that $\iota$ extends to $\iota : \Lambda' \to V(\mathcal{A}_s)$ .

In the generic fiber we have the following strengthening of (3.1).

Lemma 3.2.3. For every $\Lambda' \in \mathsf{L}(\Lambda)$ the morphism

$${}^{\circ} \mathcal{Z}(\Lambda')_\mathbb Q \to \mathcal{Z}(\Lambda)_\mathbb Q$$

is an open and closed immersion, and there is an isomorphism of $\mathbb Q$ -stacks

$$ \mathcal{Z}(\Lambda)_\mathbb Q \cong \bigsqcup_{\Lambda'\in \mathsf{L}(\Lambda)}{}^{\circ}\mathcal{Z}(\Lambda')_\mathbb Q$$

inducing the bijection (3.1) on geometric points of characteristic 0.

Proof. Proposition 2.3.2 and Remark 3.2.1 give us a decomposition

$$\mathcal{Z}(\Lambda)_\mathbb Q \cong Z(T,0) \cong \bigsqcup_{g\in G^\flat(\mathbb Q)\backslash \Xi(T,0)/K}M^\flat_g ,$$

which depends on a choiceFootnote ¹ of isometric embedding $\Lambda_\mathbb Q \hookrightarrow V.$ Under this bijection, the locally closed substack ${}^{\circ} \mathcal{Z}(\Lambda')_\mathbb Q$ is identified with the disjoint union of those $M^\flat_g$ for which g satisfies $\Lambda' = \Lambda_\mathbb Q\cap g L_{\widehat{\mathbb Z}}$ . Here the intersection is taken inside $V_{\widehat{\mathbb Z}}$ . The lemma follows immediately.

The key geometric result is the following.

1. (1) The $\mathbb Z[\Sigma^{-1}]$ -stack ${}^{\circ}\mathcal{Z}(\Lambda')$ is normal and flat, and equidimensional of dimension

   $$n-\mathrm{rank}_\mathbb Z(\Lambda) + 1 = \dim(\mathcal{M}) - \mathrm{rank}_\mathbb Z(\Lambda).$$

2. (2) For any prime $p \not\in \Sigma$ , the special fiber ${}^{\circ}\mathcal{Z}(\Lambda')_{\mathbb F_p}$ is geometrically normal and equidimensional of dimension $n-\mathrm{rank}_\mathbb Z(\Lambda)$ .

3. (3) For any prime $p\not\in \Sigma$ , the natural maps

   $$ \pi_0({}^{\circ}\mathcal{Z}(\Lambda')_{\mathbb F_p^\mathrm{alg}}) \rightarrow \pi_0({}^{\circ}\mathcal{Z}(\Lambda')_{ \mathbb Z^\mathrm{alg}_{(p)} }) \leftarrow \pi_0({}^{\circ}\mathcal{Z}(\Lambda')_{ \mathbb Q^\mathrm{alg} }) $$
   are bijections, where $\mathbb Z^\mathrm{alg}_{(p)}$ is the integral closure of $\mathbb Z_{(p)}$ in $\mathbb Q^\mathrm{alg}$ .

Proof. We use results from [Reference Howard and Madapusi PeraHMP20, 7.1] to which the reader is encouraged to refer for details. The key point is that, under our hypotheses, there exists an open substack (see [Reference Howard and Madapusi PeraHMP20, Proposition 7.1.2])

$$\mathcal{Z}^{\mathrm{pr}}(\Lambda') \subset {}^{\circ}\mathcal{Z}(\Lambda')$$

with the following properties.

1. (1) It has the same generic fiber as ${}^{\circ}\mathcal{Z}(\Lambda')$ .

2. (2) For any prime $p \not\in \Sigma$ , the special fiber $\mathcal{Z}^{\mathrm{pr}}(\Lambda')_{\mathbb F_p}$ is smooth outside of a codimension-2 substack.

Moreover, [Reference Howard and Madapusi PeraHMP20, Lemma 7.1.5] shows that the complement of $\mathcal{Z}^{\mathrm{pr}}(\Lambda')_{\mathbb F_p}$ in ${}^{\circ}\mathcal{Z}(\Lambda')_{\mathbb F_p}$ has codimension at least 2. The statement there assumes that $\Lambda$ is maximal, but this is only used to ensure that $\Lambda'$ maps to a direct summand of $V(\mathcal{A}_s)$ for every geometric point $s \to {}^{\circ}\mathcal{Z}(\Lambda')_{\mathbb F_p}$ . For us, this holds essentially by definition of ${}^{\circ}\mathcal{Z}(\Lambda')_{\mathbb F_p}$ ; see the comments after (3.1).

Combining the above with the argument of [Reference Howard and Madapusi PeraHMP20, Proposition 7.1.6] proves assertions (1) and (2). Assertion (3) follows from [Reference MadapusiMad25, Theorem B].

Proposition 3.2.4 has two consequences, which are of fundamental importance to our arguments. The first requires the following technical lemma of commutative algebra.

1. (1) R is flat over $\mathbb Z[\Sigma^{-1}]$ ;

2. (2) for every minimal prime $P\subset R$ , $R/P$ is flat over $\mathbb Z[\Sigma^{-1}]$ .

Proof. Fix an irreducible component

$$\mathcal{Z}\subset \mathcal{Z}(\Lambda)=\mathcal{Z}(T,0)$$

and a geometric generic point $s \to \mathcal{Z}$ .Footnote ²

By (3.1), there is a unique $\Lambda'\in \mathsf{L}(\Lambda)$ such that $s\to {}^{\circ}\mathcal{Z}(\Lambda')$ . By claim (1) of Proposition 3.2.4, the stack ${}^{\circ}\mathcal{Z}(\Lambda')$ is equidimensional of dimension $ \dim(\mathcal{M}) - \mathrm{rank}_\mathbb Z(\Lambda),$ and so the same is true of its Zariski closure $ \overline{ {}^{\circ}\mathcal{Z}(\Lambda') } \subset \mathcal{Z}(\Lambda). $ The inclusion $ \mathcal{Z} \subset \overline{ {}^{\circ}\mathcal{Z}(\Lambda') } $ therefore implies

$$ \dim(\mathcal{Z}) \le \dim(\mathcal{M}) - \mathrm{rank}_\mathbb Z(\Lambda). $$

As the other inequality follows from Proposition 2.4.4, we have proved both that $\mathcal{Z}$ has the expected dimension, and that it is eq2.ual to an irreducible component of $ \overline{ {}^{\circ}\mathcal{Z}(\Lambda') }$ . This latter stack is flat over $\mathbb Z[\Sigma^{-1}]$ by claim (1) of Proposition 3.2.4, and hence so is $\mathcal{Z}$ .

It now follows from Proposition 2.4.4 that $\mathcal{Z}(\Lambda)$ is a local complete intersection over $\mathbb Z[\Sigma^{-1}]$ . In particular $\mathcal{Z}(\Lambda)$ is Cohen–Macaulay, and hence flat by Lemma 3.2.5 and the flatness of its irreducible components proved above. To now see that it is reduced, it is enough to know that it is generically smooth, which follows from the complex uniformization in Proposition 2.3.2.

Proposition 3.2.7. If $ \mathrm{rank}_\mathbb Z(\Lambda)\leq (n-4)/3,$ then restriction

\begin{align*}\mathrm{CH}^1(\mathcal{Z}(\Lambda)) \to \mathrm{CH}^1( \mathcal{Z}(\Lambda)_\mathbb Q)\end{align*}

to the generic fiber is injective.

Proof. This amounts to proving the triviality of the subspace

(3.2) \begin{equation}\mathrm{CH}^1_{\mathrm{vert}}(\mathcal{Z}(\Lambda)) \subset \mathrm{CH}^1(\mathcal{Z}(\Lambda))\end{equation}

spanned by the irreducible components of $\mathcal{Z}(\Lambda)_{\mathbb F_p}$ as $p\notin \Sigma$ varies. For this we use the following lemma, which provides a parametrization of those components.

Lemma 3.2.8. For a Deligne–Mumford stack $\mathcal{X}$ , denote by $\pi_{\mathrm{irr}}(\mathcal{X})$ its set of irreducible components. For any prime $p\not\in \Sigma$ there are canonical bijections

(3.3)

characterized as follows.

1. (1) The horizontal arrow takes an irreducible component ${}^{\circ}\mathcal{Z} \subset {}^{\circ} \mathcal{Z}(\Lambda')$ to its reduction ${}^{\circ}\mathcal{Z}_{\mathbb F_p}\subset {}^{\circ} \mathcal{Z}(\Lambda')_{\mathbb F_p}$ .

2. (2) The vertical arrow on the left takes an irreducible component of the locally closed substack ${}^{\circ} \mathcal{Z}(\Lambda') \subset \mathcal{Z}(\Lambda)$ to its Zariski closure.

3. (3) The vertical arrow on the right takes an irreducible component of the locally closed substack ${}^{\circ} \mathcal{Z}(\Lambda')_{\mathbb F_p} \subset \mathcal{Z}(\Lambda)_{\mathbb F_p}$ to its Zariski closure.

Moreover, given distinct irreducible components

$${}^{\circ}\mathcal{Z}_1,{}^{\circ} \mathcal{Z}_2 \subset {}^{\circ} \mathcal{Z}(\Lambda') ,$$

the intersection of ${}^{\circ}\mathcal{Z}_1$ with the Zariski closure of ${}^{\circ} \mathcal{Z}_2$ in $\mathcal{Z}(\Lambda)$ is empty.

Proof. We first show that for any $\Lambda'\in \mathsf{L}(\Lambda)$ , all arrows in the following diagram are bijective.

Claim (3) of Proposition 3.2.4 shows that both horizontal arrows in the top row are bijective. Proposition 3.1.3 and our hypothesis on $\mathrm{rank}(\Lambda)=\mathrm{rank}(T)$ guarantee that every connected component of $\mathcal{Z}(\Lambda)_\mathbb Q = Z(T,0)$ is geometrically connected (note that $r(L)=0$ by our assumption that L is self-dual), and so the same is true of ${}^{\circ} \mathcal{Z}(\Lambda')_\mathbb Q$ by Lemma 3.2.3. This shows that the arrow labeled c is bijective. The morphism

$$ {}^{\circ}\mathcal{Z}(\Lambda')_{\mathbb Z_{(p)}} \to \mathrm{Spec}(\mathbb Z_{(p)}) $$

is flat with reduced special fiber by claims (1) and (2) of Proposition 3.2.4, and so [Sta22, Tag 055J] implies that the arrow labeled e is injective. The arrow labeled b is surjective because it is induced by a surjective morphism of stacks. It follows that all arrows in the square on the right are bijections, as is the composition $d\circ a$ . This implies the injectivity of a, and surjectivity follows by the same reasoning as for b. The arrow labeled d is bijective because, at this point, we know the bijectivity of all the other arrows.

Now we turn to the diagram (3.3). By Proposition 3.2.4 each ${}^{\circ} \mathcal{Z}(\Lambda')$ is normal and flat over $\mathbb Z[\Sigma^{-1}]$ , and so there are canonical identifications

$$\pi_{\mathrm{irr}} ( {}^{\circ} \mathcal{Z}(\Lambda'))= \pi_{\mathrm{irr}} ( {}^{\circ} \mathcal{Z}(\Lambda')_{\mathbb Z_{(p)}})= \pi_0 ( {}^{\circ} \mathcal{Z}(\Lambda')_{\mathbb Z_{(p)}}).$$

Similarly, the normality of ${}^{\circ} \mathcal{Z}(\Lambda')_{\mathbb F_p}$ implies

$$\pi_{\mathrm{irr}} ( {}^{\circ} \mathcal{Z}(\Lambda')_{\mathbb F_p})= \pi_0 ( {}^{\circ} \mathcal{Z}(\Lambda')_{\mathbb F_p }).$$

Combining this with the paragraph above yields the top horizontal bijection in (3.3). The vertical bijections in (3.3) are formal consequences of (3.1) and our dimension calculations; see the proof of Proposition 3.2.6.

The final claim follows from the normality of ${}^{\circ} \mathcal{Z}(\Lambda')$ . Suppose $s \to {}^{\circ} \mathcal{Z}_1$ is a geometric point also contained in the Zariski closure of ${}^{\circ} \mathcal{Z}_2$ in $\mathcal{Z}(\Lambda)$ . This is the same as the Zariski closure of ${}^{\circ} \mathcal{Z}_2$ in $\mathcal{Z}(\Lambda')$ , and hence any open subset of $\mathcal{Z}(\Lambda')$ containing s must intersect ${}^{\circ} \mathcal{Z}_2$ . One such open subset is ${}^{\circ} \mathcal{Z}_1$ itself, and so ${}^{\circ} \mathcal{Z}_1 \cap {}^{\circ} \mathcal{Z}_2 \neq \emptyset$ . This is impossible, as these are distinct connected components of ${}^{\circ}\mathcal{Z}(\Lambda')$ .

Lemma 3.2.8 determines a canonical bijection

(3.4) \begin{equation} \pi_{\mathrm{irr}} ( \mathcal{Z}(\Lambda) ) \to \pi_{\mathrm{irr}} ( \mathcal{Z}(\Lambda)_{\mathbb F_p} ) ,\end{equation}

but this does not send an irreducible component $\mathcal{Z} \subset \mathcal{Z}(\Lambda)$ to its reduction $\mathcal{Z}_{\mathbb F_p}\subset \mathcal{Z}(\Lambda)_{\mathbb F_p} $ . Indeed, this reduction need not be irreducible (or reduced), and an irreducible component of $\mathcal{Z}(\Lambda)_{\mathbb F_p}$ may be contained in $\mathcal{Z}_{\mathbb F_p}$ for more than one $\mathcal{Z}$ . Instead, the bijection sends $\mathcal{Z}$ to a distinguished irreducible component of $\mathcal{Z}_{\mathbb F_p}$ .

We now return to the proof of Proposition 3.2.7. Fix a prime $p\not\in \Sigma$ , a $\Lambda' \in \mathsf{L}(\Lambda)$ , and an irreducible component ${}^{\circ}\mathcal{Z} \subset {}^{\circ}\mathcal{Z}(\Lambda')$ . Using Lemma 3.2.8, we see that the Zariski closure of ${}^{\circ}\mathcal{Z}$ in $\mathcal{Z}(\Lambda)$ is an irreducible component

(3.5) \begin{equation} I(\Lambda', {}^{\circ}\mathcal{Z}) \in \pi_{\mathrm{irr}} ( \mathcal{Z}(\Lambda) ) , \end{equation}

while the Zariski closure of ${}^{\circ}\mathcal{Z}_{\mathbb F_p}$ in $\mathcal{Z}(\Lambda)_{\mathbb F_p}$ is an3. irreducible component

$$ I_p(\Lambda', {}^{\circ}\mathcal{Z}) \in \pi_{\mathrm{irr}} ( \mathcal{Z}(\Lambda)_{\mathbb F_p} )$$

contained in (3.5). These two components correspond under the bijection (3.4), and all irreducible components of $\mathcal{Z}(\Lambda)$ and $\mathcal{Z}(\Lambda)_{\mathbb F_p}$ are of this form.

To prove the triviality of the subspace (3.2), we must therefore prove the triviality of all cycle classes

(3.6) \begin{equation}I_p (\Lambda', {}^{\circ}\mathcal{Z}) \in \mathrm{CH}^1( \mathcal{Z}(\Lambda)) .\end{equation}

This will be by induction on the size of $\mathsf{L}(\Lambda)$ .

The base case is when $\mathsf{L}(\Lambda) = \{\Lambda\}$ , which happens exactly when $\Lambda$ is maximal. In this case ${}^{\circ}\mathcal{Z}(\Lambda) = \mathcal{Z}(\Lambda)$ , and so every irreducible component ${}^{\circ} \mathcal{Z}\subset {}^{\circ} \mathcal{Z}(\Lambda)$ is already Zariski closed in $\mathcal{Z}(\Lambda)$ . It follows that

$$ I_p(\Lambda', {}^{\circ}\mathcal{Z}) = {}^{\circ}\mathcal{Z}_{\mathbb F_p} = I(\Lambda', {}^{\circ}\mathcal{Z})_{\mathbb F_p},$$

which is trivial in $\mathrm{CH}^1(\mathcal{Z}(\Lambda))$ . Indeed, it is the Weil divisor of the rational function on $\mathcal{Z}(\Lambda)$ that is p on $I(\Lambda', {}^{\circ}\mathcal{Z})$ and 1 on all other irreducible components.

We now turn to the inductive step. For any $\Lambda' \in \mathsf{L}(\Lambda)$ , the inclusion $\mathcal{Z}(\Lambda') \subset \mathcal{Z}(\Lambda)$ induces a pushforward (Proposition A.1.3)

$$\mathrm{CH}^1(\mathcal{Z}(\Lambda')) \to \mathrm{CH}^1(\mathcal{Z}(\Lambda)).$$

For an irreducible component ${}^{\circ}\mathcal{Z} \subset {}^{\circ}\mathcal{Z}(\Lambda')$ we have

$$I_p(\Lambda', {}^{\circ}\mathcal{Z}) \in \pi_{\mathrm{irr}} ( \mathcal{Z}(\Lambda')_{\mathbb F_p} ) \subset \pi_{\mathrm{irr}} ( \mathcal{Z}(\Lambda)_{\mathbb F_p} )$$

by construction, and (3.6) is the pushforward of the corresponding class

$$I_p(\Lambda', {}^{\circ}\mathcal{Z}) \in \mathrm{CH}^1(\mathcal{Z}(\Lambda')).$$

If $\Lambda \subsetneq \Lambda'$ , then this last class is trivial by the induction hypothesis, and hence so is (3.6).

It now suffices to show that every $I_p(\Lambda,{}^{\circ}\mathcal{Z})$ is rationally equivalent to 0 on $\mathcal{Z}(\Lambda)$ . Consider the corresponding irreducible component $I(\Lambda,{}^{\circ}\mathcal{Z})$ of $\mathcal{Z}(\Lambda)$ . By the parametrization of the irreducible components of $\mathcal{Z}(\Lambda)_{\mathbb F_p}$ , there is an equality

(3.7) \begin{equation} I(\Lambda,{}^{\circ}\mathcal{Z})_{\mathbb F_p}= \sum_{\substack{ \Lambda' \in \mathsf{L}(\Lambda) \\ {}^{\circ}\mathcal{Z}' \in \pi_{\mathrm{irr}}( {}^{\circ} \mathcal{Z}(\Lambda'))} } m(\Lambda',{}^{\circ} \mathcal{Z}') \cdot I_p(\Lambda' , {}^{\circ}\mathcal{Z}') \in \mathscr{Z}^1(\mathcal{Z}(\Lambda))\end{equation}

for some multiplicities $ m(\Lambda',{}^{\circ} \mathcal{Z}') \in \mathbb Z$ . More precisely, $I(\Lambda,{}^{\circ}\mathcal{Z})_{\mathbb F_p}$ is an effective Cartier divisor on $\mathcal{Z}(\Lambda)$ , and the multiplicities are given by the length of its étale local rings at each of its generic points, each of which of course is the generic point of an irreducible component in $\pi_{\mathrm{irr}}(\mathcal{Z}(\Lambda)_{\mathbb F_p})$ . Note, in particular, that this means that the multiplicity $m(\Lambda,{}^{\circ}\mathcal{Z})$ is nonzero.

First we consider those terms on the right-hand side for which $\Lambda'=\Lambda$ .

Lemma 3.2.10. For any $ {}^{\circ}\mathcal{Z}' \in \pi_{\mathrm{irr}}( {}^{\circ} \mathcal{Z}(\Lambda)) $ we have

$$ m(\Lambda,{}^{\circ} \mathcal{Z}') \neq 0 \iff \mathcal{Z}'=\mathcal{Z}.$$

Proof. By construction, we have

$$ I_p(\Lambda, {}^{\circ}\mathcal{Z}) \subset I(\Lambda, {}^{\circ}\mathcal{Z})_{\mathbb F_p} ,$$

and so $ m(\Lambda ,{}^{\circ} \mathcal{Z}) \neq 0. $ Conversely, if $ m(\Lambda ,{}^{\circ} \mathcal{Z}') \neq 0$ , then

$$ I_p (\Lambda ,{}^{\circ} \mathcal{Z}') \subset I (\Lambda ,{}^{\circ} \mathcal{Z})_{\mathbb F_p}, $$

and, hence, $ {}^{\circ} \mathcal{Z}'_{\mathbb F_p} \subset I (\Lambda ,{}^{\circ} \mathcal{Z})_{\mathbb F_p}. $ In particular, ${}^{\circ} \mathcal{Z}'$ intersects the closure of ${}^{\circ} \mathcal{Z}$ in $\mathcal{Z}(\Lambda)$ , and so $ {}^{\circ} \mathcal{Z}' ={}^{\circ} \mathcal{Z}$ by the final claim of Lemma 3.2.8.

If we take the image of (3.7) in $\mathrm{CH}^1(\mathcal{Z}(\Lambda))$ , the left-hand side vanishes because it the Weil divisor of the rational function on $\mathcal{Z}(\Lambda)$ that is p on $I(\Lambda , {}^{\circ}\mathcal{Z})$ and 1 on all other irreducible components. On the right-hand side we have proven the vanishing of every term with $\Lambda \subsetneq \Lambda'$ , and of every term with $\Lambda'=\Lambda$ and ${}^{\circ}\mathcal{Z}' \neq {}^{\circ} \mathcal{Z}$ . Thus,

$$0 = m(\Lambda,{}^{\circ} \mathcal{Z}) \cdot I_p(\Lambda , {}^{\circ}\mathcal{Z})$$

in the Chow group. As our Chow groups have rational coefficients, it follows that $ I_p(\Lambda , {}^{\circ}\mathcal{Z}) =0$ , completing the proof of Proposition 3.2.7.

3.3 Geometric properties in low codimension: the general case

We now return to the consideration of the general case where L is not necessarily self-dual.

Let $r=r(L)$ and $L \hookrightarrow L^\sharp$ be as in Definition 3.1.1. Thus, $L^\sharp$ is a self-dual quadratic $\mathbb Z$ -module of signature $(n+r,2)$ , containing L as a $\mathbb Z$ -module direct summand. As in Remark 2.2.7, there is an induced finite morphism

$$\mathcal{M}\to \mathcal{M}^\sharp$$

of normal integral models over $\mathbb Z[\Sigma^{-1}]$ . The target comes with its own Kuga–Satake abelian scheme $\mathcal{A}^\sharp \to \mathcal{M}^\sharp$ , and its own family of special cycles

$$ \mathcal{Z}^\sharp( T^\sharp,\mu^\sharp )\to \mathcal{M}^\sharp. $$

Of course, we must have $\mu^\sharp =0$ , by the self-duality of $L^\sharp$ , so we abbreviate

$$ \mathcal{Z}^\sharp( T^\sharp)= \mathcal{Z}^\sharp( T^\sharp, 0). $$

Finally, we arrive at the main result of § 3.

Proposition 3.3.1. Fix $T \in \mathrm{Sym}_d(\mathbb Q)$ and $\mu \in (L^\vee/L)^d$ , and suppose

$$\mathrm{rank}(T) \leq \frac{n-2r-4}{3}.$$

The special cycle $\mathcal{Z}(T,\mu)$ is a flat, reduced, local complete intersection over $\mathbb Z[\Sigma^{-1}]$ , and is equidimensional of codimension $\mathrm{rank}(T)$ in $\mathcal{M}$ . Moreover, restriction to the generic fiber

$$\mathrm{CH}^1(\mathcal{Z}(T,\mu) ) \to \mathrm{CH}^1( Z(T,\mu))$$

is injective.

Proof. If we set $n^\sharp = n + r$ , then every $T^\sharp\in \mathsf{S}$ in Lemma 3.3.2 satisfies

$$\mathrm{rank}(T^\sharp) = \mathrm{rank}(T) +r \le \frac{n^\sharp -4}{3} .$$

Let $W^\sharp$ be the positive-definite quadratic space of rank $\mathrm{rank}(T^\sharp)$ associated to $T^\sharp$ as in (2.10), and let $\Lambda^\sharp \subset W^\sharp$ be the $\mathbb Z$ -lattice spanned by the distinguished generators $e_1,\ldots, e_{n+r} \in W^\sharp.$ As in § 3.2, there is an associated Deligne–Mumford stack

$$\mathcal{Z}^\sharp (\Lambda^\sharp) \to \mathcal{M}^{\sharp}$$

parametrizing isometric embeddings of $\Lambda^\sharp$ into $V(\mathcal{A}^\sharp)$ .

As the tuple $e^\sharp =(e_1,\ldots, e_{n+r})$ has moment matrix $T^\sharp = Q(e^\sharp)$ by construction, Remark 3.2.1 provides us with a canonical isomorphism of $\mathcal{M}^\sharp$ -stacks $\mathcal{Z}^\sharp (\Lambda^\sharp) \cong \mathcal{Z}^\sharp(T^\sharp) .$

We now need the following lemma.

Lemma 3.3.2. There exists a finite subset

$$\mathsf{S} \subset \mathrm{Sym}_{r +d}(\mathbb Q)$$

of positive-semi-definite matrices of rank $r+\mathrm{rank}(T)$ such that there is an open and closed immersion of $\mathcal{M}^\sharp$ -stacks

$$\mathcal{Z}(T,\mu)\hookrightarrow \bigsqcup_{T^\sharp\in \mathsf{S}}\mathcal{Z}^\sharp(T^\sharp) .$$

Proof. Let $\Lambda \subset L^\sharp$ be the orthogonal to $L \subset L^\sharp$ . By the self-duality of $L^\sharp$ there are canonical bijections

$$L^\vee/L \leftarrow L^\sharp/(L\oplus \Lambda)\rightarrow \Lambda^\vee/\Lambda.$$

It follows that there is a (unique) $\nu\in (\Lambda^\vee/\Lambda)^d$ such that $\mu+\nu = 0$ as elements of $ L^\sharp/(L\oplus \Lambda)$ . If we fix a lift $ \tilde{\nu}\in (\Lambda^\vee)^d$ and set

$$T_1 = T+Q(\tilde{\nu}) \in \mathrm{Sym}_d(\mathbb Q),$$

then Proposition 2.4.7 implies that there is an open and closed immersion

(3.8) \begin{equation} \mathcal{Z}(T,\mu) \hookrightarrow \mathcal{Z}^\sharp( T_1 )\times_{\mathcal{M}^\sharp}\mathcal{M}.\end{equation}

As in the proof of Proposition 2.4.7, for any scheme $S \to \mathcal{M}$ there is a canonical isometric embedding

$$V(\mathcal{A}_S)\oplus \Lambda \hookrightarrow V(\mathcal{A}^\sharp_S)$$

for any scheme $S \to \mathcal{M}$ , extending $\mathbb Q$ -linearly to an isomorphism

$$V(\mathcal{A}_S)_\mathbb Q \oplus \Lambda_\mathbb Q \cong V(\mathcal{A}^\sharp_S)_\mathbb Q.$$

In particular, we have a canonical embedding $\Lambda \to V(\mathcal{A}^\sharp_{\mathcal{M}})$ . A choice of basis $y_1,\ldots, y_r \in \Lambda$ therefore determines a morphism

(3.9) \begin{equation}\mathcal{M}\to \mathcal{Z}^\sharp(T_2) \end{equation}

of $\mathcal{M}^\sharp$ -stacks, where $T_2=Q(y)$ is the moment matrix of the tuple

$$y=(y_1,\ldots, y_r) \in \Lambda^r \subset V(\mathcal{A}^\sharp_\mathcal{M})^r.$$

This map is, in fact, an open and closed immersion. Since it is known to be finite, it is enough to know that it is an open immersion. For this, note that both source and target are normal Deligne–Mumford stacks, flat over $\mathbb Z_{(p)}$ : for $\mathcal{Z}^\sharp(T_2)$ , this follows from our numerical hypotheses and Proposition 3.2.6. By [Reference MadapusiMad25, Lemma 5.1.12], it now suffices to check that the map is generically an open immersion. This can be checked using the argument in [Reference Madapusi PeraMP16, Lemma 7.1].

Combining (3.8) and (3.9) yields an open and closed immersion

$$\mathcal{Z}(T,\mu) \to \mathcal{Z}^\sharp( T_1 )\times_{\mathcal{M}^\sharp}\mathcal{Z}^\sharp(T_2)\cong \bigsqcup_{T^\sharp = (\begin{smallmatrix}T_1 &*\\*& T_2\end{smallmatrix}{\kern-1pt})}\mathcal{Z}^\sharp(T^\sharp),$$

where we have used the product formula from Proposition 2.4.6. Explicitly, for any scheme $S\to \mathcal{M}$ , a point of $\mathcal{Z}(T,\mu)(S)$ is given by a tuple $x\in \prod_{i=1}^d V_{\mu_i}(\mathcal{A}_S)$ satisfying $Q(x) = T$ . The immersion sends x to the tuple

$$x^\sharp = (x+\tilde{\nu},y)\in V(\mathcal{A}^\sharp_{S})^d \times V(\mathcal{A}^\sharp_S)^{r}= V(\mathcal{A}^\sharp_S)^{d+r}$$

in the factor indexed by $T^\sharp = Q(x^\sharp)$ .

It remains to show that $\mathcal{Z}(T,\mu)$ only meets those $\mathcal{Z}^\sharp(T^\sharp)$ with $T^\sharp$ positive semi-definite and

$$\mathrm{rank}(T^\sharp) = \mathrm{rank}(T) + r = \mathrm{rank}(T) + \mathrm{rank}_\mathbb Z(\Lambda).$$

This follows from Remark 2.2.12 and the observation (noting that every component of $\tilde{\nu}\in \Lambda_\mathbb Q^d$ is a $\mathbb Q$ -linear combination of $y_1,\ldots, y_r\in \Lambda$ ) that the components of $x^\sharp$ and the components of (x,y) generate the same subspace of $V(\mathcal{A}^\sharp_S)_\mathbb Q = V(\mathcal{A}_S)_\mathbb Q\oplus\Lambda_\mathbb Q$ .

Using Lemma 3.3.2, the desired properties of $\mathcal{Z}(T,\mu)$ follow immediately from the corresponding properties of $\mathcal{Z}^\sharp (\Lambda^\sharp)$ proved in Propositions 3.2.6 and 3.2.7.

4.1 Jacobi forms

We recall just enough of the theory of Jacobi forms to fix our conventions, as these differ slightly from [Reference ZhangZha09] and [Reference Bruinier and Westerholt-RaumBW-R15].

Fix an integer $g\ge 1$ . The Siegel modular group $\Gamma_g=\mathrm{Sp}_{2g}(\mathbb Z)$ acts on the Siegel half-space $\mathcal{H}_g \subset \mathrm{Sym}_g(\mathbb C)$ via the usual formula

$$\gamma \cdot \tau = (A\tau+B)(C\tau+D)^{-1},$$

where we have written

$$\gamma= \Big(\begin{matrix}A\quad &B\\C\quad &D \end{matrix} \Big) \in \Gamma_g.$$

Let $\Gamma_1=\mathrm{SL}_2(\mathbb Z)$ act on the space of matrices $M_{2,g-1}(\mathbb Z)$ by left multiplication. Following [Reference Bruinier and Westerholt-RaumBW-R15], we regard the Jacobi group

$$J_g \stackrel{\mathrm{def}}{=} \Gamma_1 \ltimes M_{2,g-1}(\mathbb Z)$$

as a subset of $\Gamma_g$ using the injective function (not a group homomorphism)

(4.1) \begin{equation}\Big( \Big(\begin{array}{c@{\quad }c}a& b\\c&d \end{array}\Big), \Big(\begin{matrix} {}^tx \\ {}^ty \end{matrix}\Big) \Big)\mapsto\left(\begin{array}{c@{\quad }c | c@{\quad }c}1_{g-1} & - y & 0_{g-1} & x \\0 & a & a\, {}^tx+b\, {}^ty & b \\ \hline0_{g-1} & 0 & 1_{g-1} & 0 \\0 & c & c \, {}^t x + d\, {}^ty & d\end{array}\right)\end{equation}

for column vectors $x,y\in \mathbb Z^{g-1}$ . Note that the restriction of this injective map to the subgroup $\Gamma_1\subset J_g$ is actually a group homomorphism.

Remark 4.1.1. As in [Reference Westerholt-RaumWR15, 4], there is an extended Jacobi group $J_g^{ \mathrm{ext} }$ , which can be realized both as a subgroup of $\Gamma_g$ , and as a central extension

$$1 \to \mathbb Z \to J_g^{ \mathrm{ext} } \to J_g \to 1$$

with the property that the surjection to $J_g$ admits a set-theoretic section whose image generates $J_g^{\mathrm{ext}}$ . The use of the function (4.1) is a convenient way of hiding the presence of the larger extended Jacobi group, as the subset $J_g \subset \Gamma_g$ generates a subgroup isomorphic to $J_g^{ \mathrm{ext} }$ .

The metaplectic double cover of the Siegel modular group is denoted by $\widetilde{\Gamma}_g.$ Its elements are pairs

(4.2) \begin{equation}\tilde{\gamma} = ( \gamma , j_\gamma ) \in \widetilde{\Gamma}_g,\end{equation}

consisting of a $\gamma \in \Gamma_g$ and a holomorphic function $j_\gamma(\tau)$ on $\mathcal{H}_g$ whose square is $\det(C\tau+D)$ . As in [Reference Bruinier and Westerholt-RaumBW-R15, (5)], the metaplectic Jacobi group

$$\widetilde{J}_g \stackrel{\mathrm{def}}{=} \widetilde{\Gamma}_1 \ltimes M_{2,g-1}(\mathbb Z)$$

can be identified with a subset of $\widetilde{\Gamma}_g$ , using an injection lifting (4.1). Moreover, the restriction of this embedding to $\widetilde{\Gamma}_1$ is also a group homomorphism. In this way, we can view $\widetilde{\Gamma}_1$ as a subgroup of $\widetilde{\Gamma}_g$ .

In the following definition, taken from [Reference Bruinier and Westerholt-RaumBW-R15, 2.2], we write elements of the Siegel half-space as

$$\tau = \Big(\begin{array}{c@{\quad }c}\tau'' & z \\{}^t z & \tau' \end{array}\Big) \in \mathcal{H}_g\quad \mbox{with} \ \tau' \in \mathcal{H}_1, \ \tau'' \in \mathcal{H}_{g-1},$$

and $z\in \mathbb C^{g-1}$ a column vector.

Definition 4.1.2. Suppose $ \rho : \widetilde{\Gamma}_g \to \mathrm{GL}(V)$ is a finite-dimensional representation with finite kernel. A holomorphic function

$$\phi_T : \mathcal{H}_1 \times \mathbb C^{g-1} \to V$$

is a Jacobi form of half-integral weight k, index $T\in \mathrm{Sym}_{g-1}(\mathbb Q)$ , and representation $ \rho$ if the function

(4.3) \begin{equation}\Phi_T(\tau) \stackrel{\mathrm{def}}{=} \phi_T(\tau',z)\cdot e^{2\pi i \mathrm{Tr}( T\cdot \tau'')}\end{equation}

on $\mathcal{H}_g$ satisfies the transformation law

$$\Phi_T(\gamma \cdot \tau) = j_\gamma (\tau)^{2k} \rho( \tilde{ \gamma} ) \cdot \Phi_T(\tau)$$

for all elements (4.2) in the subset $\widetilde{J}_g \subset \widetilde{\Gamma}_g$ , and if for all $\alpha,\beta \in \mathbb Q^{g-1}$ the function $\phi_T( \tau' , \tau' \alpha +\beta)$ of $\tau' \in \mathcal{H}_1$ is holomorphic at $\in fty$ .

Any Jacobi form of representation $\rho$ has a Fourier expansion

$$ \phi_T( \tau' , z ) = \sum_{ \substack{ m \in \mathbb Q \\ \alpha \in \mathbb Q^{g-1} } } c(m,\alpha) \cdot q^m \xi_1^{\alpha_1} \cdots \xi_{g-1}^{\alpha_{g-1} }, $$

where we have set $q^m = e^{2\pi i m \tau'}$ and $\xi_i^{\alpha_i} = e^{2\pi i \alpha_i z_i}$ .

Remark 4.1.3. Suppose $ \rho : \widetilde{\Gamma}_g \to \mathrm{GL}(V)$ is a finite-dimensional representation with finite kernel. Given a holomorphic Siegel modular form $\phi : \mathcal{H}_g \to V$ of half-integer weight k and representation $\rho$ , there is a Fourier–Jacobi expansion

$$\phi(\tau ) = \sum_{ T \in \mathrm{Sym}_{g-1} (\mathbb Q)} \phi_T( \tau', z ) \cdot e^{2\pi i \mathrm{Tr}(T\cdot \tau'')}$$

in which each coefficient $\phi_T$ is a Jacobi form of the weight k, index T, and representation $\rho$ .

In practice, the representation $\rho$ will always be a form of the Weil representation. Let N be a free $\mathbb Z$ -module of finite rank endowed with a (nondegenerate) quadratic form Q. Denote by

$$S_{N,g} = \mathbb C [ ( N^\vee / N)^g ] \cong \mathbb C[ N^\vee/N]^{\otimes g}$$

the finite-dimensional vector space of $\mathbb C$ -valued functions on $(N^\vee/N)^g$ , and by $S_{N,g}^*$ its $\mathbb C$ -linear dual. For any $\mu \in (N^\vee / N)^g$ we denote by $\phi_\mu \in S_{N, g}$ the characteristic function of $\mu$ . As $\mu$ varies these form a basis of $S_{N,g}$ , and we denote by $\phi^*_\mu \in S^*_{N, g}$ the dual basis vectors.

Denote by

(4.4) \begin{equation}\omega_{N,g} : \widetilde{\Gamma}_g \to \mathrm{GL}( S_{N,g} )\quad \mbox{and}\quad \omega^*_{N,g} : \widetilde{\Gamma}_g \to \mathrm{GL}( S^*_{N,g})\end{equation}

the Weil representation and its contragredient. To resolve some confusion in the literature, we now pin down the precise normalization of (4.4). Suppose N has signature (p,q), and $\tilde{\gamma}=(\gamma, j_\gamma)$ is as in (4.2).

1. (1) If $\gamma = (\!\begin{smallmatrix}A & 0 \\0 & D\end{smallmatrix}\!)$ , then, noting that $j_\gamma$ is a constant function (in fact, a square root of $\det(A)=\det(D)=\pm 1$ ),

   $$\omega_{N,g}( \tilde{\gamma} ) \cdot \phi_\mu = j_\gamma^{p-q} \cdot\phi_{\mu \cdot A^{-1}} .$$

2. (2) If $ \gamma = (\!\begin{smallmatrix}I & B \\0 & I\end{smallmatrix}\!)$ and $j_\gamma =1$ , then

   $$\omega_{N,g}( \tilde{\gamma} ) \cdot \phi_\mu = e^{-2\pi i \mathrm{Tr}( Q( \tilde{\mu} ) B)} \cdot \phi_{\mu} ,$$
   where $Q( \tilde{\mu})\in \mathrm{Sym}_g(\mathbb Q)$ is the moment matrix of any lift $\tilde{\mu} \in (N^\vee)^g$ .

3. (3) If $ \gamma = (\!\begin{smallmatrix}0 & -I \\I & 0\end{smallmatrix}\!)$ and the square root $j_\gamma(\tau) = \sqrt{\det(\tau)}$ is the standard branch determined by $j_\gamma( i I_g ) = e^{ g \pi i /4} ,$ then

   $$\omega_{N,g}( \tilde{\gamma}) \cdot \phi_\mu = \frac{e^{ \pi i g(p-q)/4}}{ [ N^\vee : N]^{g/2} }\sum_{ \nu \in (N^\vee/N)^g } e^{2\pi i [ \mu,\nu] } \cdot \phi_{\nu} .$$

These relations determine $\omega_{N,g}$ uniquely. The normalization is such that if N is positive definite of rank n, then the theta series

$$\vartheta_{N,g}(\tau) = \sum_{ \mu \in (N^\vee/N)^g }\bigg( \sum_{ x\in \mu+ N^g} e^{ 2\pi i \mathrm{Tr}(\tau Q(x)) } \bigg)\phi^*_\mu$$

is a Siegel modular form of weight $n/2$ and representation $\omega^*_{N,g}$ .

Remark 4.1.4. Our Weil representation does not agree with the Weil representation $\rho_{N,g}$ of [Reference ZhangZha09, Definition 2.2]. Instead, the isomorphism

$$ S_{N,g} \xrightarrow{\phi_\mu \mapsto \phi_\mu^* } S_{N,g}^{*} $$

identifies $\rho_{N,g}$ with $\omega_{N,g}^*$ . Alternatively, our $\omega_{N,g}$ agrees with Zhang’s $\rho_{-N,g}$ , where $-N$ has the same underlying $\mathbb Z$ -module as N, but is endowed with the signature (q,p) quadratic form $-Q$ .

Remark 4.1.6. When $g=1$ we omit it from the notation, so that

$$\widetilde{\Gamma} = \widetilde{\Gamma}_1\quad \mbox{and}\quad \mathcal{H}=\mathcal{H}_1,$$

and the Weil representation and its contragredient are

$$\omega_{N} : \widetilde{\Gamma} \to \mathrm{GL}( S_{N} )\quad \mbox{and}\quad \omega^*_{N} : \widetilde{\Gamma} \to \mathrm{GL}( S^*_{N} ).$$

4.2 Statement of modularity in low codimension

Throughout the remainder of § 4 we impose the following two hypotheses.

Hypothesis 4.2.2. Recalling the integer $r(L) \ge 0$ of Definition 3.1.1, we assume that d is a positive integer satisfying

$$d+1 \leq \frac{n-2r(L)-4}{3} .$$

The first hypothesis is neededFootnote ³ to make sense of (4.5), which requires a well-defined intersection product on the rational Chow groups of $\mathcal{M}$ . This is available to us if $\mathcal{M}$ is regular, as explained in § A.2. The second hypothesis is imposed so that we may make use of Proposition 3.3.1.

Suppose $T' \in \mathrm{Sym}_{d+1}(\mathbb Q)$ and $\mu' \in (L^\vee/L)^{d+1}$ . Hypothesis 4.2.2 and Proposition 3.3.1 imply that the special cycle

$$\mathcal{Z}(T',\mu') \to \mathcal{M}$$

is flat over $\mathbb Z[\Sigma^{-1}]$ , and equidimensional of codimension $\mathrm{rank}(T')$ in $\mathcal{M}$ . By Definition 1.4, there is an associated naive cycle class

\begin{equation*}[\mathcal{Z}(T',\mu')] \in \mathrm{CH}^{\mathrm{rank}(T')}( \mathcal{M} ).\end{equation*}

Define the corrected cycle class

(4.5) \begin{equation} \mathcal{C}(T',\mu') \stackrel{\mathrm{def}}{=} \underbrace{c_1(\omega^{-1}) \cdots c_1(\omega^{-1})}_{d+1-\mathrm{rank}(T') } \cdot{\kern1.5pt} [\mathcal{Z}(T',\mu')] \in \mathrm{CH}^{d+1} (\mathcal{M}) , \end{equation}

where $c_1(\omega^{-1})$ is the image of the inverse tautological line bundle (2.6) under the first Chern class map of Definition A.3.1. Abbreviate

$$\mathcal{C}(T') = \sum_{\mu'\in (L^\vee/L)^{d+1}}\mathcal{C}(T',\mu') \otimes \phi^*_{\mu'}\in \mathrm{CH}^{d+1}(\mathcal{M})\otimes S^*_{L,{d+1}}.$$

The remainder of § 4 is devoted to the proof of the following result.

Proposition 4.2.3. For any fixed $T\in \mathrm{Sym}_d(\mathbb Q)$ , the formal generating series

$$ \sum_{ \substack{ m\in \mathbb Q \\ \alpha\in \mathbb Q^d } }\mathcal{C} \Big(\begin{array}{c@{\quad }c}T & \tfrac{\alpha}{2} \\\frac{{}^t \alpha}{2} & m \end{array}\Big) \cdot q^m \xi_1^{\alpha_1}\cdots \xi_d^{\alpha_d}$$

with coefficients in $\mathrm{CH}^{d+1}( \mathcal{M} ) \otimes S^*_{L,d+1}$ is a Jacobi form of index T, weight $1+({n}/{2})$ , and representation

$$\omega^*_{L,d+1}: \widetilde{\Gamma}_{d+1} \to \mathrm{GL}(S_{L,d+1}^*) .$$

Here Jacobi modularity is understood, as in Theorem A, after applying any $\mathbb Q$ -linear functional $\mathrm{CH}^{d+1}( \mathcal{M} ) \to \mathbb C$ .

Proposition 4.2.3, when combined with the main result of [Reference Bruinier and Westerholt-RaumBW-R15], is already enough to show that

$$ \sum_{ T' \in \mathrm{Sym}_{d+1} (\mathbb Q) } \mathcal{C}(T') \cdot q^{T'}$$

is the q-expansion of a Siegel modular form of representation $\omega^*_{L,d+1} $ . See Theorem 6.2.1 and its proof for details.

4.3 Auxiliary cycle classes

In this subsection we work with a fixed positive-semi-definite $T\in \mathrm{Sym}_d(\mathbb Q)$ and $\mu=(\mu_1,\ldots, \mu_d) \in (L^\vee/L)^d .$

Given a $T'\in \mathrm{Sym}_{d+1}(\mathbb Q)$ with upper left $d\times d$ block T, and a $ \mu_{d+1} \in L^\vee/L $ , there is a morphism

(4.6) \begin{equation}\mathcal{Z} ( T' , \mu' ) \to \mathcal{Z} (T, \mu)\end{equation}

of special cycles on $\mathcal{M}$ , where $\mu'=( \mu_1,\ldots,\mu_{d+1})$ . This morphism sends an S-valued point

$$(x_1,\ldots, x_{d+1}) \in V_{\mu_1}(\mathcal{A}_S) \times \cdots \times V_{\mu_{d+1}}(\mathcal{A}_S)$$

of the source to its truncation

$$(x_1,\ldots, x_d) \in V_{\mu_1}(\mathcal{A}_S) \times \cdots \times V_{\mu_d}(\mathcal{A}_S) .$$

The morphism (4.6) is finite and unramified, by the same argument as in the proof of [Reference Andreatta, Goren, Howard and Madapusi PeraAGH+17, Proposition 2.7.2].

Recall from (2.10) that T determines a positive-definite quadratic space W of dimension $\mathrm{rank}(T)$ , together with distinguished generators $e_1,\ldots, e_d \in W$ . As in (2.11), a functorial point $S\to \mathcal{Z}(T,\mu)$ determines an isometric embedding

(4.7) \begin{equation}W\xrightarrow{e_i \mapsto x_i} V(\mathcal{A}_S )_\mathbb Q.\end{equation}

This allows us to define a family of finite unramified stacks

(4.8) \begin{equation}\mathcal{Y} (m,\mu_{d+1},w)\to \mathcal{Z}(T,\mu)\end{equation}

indexed by $m\in \mathbb Q$ , $\mu_{d+1}\in L^\vee/L$ , and $w\in W$ , with functor of points

$$\mathcal{Y}(m,\mu_{d+1},w) (S)= \Big\{x_{d+1}\in V_{\mu_{d+1}}(\mathcal{A}_S): \begin{array}{c} Q(x_{d+1}-w) = m \\{ [x_{d+1},e_i] = [w,e_i] \ \forall 1\le i \le d } \end{array} \Big\}.$$

Note that the conditions $[ x_{d+1} , e_i] = [w,e_i]$ are equivalent to w being the orthogonal projection of $x_{d+1}$ to $W \subset V(\mathcal{A}_S)_\mathbb Q$ .

The following Proposition shows that the new stacks (4.8) are not really new at all. They are special cycles we already know, but indexed in a different way.

Proposition 4.3.1. There is an isomorphism of $\mathcal{Z}(T,\mu)$ -stacks

(4.9) \begin{equation} \mathcal{Y}(m,\mu_{d+1},w) \cong \mathcal{Z} (T' , \mu' ),\end{equation}

where $\mu'=(\mu_1,\ldots, \mu_{d+1})$ , and

$$T' = \begin{pmatrix}T&\tfrac{\alpha}{2}\\\frac{{}^t \alpha}{2}\quad & m + Q(w)\end{pmatrix} \!,$$

for the column vector $\alpha\in \mathbb Q^d$ with components $\alpha_i = [w,e_i]$ . Moreover,

$$m\ne 0 \iff \mathrm{rank}(T') = \mathrm{rank}(T)+1,$$

and when these conditions hold both (4.6) and (4.8) are generalized Cartier divisors (Definition 2.4.1).

Proof. Let W’ be the quadratic space of dimension $\mathrm{rank}(T')$ determined by T’, exactly as the space W of (2.10) was determined by T. As we do not assume that T’ is positive semi-definite, the quadratic space W’ need not be positive definite, but it is nondegenerate (by construction). There are distinguished vectors $e_1,\ldots, e_{d+1}$ that span W’, the vectors $e_1,\ldots, e_d$ span a positive-definite subspace $W\subset W'$ isometric to (2.10), and the tuples

$$e =(e_1,\ldots, e_d) \in W^d\quad \mbox{and}\quad e' =(e_1,\ldots, e_{d+1}) \in (W')^{d+1}$$

satisfy $Q(e)=T$ and $Q(e') = T'$ . Using the relation between T’ and w, one checks first that w is the orthogonal projection of $e_{d+1}$ to W, and then that

$$0 = [ e_{d+1}-w,w] = Q(e_{d+1}) -Q( e_{d+1}-w) - Q(w) .$$

In particular,

(4.10) \begin{equation}Q( e_{d+1}-w) =m.\end{equation}

Now we construct the isomorphism (4.9). Fix an $\mathcal{M}$ -scheme S and a tuple

$$x=(x_1,\ldots, x_d) \in V_{\mu_1}(\mathcal{A}_S) \times \cdots \times V_{\mu_d}(\mathcal{A}_S).$$

with moment matrix $Q(x)=T$ . This determines a point $x\in \mathcal{Z}(T,\mu)(S)$ , and an isometric embedding $W \to V(\mathcal{A}_S)_\mathbb Q$ by (4.7).

A lift of x to $\mathcal{Z}(T',\mu')(S)$ determines a special quasi-endomorphism

$$x_{d+1} \in V_{\mu_{d+1}} (\mathcal{A}_S),$$

which then determines an extension of $W \to V(\mathcal{A}_S)_\mathbb Q$ to

$$W' \xrightarrow{e_i \mapsto x_i} V(\mathcal{A}_S)_\mathbb Q.$$

The calculation (4.10) shows that $Q(x_{d+1}-w)=m$ , and combining this with $[x_{d+1} , e_i] = [x_{d+1},x_i] = \alpha_i$ shows that $x_{d+1}$ defines a lift of x to $\mathcal{Y}(m,\mu_{d+1},w)(S)$ .

Conversely, any lift of x to $\mathcal{Y}(m,\mu_{d+1},w)(S)$ corresponds to an $x_{d+1} \in V_{\mu_{d+1}} (\mathcal{A}_S)$ with the property that $Q(x_{d+1}-w)=m$ , and the orthogonal projection of $x_{d+1}$ to $W \subset V(\mathcal{A}_S)_\mathbb Q$ is w. An elementary linear algebra argument shows that $x'=(x_1,\ldots, x_{d+1})$ has moment matrix T’, so determines a lift of x to $\mathcal{Z}(T',\mu' )(S)$ . This establishes the isomorphism (4.9).

If $m=0$ , then (4.10) implies that $e_{d+1}-w$ is an isotropic vector orthogonal to W, which is therefore contained in the radical of the quadratic form on W’. As this radical is trivial, $e_{d+1}=w\in W$ . It follows that $W'=W$ , and $\mathrm{rank}(T') = \mathrm{rank}(T)$ .

Now suppose $m\neq 0$ . It follows from (4.10) that $e_{d+1}\neq w$ , hence $e_{d+1} \neq W$ and

$$\mathrm{rank}(T') = \dim(W') = \dim(W)+1=\mathrm{rank}(T)+1.$$

It remains to show that (4.8) is a generalized Cartier divisor. In light of the isomorphism (4.9), it suffices to show the same for (4.6).

By Remark 2.2.12 we may assume that T’ is positive semi-definite, as otherwise $\mathcal{Z}(T',\mu') = \emptyset$ and the claim is vacuous. This assumption implies that W’ is a positive-definite quadratic space, and so (4.10) implies $m \gt 0$ . In particular, $m+Q(w) \gt 0$ . By Proposition 2.4.3, the right vertical arrow in the following diagram

is a generalized Cartier divisor. Proposition 2.4.6 allows us to realize $\mathcal{Z}(T',\mu')$ as an open and closed substack of the upper left corner, and so there exists an étale cover $U\to \mathcal{Z}(T,\mu)$ such that $ \mathcal{Z}(T',\mu')_U \to U$ is a disjoint union of closed immersions $Z_i\to U$ , each of which is locally defined by the vanishing of a section of $\mathcal O_U$ . To see that $Z_i$ is an effective Cartier divisor on U, we must show that this section is not a zero divisor. This relies on the following lemma of commutative algebra.

Proof. By Krull’s Hauptidealsatz [Sta22, Tag 00KV], we have

$$\dim ( S/(a) ) \ge \dim (S) - 1,$$

with equality holding exactly when a is not contained in any minimal prime of S. On the other hand, saying that a is a nonzero divisor is equivalent to saying that a is not contained in any associated prime of S. Since S is Cohen–Macaulay, its associated primes are precisely its minimal ones, and so the lemma follows.

Recall that Hypothesis (4.2.2) guarantees that $\mathcal{Z}(T',\mu')$ has dimension

$$\dim( \mathcal{M} ) - \mathrm{rank}(T') = \dim (\mathcal{M} )- \mathrm{rank}(T) - 1 = \dim( \mathcal{Z}(T,\mu) ) - 1.$$

As $\mathcal{Z}(T,\mu)$ is Cohen–Macaulay by Proposition 3.3.1, the desired conclusion now follows from Lemma 4.3.2.

If $m\neq 0$ , Proposition 4.3.1 allows us to define, using Remark 2.4.2,

(4.11) \begin{equation}[ \mathcal{Y}(m,\mu_{d+1},w) ] \in \mathrm{CH}^1 (\mathcal{Z}(T,\mu))\end{equation}

as the cycle class associated to the generalized Cartier divisor (4.8). More precisely, it is the first Chern class (Definition A.3.1) of the associated line bundle.

We next extend the definition to $m=0$ . In this case the condition $Q(x_{d+1}-w) =0$ imposed in the definition of the domain of (4.8) can be satisfied by at most the vector $x_{d+1}=w$ , and so

$$\mathcal{Y}(0,\mu_{d+1},w) (S)= \begin{cases}\mathcal{Z}(T,\mu)(S) & \mbox{if } w \in V_{ \mu_{d+1} } ( \mathcal{A}_S ) \\\emptyset & \mbox{otherwise.}\end{cases}$$

This implies that the morphism (4.8) is a closed immersion. On the other hand, Proposition 4.3.1 tells us that there is a distinguished choice of $(T',\mu')$ for which $\mathrm{rank}(T) = \mathrm{rank}(T')$ and

$$ \mathcal{Y}(0,\mu_{d+1},w) \cong \mathcal{Z}(T',\mu'). $$

We deduce that for this choice of $(T',\mu')$ the morphism (4.6) is a closed immersion between stacks of the same dimension. Now form the first Chern class

$$c_1(\omega^{-1}|_{\mathcal{Z}(T',\mu')})\in \mathrm{CH}^1(\mathcal{Z}(T',\mu')),$$

where $\omega \in \mathrm{Pic}(\mathcal{M})$ is the tautological bundle, and define

(4.12) \begin{equation}[ \mathcal{Y}(0,\mu_{d+1},w) ] \in \mathrm{CH}^1(\mathcal{Z}(T,\mu))\end{equation}

to be its pushforward via (4.6).

Let $Y (m,\mu_{d+1},w)$ be the generic fiber of (4.8), and let

(4.13) \begin{equation}[ Y (m,\mu_{d+1},w) ] \in \mathrm{CH}^1 ( Z(T ,\mu ) )\end{equation}

be the restriction of (4.11) and (4.12) to the generic fiber of $\mathcal{Z}(T,\mu)$ . The following proposition is our version of [Reference ZhangZha09, Proposition 2.6]. One should regard it as a corollary of a theorem of Borcherds [Reference BorcherdsBor99].

Proposition 4.3.3. The formal generating series

$$\sum_{ \substack{ m \in \mathbb Q \\w \in W \\ \mu_{d+1} \in L^\vee/L } } [ Y ( m , \mu_{d+1} , w ) ] \otimes \phi^*_ {\mu_{d+1}}\cdot q^{ m+ Q(w) } \xi_1^{ [ w , e_1] } \cdots \xi_{d}^{ [ w , e_d]}$$

convergesFootnote ⁴ to a holomorphic function

$$\phi_T(\tau', z) : \mathcal{H} \times \mathbb C^d \to \mathrm{CH}^1(Z(T,\mu)) \otimes S_L^*.$$

The corresponding function (4.3) on $\mathcal{H}_{d+1}$ satisfies

$$\Phi_T(\gamma \cdot \tau) = j_\gamma (\tau)^{2+n} \omega_{L}^*( \tilde{ \gamma} ) \cdot \Phi_T(\tau)$$

for all elements $( \gamma , j_\gamma) \in \widetilde{\Gamma} \subset \widetilde{\Gamma}_{d+1}$ as in (4.2).

Proof. The claim is vacuously true if $Z(T,\mu)$ is empty. Hence, as in § 2.3, we may fix an orthogonal decomposition $V = V^\flat \oplus W$ and use this to express

(4.14) \begin{equation} Z(T, \mu) =\bigsqcup_{ g \in G^\flat(\mathbb Q)\backslash \Xi(T,\mu)/K } M^\flat_g .\end{equation}

as a disjoint union of smaller Shimura varieties. As in (2.12), each $g\in \Xi(T,\mu)$ determines lattices $L_g^\flat \subset V^\flat$ and $\Lambda_g \subset W$ .

The Shimura variety $M^\flat_g$ has its own tautological line bundle $\omega^\flat_g$ , its own Kuga–Satake abelian scheme $A^\flat_g \to M^\flat_g$ , and its own family of special cycles

$$ Z^\flat_g(m,\nu) \to M^\flat_g $$

indexed by $m\in \mathbb Q$ and $\nu \in L_g^{\flat,\vee} / L^\flat_g$ . When $m\neq 0$ there is an associated class

$$ [ Z^\flat_g(m,\nu)] \in \mathrm{CH}^1(M^\flat_g) $$

by Remark 2.4.2 and Proposition 2.4.3. When $m=0$ we define

$$ [ Z^\flat_g(0,\nu)] = \begin{cases} c_1( \omega^{\flat,-1}_g) & \mbox{if }\nu =0, \\ 0 & \mbox{otherwise}. \end{cases}$$

Lemma 4.3.5. For every $g\in \Xi(T,\mu)$ there is a canonical isomorphism of $M_g^\flat$ -stacks

(4.15) \begin{equation}Y(m,\mu_{d+1},w) \times_{Z(T,\mu)} M^\flat_g \cong \bigsqcup_{\substack{ \nu \in L_g^{\flat,\vee} / L^\flat_g \\ \nu+w \in g \cdot( \mu_{d+1} +L_{\widehat{\mathbb Z}}) } } Z^\flat_g(m,\nu) .\end{equation}

Here we regard $\mu_{d+1} +L_{\widehat{\mathbb Z}} \subset V_{\mathbb A_f}$ . Moreover,

(4.16) \begin{equation}[Y(m,\mu_{d+1},w)] \vert_{M^\flat_g} = \sum_{\substack{ \nu \in L_g^{\flat,\vee} / L^\flat_g \\ \nu+w \in g \cdot( \mu_{d+1} +L_{\widehat{\mathbb Z}}) } }[ Z^\flat_g(m,\nu) ] \in \mathrm{CH}^1(M^\flat_g)\end{equation}

where the left-hand side is the projection of (4.13) to the g-summand in

$$ \mathrm{CH}^1 ( Z(T ,\mu ) ) \cong \bigoplus_{g \in G^\flat(\mathbb Q)\backslash \Xi(T,\mu)/K }\mathrm{CH}^1(M_g^\flat).$$

Proof. Applying the proof of Proposition 2.4.7 to the morphism $M^\flat_g \to M$ of Remark 2.5, we see that for any $M^\flat_g$ -scheme S there is a canonical isometry

$$V(A_S)_\mathbb Q = V(A^\flat_{g,S})_\mathbb Q \oplus W,$$

which restricts to a bijection

(4.17) \begin{align} V_{\mu_{d+1}}( A_S) = \bigsqcup_{ \substack{ {\nu\in L_g^{\flat,\vee}/L_g^\flat } \\ { \lambda\in \Lambda_g^\vee/\Lambda_g} \\ { \nu+\lambda \in g \cdot ( \mu_{d+1} + L_{\widehat{\mathbb Z}} ) } }} V_\nu(A^\flat_{g,S})\times (\lambda + \Lambda_g). \end{align}

This isometric embedding $W \to V(A_S)_\mathbb Q$ agrees with that of (4.7).

An S-point of the left-hand side of (4.15) is an S-point of $M_g^\flat \subset Z(T,\mu)$ , together with a special quasi-endomorphism

$$ x_{d+1}\in V_{\mu_{d+1}}(A_S) $$

whose orthogonal projection to W is w, and such that $Q(x_{d+1}-w) = m$ . Using (4.17), we find that there is a unique pair of cosets

$$ \nu\in L_g^{\flat,\vee}/L_g^\flat \quad \mbox{and}\quad \lambda\in \Lambda_g^\vee/\Lambda_g $$

such that $\nu +\lambda \in g\cdot (\mu_{d_1}+L_{\widehat{\mathbb Z}})$ , and such that

$$ x_{d+1}-w \in V_\nu(A^\flat_{g,S}) \quad \mbox{and}\quad w\in \lambda + \Lambda_g.$$

In particular, $x_{d+1}-w$ determines an S-point of $Z_g^\flat(m,\nu)$ .

This construction establishes the isomorphism (4.15), from which (4.16) follows directly. We note that when $m=0$ both sides of (4.16) are equal to

$$ \begin{cases} c_1(\omega^{-1}_{M^\flat_g}) & \mbox{if }w \in g \cdot (\mu_{d+1} + L_{\widehat{\mathbb Z}} ), \\ 0 & \mbox{otherwise.} \end{cases} $$

Dualizing the tautological map $S_{\Lambda_g} \otimes S^*_{\Lambda_g} \to \mathbb C$ yields a homomorphism

(4.18) \begin{equation}\mathbb C \to S^*_{\Lambda_g} \otimes S_{\Lambda_g}\end{equation}

sending $1\mapsto \sum_{ w\in \Lambda_g^\vee / \Lambda_g } \phi^*_w \otimes \phi_w$ . On the other hand, if we abbreviate

$$L_g = V \cap g L_{\widehat{\mathbb Z}},$$

we may use the inclusions

\begin{equation*}L^\flat_g \oplus \Lambda_g \subset L_g \subset L_g^\vee \subset L_g^{\flat,\vee} \oplus \Lambda_g^\vee\end{equation*}

to define a homomorphism $ S^*_{L^\flat_g} \otimes S^*_{\Lambda_g} \to S_{L_g}^*$ by

$$\phi^*_\nu \otimes \phi^*_w \mapsto \begin{cases}\phi^*_{ \nu + w } & \mbox{if } \nu+w \in L_g^\vee,\\0 & \mbox{otherwise}\end{cases}$$

for all $\nu \in L_g^{\flat,\vee}/L^\flat_g$ and $w\in \Lambda_g^\vee/\Lambda_g$ . This defines the second arrow in the $\widetilde{\Gamma}_1$ -equivariant composition

(4.19)

The third arrow is defined using the isomorphism $S^*_{L_g} \cong S^*_L $ induced by $g^{-1} : L^\vee_g/L_g \to L^\vee/L$ .

It is a theorem of Borcherds [Reference BorcherdsBor99] that the formal generating series

$$ \sum_{ \substack{ m \ge 0 \\ \nu \in L_g^{\flat,\vee} / L^\flat_g } } [ Z^\flat_g(m,\nu) ] \otimes \phi^*_\nu \cdot q^m$$

with coefficients in $ \mathrm{CH}^1( M^\flat_g) \otimes S^*_{ L^\flat_g}$ is a modular form on $\mathcal{H}$ of weight $1+{n^\flat}/{2}$ and representation

$$ \omega^*_{L^\flat_g} : \widetilde{\Gamma} \to \mathrm{GL}( S^*_{L^\flat_g} ) . $$

More precisely, as Borcherds works only in the complex fiber, one should use [Reference Howard and Madapusi PeraHMP20, Theorem B] for the corresponding modularity statement on the canonical model.

Applying (4.19) to this last generating series coefficient-by-coefficient yields the formal generating series

(4.20) \begin{equation} \sum_{ \substack{ m \ge 0 \\ \nu \in L_g^{\flat,\vee} / L^\flat_g } } \sum_{ \substack{ \mu_{d+1} \in L^\vee/L \\ w \in \Lambda^\vee_g/\Lambda_g \\ \nu+w \in g\cdot( \mu_{d+1} + L_{\widehat{\mathbb Z}}) } }[ Z^\flat_g ( m , \nu ) ] \otimes \phi^*_{\mu_{d+1}} \otimes \phi_ w \cdot q^m\end{equation}

with coefficients in $\mathrm{CH}^1(M^\flat_g) \otimes S^*_L \otimes S_{ \Lambda_g }$ , which is therefore a modular form on $\mathcal{H}$ of weight $1+{n^\flat}/{2}$ and representation

$$ \omega^*_{L} \otimes \omega_{ \Lambda_g } : \widetilde{\Gamma} \to \mathrm{GL}( S^*_{L} \otimes S_{ \Lambda_g } ) . $$

Consider the theta function $ \mathcal{H} \times \mathbb C^d \to S_{\Lambda_g}^*$ defined by

(4.21) \begin{equation}\vartheta_w (\tau',z) = \sum_{ w \in \Lambda_g^\vee } \phi^*_w \cdot q^{Q(w) } \xi_1^{ [ w , e_1] } \cdots \xi_d^{ [ w , e_d] } .\end{equation}

If, as in Definition 4.1.2, we define a function on $\mathcal{H}_{d+1}$ by

$$\Theta_w( \tau) = \vartheta_w (\tau',z)\cdot e^{2\pi i \mathrm{Tr}( T\cdot \tau'' ) },$$

then [Reference ZhangZha09, Lemma 2.8] implies the equality

$$ \Theta_w( \gamma \cdot \tau) = j_\gamma(\tau)^{n-n^\flat} \omega^*_{\Lambda_g}( \tilde{\gamma}) \cdot \Theta_w (\tau) $$

for all (4.2) in the subgroup $\widetilde{\Gamma} \subset \widetilde{\Gamma}_{d+1}$ .

Now use the tautological pairing $S_{\Lambda_g} \otimes S_{\Lambda_g}^* \to \mathbb C$ to multiply (4.20) with (4.21). Lemma 4.3.5 implies that the resulting generating series is

$$ \sum_{ \substack{ m \in \mathbb Q \\ w \in W \\ \mu_{d+1} \in L^\vee/L } }[ Y ( m , \mu_{d+1} , w ) ]\rvert_{M^\flat_g} \otimes \phi^*_ {\mu_{d+1}}\cdot q^{ m+ Q(w) } \xi_1^{ [ w , e_1] } \cdots \xi_d^{ [ w , e_d] } ,$$

which therefore satisfies the transformation law stated in Proposition 4.3.3. Varying g and using (4.14) completes the proof of that proposition.

Corollary 4.3.6. The generating series

$$\sum_{ \substack{ m \in \mathbb Q \\ w \in W \\ \mu_{d+1} \in L^\vee/L } }[ \mathcal{Y} ( m , \mu_{d+1} , w ) ] \otimes \phi^*_ {\mu_{d+1}}\cdot q^{ m+ Q(w) } \xi_1^{ [ w , e_1] } \cdots \xi_{d}^{ [ w , e_d]}$$

defines a holomorphic function

$$\mathcal{H} \times \mathbb C^d \to \mathrm{CH}^1(\mathcal{Z}(T,\mu)) \otimes S_L^*$$

satisfying the same transformation law under $\widetilde{\Gamma} \subset \widetilde{\Gamma}_{d+1}$ as the generating series of Proposition 4.3.3.

Proof. Proposition 3.3.1 (which applies, thanks to Hypothesis 4.2.2) implies the injectivity of the restriction map

$$\mathrm{CH}^1(\mathcal{Z}(T,\mu)) \to \mathrm{CH}^1(Z(T,\mu)),$$

and so the claim is immediate from Proposition 4.3.3.

4.4 Proof of Proposition 4.2.3

Proposition 4.2.3 is vacuously true if $T\in \mathrm{Sym}_d(\mathbb Q)$ is not positive semi-definite. Indeed, in this case $\mathcal{Z}(T,\mu)= \emptyset$ by Remark 2.2.12, and it follows from (4.5) and (4.6) that the generating series of Proposition 4.2.3 vanishes coefficient-by-coefficient. Thus we may assume, as in § 4.3 that $T\in \mathrm{Sym}_d(\mathbb Q)$ is positive semi-definite, and let $e_1,\ldots, e_d \in W$ be as in (2.10).

By Hypothesis 4.2.2 and Proposition 3.3.1, for any $\mu=(\mu_1,\ldots, \mu_d) \in (L^\vee/L)^d$ the canonical finite unramified map

$$f^{(T,\mu)} : \mathcal{Z}(T,\mu) \to \mathcal{M}$$

has equidimensional image of codimension $\mathrm{rank}(T)$ , and so induces (Proposition A.1.3) a pushforward

$$f^{(T,\mu)}_*: \mathrm{CH}^1 (\mathcal{Z}(T ,\mu)) \to \mathrm{CH}^{ \mathrm{rank}(T) + 1 } (\mathcal{M}).$$

The following lemma relates the images of the cycle classes (4.11) and 4.12 under this map to the coefficients appearing in Proposition 4.2.3.

Lemma 4.4.1 For any $m\in \mathbb Q$ , $w \in W$ , and $\mu_{d+1} \in L^\vee/L$ we have the equality

$$\underbrace{ c_1(\omega^{-1}) \cdots c_1(\omega^{-1}) }_{ d - \mathrm{rank}(T ) } \cdot{\kern1.5pt}f^{(T,\mu)}_* [ \mathcal{Y}(m,\mu_{d+1},w) ]= \mathcal{C}( T' , \mu')$$

in $ \mathrm{CH}^{d+1}(\mathcal{M})$ . On the right-hand side, $T' \in \mathrm{Sym}_{d+1}(\mathbb Q)$ and $\mu' \in ( L^\vee/ L )^{d+1}$ have the same meaning as in Proposition 4.3.1.

Proof. First suppose $m\neq 0$ , so that $\mathrm{rank}(T') = \mathrm{rank}(T)+1$ by Proposition 4.3.1. It follows directly from the definitions and the commutative diagram

that

$$f^{(T,\mu)}_* [ \mathcal{Y}(m,\mu_{d+1},w)]= [\mathcal{Z}(T',\mu') ]$$

in the codimension- $\mathrm{rank}(T)+1$ Chow group of $\mathcal{M}$ , and intersecting both sides with $d-\mathrm{rank}(T)$ copies of $c_1(\omega^{-1})$ proves the claim.

Suppose now that $m=0$ , so that $\mathrm{rank}(T') = \mathrm{rank}(T)$ by Proposition 4.3.1. In this case

$$f^{(T,\mu)}_* [ \mathcal{Y}(0,\mu_{d+1},w) ]=f^{(T',\mu')}_* ( \omega^{-1}|_{\mathcal{Z}(T' , \mu' )} ) = c_1(\omega^{-1})\cdot [ \mathcal{Z}(T' ,\mu' )],$$

holds in the codimension- $\mathrm{rank}(T)+1$ Chow group, where the first equality is by the definition of (4.12), and the second is by Proposition A.3.3. Once again, the claim follows.

Lemma 4.4.2. Suppose $m\in \mathbb Q$ and $\alpha\in \mathbb Q^d$ . If

$$\mathcal{C} \bigg(\begin{matrix} T\quad & \tfrac{\alpha}{2} \\\frac{{}^t \alpha}{2}\quad & m\end{matrix}\bigg) \neq 0,$$

then there exists a unique $w\in W$ such that $\alpha_i = [w,e_i]$ for all $i=1,\ldots,d$ .

Proof. Our assumption implies that there is some $\mu' \in (L^\vee/L)^{d+1}$ for which

$$\mathcal{Z}\bigg(\bigg(\begin{matrix} T\quad & \tfrac{\alpha}{2} \\\frac{{}^t \alpha}{2}\quad & m\end{matrix}\bigg),\mu ' \bigg) \neq \emptyset.$$

Any non-empty scheme S mapping to it determines special quasi-endomorphisms

$$ x_1,\ldots,x_{d+1} \in V(\mathcal{A}_S)_\mathbb Q.$$

The first d-coordinates $x=(x_1,\ldots, x_d)$ satisfy $Q(x)=T$ , and so determine an isometric embedding

$$ W\xrightarrow{e_i\mapsto x_i}V(\mathcal{A}_S)_\mathbb Q . $$

Using the relation $[x_{d+1},x_i] = \alpha_i$ for $i=1,\ldots,d$ , we see that the orthogonal projection of $x_{d+1}$ to $W \subset V(\mathcal{A}_S)_\mathbb Q$ is a vector $w\in W$ with the desired properties. The uniqueness is clear, as $e_1,\ldots, e_d$ span the positive definite quadratic space W.

Lemma 4.4.3. For any $m\in \mathbb Q$ and column vectors $\alpha \in \mathbb Q^d$ and $x,y \in \mathbb Z^d$ we have

$$\omega_{L,d+1}^*(\tilde{\gamma}) \cdot \mathcal{C} \bigg(\begin{matrix} T\quad &\tfrac{\alpha}{2} \\ \frac{{}^t \alpha}{2}\quad & m \end{matrix}\bigg) = e^{2\pi i \, {}^tx \alpha } \cdot \mathcal{C} \bigg(\begin{matrix} T\quad & T y + \tfrac{\alpha}{2} \\ {}^ty \, {}^tT + \frac{{}^t \alpha}{2}\quad & {}^ty \,T \, y + {}^t y\,\alpha + m \end{matrix}\bigg) ,$$

where

(4.22) \begin{equation}\tilde{\gamma} = \bigg( \mathrm{id}_{ \widetilde{\Gamma}} ,\begin{pmatrix} {}^tx \\ {}^ty \end{pmatrix}\!\! \bigg)\in \widetilde{\Gamma} \ltimes M_{2,d}(\mathbb Z) = \widetilde{J}_{d+1} .\end{equation}

Proof. By the explicit formulas for (4.4), if we set

$$A= \begin{pmatrix}I_d\quad & y \\0\quad & 1 \end{pmatrix} \in \mathrm{GL}_{d+1}(\mathbb Z),$$

then $\tilde{\gamma}$ acts on $S^*_{L,d+1}$ as

$$\omega_{L,d+1}^*(\tilde{\gamma}) \cdot \phi_{\mu' }^*= e^{2\pi i x_1 [ \mu_1 , \mu_{d+1} ] } \cdots e^{2\pi i x_d [ \mu_d , \mu_{d+1} ] } \cdot \phi_{ \mu' A }^*$$

for any $\mu' \in (L^\vee / L)^{d+1}$ . It follows that

$$ \mathcal{C} \bigg(\!\! \begin{pmatrix} T\quad &\tfrac{\alpha}{2} \\ \frac{{}^t \alpha}{2}\quad & m \end{pmatrix} \!, \mu' \bigg) \otimes ( \omega_{L,d+1}^*(\tilde{\gamma}) \cdot \phi^*_{\mu'} ) = e^{2\pi i \, {}^tx \alpha } \mathcal{C} \bigg(\!\! \begin{pmatrix} T\quad &\tfrac{\alpha}{2} \\ \frac{{}^t \alpha}{2}\quad & m \end{pmatrix}\!, \mu' \bigg) \otimes\phi^*_{\mu' A}.$$

Indeed, the essential point here is that

$$ \mathcal{C} \bigg(\!\! \begin{pmatrix}T\quad &\tfrac{\alpha}{2}\\\frac{{}^t \alpha}{2}\quad & m\end{pmatrix} \!, \mu' \bigg) \neq 0 \implies \mathcal{Z} \bigg(\!\! \begin{pmatrix}T\quad &\tfrac{\alpha}{2}\\\frac{{}^t \alpha}{2}\quad & m\end{pmatrix} \!,\mu' \bigg) \neq \emptyset,$$

which implies, using Remark 2.2.8, that $[ \mu_i , \mu_{d+1} ] \equiv \alpha_i \pmod{\mathbb Z} $ .

The preceding paragraph allows us to compute

\begin{align*}\omega_{L,d+1}^*(\tilde{\gamma}) \cdot \mathcal{C} \begin{pmatrix} T\quad &\tfrac{\alpha}{2} \\ \frac{{}^t \alpha}{2}\quad & m \end{pmatrix} & = e^{2\pi i \, {}^tx \alpha } \sum_{ \mu' \in (L^\vee/L)^{d+1}} \mathcal{C} \bigg(\!\!\begin{pmatrix} T\quad &\tfrac{\alpha}{2} \\ \frac{{}^t \alpha}{2}\quad & m \end{pmatrix}\!, \mu' \bigg) \otimes\phi^*_{\mu' A} \\ & = e^{2\pi i \, {}^tx \alpha } \sum_{ \mu' \in (L^\vee/L)^{d+1}} \mathcal{C} \bigg(\!\! \begin{pmatrix} T\quad &\tfrac{\alpha}{2} \\ \frac{{}^t \alpha}{2}\quad & m \end{pmatrix}\!, \mu' A^{-1} \bigg) \otimes\phi^*_{\mu' } \\ & = e^{2\pi i \, {}^tx \alpha } \sum_{ \mu' \in (L^\vee/L)^{d+1}} \mathcal{C} \bigg( {}^tA \begin{pmatrix} T\quad &\tfrac{\alpha}{2} \\ \frac{{}^t \alpha}{2}\quad & m \end{pmatrix} A , \mu' \bigg) \otimes\phi^*_{\mu' } \\ & = e^{2\pi i \, {}^tx \alpha } \cdot \mathcal{C} \bigg( {}^tA \begin{pmatrix} T\quad &\tfrac{\alpha}{2} \\ \frac{{}^t \alpha}{2}\quad & m \end{pmatrix} A \bigg) .\end{align*}

In the third equality we have used the linear invariance of special cycles proved in Proposition 2.4.5, which implies the same invariance for the corrected cycle classes (4.5).

Proof of Proposition 4.2.3. For a fixed $\mu \in (L^\vee/L)^d$ , consider the generating series

$$ \sum_{ \substack{ m \in \mathbb Q \\ w \in W \\ \mu_{d+1} \in L^\vee/L } }f^{(T,\mu)}_* [ \mathcal{Y}( m , \mu_{d+1} , w ) ] \otimes \phi^*_ {\mu_{d+1}}\cdot q^{ m+ Q(w) } \xi_1^{ [ w , e_1] } \cdots \xi_d^{ [ w , e_d] } .$$

This agrees with the pushforward via $\mathcal{Z}(T,\mu) \to \mathcal{M}$ of the generating series of Corollary 4.3.6, and so converges to a holomorphic function

$$\mathcal{H} \times \mathbb C^d \to \mathrm{CH}^{\mathrm{rank}(T) +1}(\mathcal{M}) \otimes S_L^*$$

transforming under $\widetilde{\Gamma} \subset \widetilde{J}_{d+1}$ as in Proposition 4.3.3.

The linear map $S_L^* \to S^*_{L,d} \otimes S^*_{L} \cong S^*_{L,d+1}$ sending

$$\phi^*_{\mu_{d+1}} \mapsto\phi^*_{\mu} \otimes\phi^*_{\mu_{d+1}}$$

is $\widetilde{\Gamma}$ -equivariant, where the action on the source is via $\omega^*_L$ , and the action on the target is via the restriction of $\omega^*_{L,d+1}$ to $\widetilde{\Gamma} \subset \widetilde{\Gamma}_{d+1}$ . Applying this map to the above generating series, summing over $\mu$ , and using Lemma 4.4.1, we find that

(4.23) \begin{equation} \sum_{ \substack{ m \in \mathbb Q \\ w \in W } } \mathcal{C} \begin{pmatrix}T\quad &\tfrac{\alpha}{2}\\\frac{{}^t \alpha}{2}\quad & m + Q(w)\end{pmatrix}\cdot q^{ m+ Q(w) } \xi_1^{ \alpha_1 } \cdots \xi_d^{ \alpha_d }\end{equation}

(inside the sum, $\alpha\in \mathbb Q^d$ has components $\alpha_i = [ w,e_i]$ ) defines a holomorphic function

$$ \mathcal{H} \times \mathbb C^d \to \mathrm{CH}^{d+1}(\mathcal{M}) \otimes S_{L,d+1}^* $$

transforming under the subgroup $\widetilde{\Gamma} \subset \widetilde{\Gamma}_{d+1}$ in the same way as a Jacobi form of index T, weight $1+{n}/{2}$ , and representation

$$ \omega^*_{L,d+1}:\widetilde{\Gamma}_{d+1} \to \mathrm{GL}(S_{L,d+1}^*).$$