Structure of the paper

In § 1, we introduce the $D$ -tower of period spaces $\{\mathscr{F}(N)\}_{N\geqslant 3}$ . The focus here is on the arithmetic of $D$ -lattices. We define the main Heegner divisors, namely the nodal, hyperelliptic, and unigonal divisors in $\mathscr{F}(N)$ , denoted respectively $H_{n}(N)$ , $H_{h}(N)$ , and $H_{u}(N)$ . If $N=19$ (the period space for quartic surfaces), they have a well-known geometric meaning and in other dimensions they are the arithmetic analogue of the divisors for $N=19$ . The salient point in the $D$ -tower is that $\mathscr{F}(N-1)$ is isomorphic to the hyperelliptic Heegner divisor $H_{h}(N)$ of $\mathscr{F}(N)$ . Our boundary divisor $\unicode[STIX]{x1D6E5}(N)$ is equal to $H_{h}(N)/2$ except when $N\equiv 3,4\hspace{0.6em}({\rm mod}\hspace{0.2em}8)$ , in which case $\unicode[STIX]{x1D6E5}(N)=(H_{h}(N)+H_{u}(N))/2$ . (Note that the factor $\frac{1}{2}$ appears because the quotient map from the relevant Type IV domain to $\mathscr{F}(N)$ is ramified over the hyperelliptic divisor, and also over the unigonal divisor, if $N\equiv 3,4\hspace{0.6em}({\rm mod}\hspace{0.2em}8)$ .) This leads to an inductive behavior.

In the following § 2, we go through results which are (more or less) known, namely that $\mathscr{F}(19)$ is the period space of degree- $4$ polarized $K3$ surfaces, that $\mathscr{F}(18)$ is the period space of hyperelliptic quartics, and that $\mathscr{F}(20)$ is the period space of double EPW sextics up to the duality involution (alternatively, it is the period space of EPW cubes).

In § 3, we study the quasi-pull-back of Borcherds’ celebrated automorphic form given by two embeddings of the lattice $\unicode[STIX]{x1D6EC}_{N}$ into $\text{II}_{2,26}$ . The resulting relations between the automorphic line bundle on $\mathscr{F}(N)$ , and the Heegner divisors $H_{n}(N)$ , $H_{h}(N)$ , and $H_{u}(N)$ (see Theorems 3.1.1 and 3.1.2), are key results for our work.

Section 4 contains a proof that $\mathscr{F}(19,1)=\text{Proj}\,R(\mathscr{F}(19),\unicode[STIX]{x1D706}(19)+\unicode[STIX]{x1D6E5}(19))$ is isomorphic to the GIT moduli space $\mathfrak{M}(19)$ . An analogous proof shows that $\mathscr{F}(18,1)$ is isomorphic to the GIT moduli space $\mathfrak{M}(18)$ and we expect that a similar result holds for $N=20$ , i.e. that $\mathscr{F}(20,1)$ is isomorphic to $\mathfrak{M}(20)$ modulo the duality involution. In other words, if $N\in \{18,19\}$ , then the schemes $\mathscr{F}(N,\unicode[STIX]{x1D6FD})$ , for $\unicode[STIX]{x1D6FD}\in (0,1)\cap \mathbb{Q}$ , interpolate between the Baily–Borel compactification $\mathscr{F}(N)^{\ast }$ and the GIT compactification $\mathfrak{M}(N)$ .

In the last section (§ 5), we identify the critical values of $\unicode[STIX]{x1D6FD}$ and the corresponding centers. To describe our work, let us review the basic picture in Looijenga. The starting point for him is to note that frequently the GIT quotients $\mathfrak{M}$ are obtained by contracting a certain Heegner divisor $H$ in $\mathscr{F}$ . Since the divisor $\unicode[STIX]{x1D706}+H$ restricts trivially to $H$ , one gets that the GIT quotient is the $\text{Proj}\,$ of a ring of meromorphic automorphic forms with poles on $H$ . The complication of recovering the GIT quotient, say $\mathfrak{M}\cong \text{Proj}\,R(\mathscr{F},\unicode[STIX]{x1D706}+H)$ , from $\mathscr{F}^{\ast }=\text{Proj}\,R(\mathscr{F},\unicode[STIX]{x1D706})$ comes from the fact that the hyperplanes in the arrangement $\mathscr{H}$ (defining the Heegner divisor $H=\mathscr{H}/\unicode[STIX]{x1D6E4}$ ) intersect and, according to [Reference LooijengaLoo03b], the intersection strata have to be flipped starting with the largest codimension. While this flipping behavior is easy to understand if the strata are not too deep (e.g. for degree- $2$   $K3$ surfaces or cubic fourfolds), it is clear that for the quartic examples, or other examples where lots of intersections occur, it is wise to consider a continuous variation of the parameter $\unicode[STIX]{x1D706}+\unicode[STIX]{x1D6FD}H$ . Roughly, $\unicode[STIX]{x1D6FD}=0$ is the known case, $\unicode[STIX]{x1D6FD}=1$ is the target, and in between we understand the flipping behavior by wall crossings.

By taking the variational point of view described above, we are able to give a quantitative refinement of the flipping behavior predicted by Looijenga. Specifically, our computations show that the critical value of $\unicode[STIX]{x1D6FD}$ for which a specified intersection stratum $Z$ should be flipped (in the variation of models given by $\unicode[STIX]{x1D706}+\unicode[STIX]{x1D6FD}H$ ) is determined by the log canonical threshold of $Z\subset \mathbb{P}^{N}$ . The precise formulae (see § 5.1.1) are more complicated as one has to take into account the ramification behavior. While these arguments and computations are not explicitly contained in [Reference LooijengaLoo03b], it is natural to call the computations of the (potential) critical $\unicode[STIX]{x1D6FD}$ the Looijenga predictions. The surprising aspect that we discovered in our study is that these are only first-order predictions. Namely, as explained below, some arithmetic corrections are needed. Our final results are discussed in § 5.1.2 (especially Prediction 5.1.1).

By comparing the GIT quotient for quartic surfaces of Shah [Reference ShahSha81] with the predicted Looijenga [Reference LooijengaLoo03b] model, we have observed a puzzling thing: the flips associated to intersections of components of the hyperelliptic arrangement predicted by Looijenga do occur, but, roughly speaking, only up to half the dimension. The explanation for this is in the Borcherds–Gritsenko relations between the automorphic line bundle and the Heegner divisors that were established in § 3. In short, while, for large $N$ , $H_{h}(N)$ is birationally contractible in $\mathscr{F}(N)$ , this is far from true for $N$ small: in fact, $H_{h}(N)$ is ample on $\mathscr{F}(N)$ (and proportional to $\unicode[STIX]{x1D706}(N)$ ) for $N\leqslant 10$ . (This last fact was previously observed by Gritsenko [Reference GritsenkoGri18] in a different context.) This leads to a different behavior for the flips associated to high-codimension intersection strata. Very roughly, all the sufficiently deep strata (in the $D$ -tower example) will be flipped at once. In other words, the Looijenga predictions come from combinatorics of the hyperplane arrangement: this is the dominating factor for small codimension, while for high codimension there are arithmetic corrections that are dominating.

The essential ingredient that is missing for making our predictions/conjectures into theorems is to show that all the centers of the birational transformations that occur in a variation of models on $\mathscr{F}=\mathscr{D}/\unicode[STIX]{x1D6E4}$ associated to varying $\unicode[STIX]{x1D706}+\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D6E5}$ are of Shimura type (i.e. $\mathscr{D}^{\prime }/\unicode[STIX]{x1D6E4}^{\prime }$ ). This a Fulton-type arithmetic conjecture. As mentioned, we have checked this (in the $D$ -tower case) for $N=18$  [Reference Laza and O’GradyLO18a]; also, the case $N=19$ is essentially complete [Reference Laza and O’GradyLO18b].

Notation and conventions

We follow established conventions on integral lattices following Nikulin [Reference NikulinNik80]. In particular, the lattice $\unicode[STIX]{x1D6EC}_{N}=D_{N-2}\oplus U^{2}$ plays an essential role in our paper.

Throughout the paper, $\mathscr{F}=\mathscr{D}/\unicode[STIX]{x1D6E4}$ is a locally symmetric variety of Type IV, i.e. the quotient of a bounded symmetric domain of Type IV modulo an arithmetic group (see § 1.2 for a brief review). The notation $\mathscr{F}$ might come with various additional decorations, which help specify it, e.g. $\mathscr{F}_{\unicode[STIX]{x1D6EC}}(\unicode[STIX]{x1D6E4})$ indicates the underlying lattice $\unicode[STIX]{x1D6EC}$ of signature $(2,\ast )$ and arithmetic group $\unicode[STIX]{x1D6E4}\subset O(\unicode[STIX]{x1D6EC})$ . Starting with § 2, $\mathscr{F}(N)$ is a specific $N$ -dimensional locally symmetric variety (see Proposition 1.2.3) associated to the lattice $\unicode[STIX]{x1D6EC}_{N}$ . There are natural inclusion maps $\mathscr{F}(N-1){\hookrightarrow}\mathscr{F}(N)$ (cf. (1.7.1)); our main focus is $\mathscr{F}(19)\cong \mathscr{F}_{4}$ , the moduli space of quartic $K3$ surfaces. The Baily–Borel compactifications are indicated by a superscript $^{\ast }$ , e.g. $\mathscr{F}(N)^{\ast }$ . The GIT quotients are typically denoted by $\mathfrak{M}$ with various identifying decorations. We view the period map $\mathfrak{p}$ as a birational map going from an appropriate GIT quotient $\mathfrak{M}$ to a Baily–Borel compactification $\mathscr{F}^{\ast }$ . The notation $\unicode[STIX]{x1D6FD}$ is used to denote a rational number in $[0,1]$ , which should be understood as an interpolating parameter between a GIT quotient ( $\unicode[STIX]{x1D6FD}=1$ ) and a Baily–Borel compactification ( $\unicode[STIX]{x1D6FD}=0$ ). In the final § 5, $\mathscr{F}(N,\unicode[STIX]{x1D6FD})$ denotes a birational modification of $\mathscr{F}(N)^{\ast }$ .

We use repeatedly the Hodge bundle $\unicode[STIX]{x1D706}$ on $\mathscr{F}$ and the boundary divisor $\unicode[STIX]{x1D6E5}$ (Definition 1.3.6), which is a Heegner divisor on $\mathscr{F}$ . We use various other Heegner divisors $H=\mathscr{H}/\unicode[STIX]{x1D6E4}\subset \mathscr{F}=\mathscr{D}/\unicode[STIX]{x1D6E4}$ (see § 1.3.1 for a quick review), especially the nodal $H_{n}$ , the hyperelliptic $H_{h}$ , and the unigonal $H_{u}$ divisors (see Definition 1.3.4). Typically, the divisors have the same decorations as the ambient locally symmetric variety (e.g. $H_{h}(N)$ is a divisor in $\mathscr{F}(N)$ ). In general, Borcherds’ relations (e.g. (3.1.1)) refer to a linear equivalence between the Hodge bundle $\unicode[STIX]{x1D706}$ and some Heegner divisors on $\mathscr{F}$ (viewed in $\operatorname{Pic}(\mathscr{F})_{\mathbb{Q}}$ ).

1.1 $D$ -lattices

Let $\unicode[STIX]{x1D6EC}$ be a lattice, i.e. a finitely generated torsion-free abelian group equipped with an integral non-degenerate bilinear symmetric form $(\,,\,)$ . If $v\in \unicode[STIX]{x1D6EC}$ , we let $v^{2}=(v,v)$ . Given a ring $R$ , we let $\unicode[STIX]{x1D6EC}_{R}:=\unicode[STIX]{x1D6EC}\otimes _{\mathbb{Z}}R$ and we extend by linearity the quadratic form to $\unicode[STIX]{x1D6EC}_{R}$ . We let $A_{\unicode[STIX]{x1D6EC}}:=\operatorname{Hom}(\unicode[STIX]{x1D6EC},\mathbb{Z})/\unicode[STIX]{x1D6EC}$ be the discriminant group of $\unicode[STIX]{x1D6EC}$ . The quadratic form defines an embedding $\operatorname{Hom}(\unicode[STIX]{x1D6EC},\mathbb{Z})\subset \unicode[STIX]{x1D6EC}_{\mathbb{Q}}$ . Let $v\in \unicode[STIX]{x1D6EC}$ . The divisibility of $v$ is defined to be the positive generator of $(v,\unicode[STIX]{x1D6EC})$ and is denoted $\operatorname{div}(v)$ . Thus, $v/\text{div}(v)$ is an element of $\unicode[STIX]{x1D6EC}_{\mathbb{Q}}$ which belongs to $\operatorname{Hom}(\unicode[STIX]{x1D6EC},\mathbb{Z})$ and hence it determines an element $v^{\ast }\in A_{\unicode[STIX]{x1D6EC}}$ . Now suppose that $\unicode[STIX]{x1D6EC}$ is an even lattice. The embedding $\operatorname{Hom}(\unicode[STIX]{x1D6EC},\mathbb{Z})\subset \unicode[STIX]{x1D6EC}_{\mathbb{Q}}$ induces a discriminant quadratic form $q_{\unicode[STIX]{x1D6EC}}:A_{\unicode[STIX]{x1D6EC}}\rightarrow \mathbb{Q}/2\mathbb{Z}$ . We recommend the classical paper by Nikulin [Reference NikulinNik80] as a reference for definitions and results on lattices.

For us, $U$ is the standard hyperbolic plane and the standard $ADE$ root lattices are negative definite. Thus, $E_{8}$ is the unique even unimodular negative-definite lattice of rank $8$ ; in the literature, sometimes, it is denoted by $E_{8}(-1)$ or $-E_{8}$ . If $0<p\leqslant q$ and $p\equiv q\hspace{0.6em}({\rm mod}\hspace{0.2em}8)$ , then $\text{II}_{p,q}$ stands for an even unimodular lattice of signature $(p,q)$ ; such a lattice is unique up to isomorphism. For $m\in \mathbb{Z}$ , we let $(m)$ be the rank- $1$ lattice with quadratic form that takes the value $m$ on a generator. We let

(1.1.1) $$\begin{eqnarray}D_{n}:=\bigg\{x\in \mathbb{Z}^{n}\biggm\vert\mathop{\sum }_{i}x_{i}\equiv 0\quad (\text{mod}~2)\bigg\}\subset \mathbb{Z}^{n}\end{eqnarray}$$

and we equip $D_{n}$ with the restriction of the negative of the standard Euclidean pairing on $\mathbb{Z}^{n}$ . Thus, $D_{1}\cong (-4)$ , $D_{2}\cong A_{1}\oplus A_{1}$ , $D_{3}=A_{3}$ , and $D_{n}$ is the negative-definite root lattice with Dynkin diagram $D_{n}$ if $n\geqslant 4$ . Let $\unicode[STIX]{x1D6FC}_{n},\unicode[STIX]{x1D6FD}_{n}\in A_{D_{n}}$ be the classes of $(1/2,1/2,\ldots ,1/2)$ and $(-1/2,1/2,\ldots ,1/2)$ , respectively. If $n$ is odd, then $4\unicode[STIX]{x1D6FC}_{n}=4\unicode[STIX]{x1D6FD}_{n}=0$ and $\unicode[STIX]{x1D6FC}_{n}+\unicode[STIX]{x1D6FD}_{n}=0$ . If $n$ is even, then $2\unicode[STIX]{x1D6FC}_{n}=2\unicode[STIX]{x1D6FD}_{n}=0$ . Moreover,

(1.1.2) $$\begin{eqnarray}q_{D_{n}}(\unicode[STIX]{x1D6FC}_{n})=q_{D_{n}}(\unicode[STIX]{x1D6FD}_{n})\equiv -n/4\hspace{0.6em}({\rm mod}\hspace{0.2em}2\mathbb{Z}),\quad q_{D_{n}}(\unicode[STIX]{x1D6FC}_{n}+\unicode[STIX]{x1D6FD}_{n})\equiv n+1\hspace{0.6em}({\rm mod}\hspace{0.2em}2\mathbb{Z}).\end{eqnarray}$$

The following result is an easy exercise.

For $N\geqslant 3$ , let

(1.1.3) $$\begin{eqnarray}\unicode[STIX]{x1D6EC}_{N}:=U^{2}\oplus D_{N-2}.\end{eqnarray}$$

Notice that

(1.1.4) $$\begin{eqnarray}(A_{\unicode[STIX]{x1D6EC}_{N}},q_{\unicode[STIX]{x1D6EC}_{N}})\cong (A_{D_{N-2}},q_{D_{N-2}}).\end{eqnarray}$$

Below is the key definition of the present subsection.

Remark 1.1.4. If $N\geqslant 3$ , then $\unicode[STIX]{x1D6EC}_{N}\oplus E_{8}\cong \unicode[STIX]{x1D6EC}_{N+8}$ (both are even unimodular lattices, of signature $(2,N+8)$ ). More generally, writing $N-2=8k+a$ with $0\leqslant k$ and $a\in \{0,\ldots ,7\}$ , we get

(1.1.5) $$\begin{eqnarray}\unicode[STIX]{x1D6EC}_{N}\cong \text{II}_{2,2+8k}\oplus D_{a}.\end{eqnarray}$$

1.2 Locally symmetric varieties of Type IV

Suppose that $\unicode[STIX]{x1D6EC}$ is a lattice of signature $(2,m)$ . We let

$$\begin{eqnarray}\mathscr{D}_{\unicode[STIX]{x1D6EC}}:=\{[\unicode[STIX]{x1D70E}]\in \mathbb{P}(\unicode[STIX]{x1D6EC}_{\mathbb{C}})\mid \unicode[STIX]{x1D70E}^{2}=0,~(\unicode[STIX]{x1D70E}+\overline{\unicode[STIX]{x1D70E}})^{2}>0\}.\end{eqnarray}$$

Then $\mathscr{D}_{\unicode[STIX]{x1D6EC}}$ is a complex manifold of dimension $m$ , with two connected components $\mathscr{D}_{\unicode[STIX]{x1D6EC}}^{\pm }$ , interchanged by complex conjugation (the ‘real part’ projection $\unicode[STIX]{x1D6EC}_{\mathbb{C}}\rightarrow \unicode[STIX]{x1D6EC}_{\mathbb{R}}$ identifies $\mathscr{D}_{\unicode[STIX]{x1D6EC}}$ with the set of oriented positive-definite two-dimensional subspaces of $\unicode[STIX]{x1D6EC}_{\mathbb{R}}$ ). Each of $\mathscr{D}_{\unicode[STIX]{x1D6EC}}^{\pm }$ is a bounded symmetric domain of Type IV. The orthogonal group $O(\unicode[STIX]{x1D6EC})$ acts naturally on $\mathscr{D}_{\unicode[STIX]{x1D6EC}}$ (left action). Let $O^{+}(\unicode[STIX]{x1D6EC})<O(\unicode[STIX]{x1D6EC})$ be the subgroup of elements fixing each of $\mathscr{D}_{\unicode[STIX]{x1D6EC}}^{\pm }$ . By definition, $O^{+}(\unicode[STIX]{x1D6EC})$ acts on $\mathscr{D}_{\unicode[STIX]{x1D6EC}}^{+}$ . For a finite-index subgroup $\unicode[STIX]{x1D6E4}<O^{+}(\unicode[STIX]{x1D6EC})$ , we let

$$\begin{eqnarray}\mathscr{F}_{\unicode[STIX]{x1D6EC}}(\unicode[STIX]{x1D6E4}):=\unicode[STIX]{x1D6E4}\backslash \mathscr{D}_{\unicode[STIX]{x1D6EC}}^{+}.\end{eqnarray}$$

Then $\mathscr{F}_{\unicode[STIX]{x1D6EC}}(\unicode[STIX]{x1D6E4})$ is naturally a complex space of dimension $n$ . By a celebrated result of Baily and Borel, there exists a projective variety $\mathscr{F}_{\unicode[STIX]{x1D6EC}}(\unicode[STIX]{x1D6E4})^{\ast }$ (the Baily–Borel compactification of $\mathscr{F}_{\unicode[STIX]{x1D6EC}}(\unicode[STIX]{x1D6E4})$ ) containing an open dense subset isomorphic to $\mathscr{F}_{\unicode[STIX]{x1D6EC}}(\unicode[STIX]{x1D6E4})$ as analytic space. In particular, $\mathscr{F}_{\unicode[STIX]{x1D6EC}}(\unicode[STIX]{x1D6E4})$ has a compatible structure of a quasi-projective variety.

Definition 1.2.1. Let $(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})$ be a dimension- $N$ decorated $D$ -lattice. Then $O(\unicode[STIX]{x1D6EC})$ acts naturally on $A_{\unicode[STIX]{x1D6EC}}$ and hence we may define

$$\begin{eqnarray}\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D709}}:=\{\unicode[STIX]{x1D719}\in O^{+}(\unicode[STIX]{x1D6EC})\mid \unicode[STIX]{x1D719}(\unicode[STIX]{x1D709})=\unicode[STIX]{x1D709}\}.\end{eqnarray}$$

We let $\mathscr{F}_{\unicode[STIX]{x1D6EC}}(\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D709}})$ be the associated locally symmetric variety (of dimension $N$ ): this is a $D$  locally symmetric variety. To simplify notation, we will write $\mathscr{F}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})$ for $\mathscr{F}_{\unicode[STIX]{x1D6EC}}(\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D709}})$ .

We will compare $\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D709}}$ to more familiar subgroups of $O^{+}(\unicode[STIX]{x1D6EC})$ . First, we recall that if $\unicode[STIX]{x1D6EC}$ is an even lattice, the stable orthogonal subgroup $\widetilde{O}(\unicode[STIX]{x1D6EC})<O(\unicode[STIX]{x1D6EC})$ is the kernel of the natural homomorphism $\widetilde{O}(\unicode[STIX]{x1D6EC})\rightarrow O(A_{\unicode[STIX]{x1D6EC}})$ . We let $\widetilde{O}^{+}(\unicode[STIX]{x1D6EC}):=O^{+}(\unicode[STIX]{x1D6EC})\cap \widetilde{O}(\unicode[STIX]{x1D6EC})$ . Notice that $\widetilde{O}^{+}(\unicode[STIX]{x1D6EC})$ is of finite index in $O(\unicode[STIX]{x1D6EC})$ .

Example 1.2.2. Let $\unicode[STIX]{x1D6EC}$ be an even lattice and $r\in \unicode[STIX]{x1D6EC}$ non-isotropic. Let

(1.2.1) $$\begin{eqnarray}\begin{array}{@{}ccc@{}}\unicode[STIX]{x1D6EC}_{\mathbb{Q}} & \overset{\unicode[STIX]{x1D70C}_{r}}{\longrightarrow } & \unicode[STIX]{x1D6EC}_{\mathbb{Q}}\\ x & \mapsto & x-{\displaystyle \frac{2(x,r)}{r^{2}}}r\end{array}\end{eqnarray}$$

be the reflection in the hyperplane $r^{\bot }$ . If $r$ is primitive, then $\unicode[STIX]{x1D70C}_{r}(\unicode[STIX]{x1D6EC})\subset \unicode[STIX]{x1D6EC}$ if and only if $r^{2}|2\operatorname{div}(r)$ ; if this is the case we use the same symbol $\unicode[STIX]{x1D70C}_{r}$ for the corresponding element of $O(\unicode[STIX]{x1D6EC})$ . One checks easily the following results:

1. (1) $\unicode[STIX]{x1D70C}_{r}\in O^{+}(\unicode[STIX]{x1D6EC})$ if and only if $r^{2}<0$ ;

2. (2) if $r^{2}=\pm 2$ , then $\unicode[STIX]{x1D70C}_{r}\in \widetilde{O}(\unicode[STIX]{x1D6EC})$ .

Let $(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})$ be a decorated $D$ -lattice; then

(1.2.2) $$\begin{eqnarray}\widetilde{O}^{+}(\unicode[STIX]{x1D6EC})<\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D709}}<O^{+}(\unicode[STIX]{x1D6EC}).\end{eqnarray}$$

1.3 Heegner divisors on $D$ locally symmetric varieties

1.3.1 Divisor classes on locally symmetric variety of Type IV

Let $X:=\mathscr{F}_{\unicode[STIX]{x1D6EC}}(\unicode[STIX]{x1D6E4})$ be a locally symmetric variety of Type IV. The Hodge bundle (or automorphic bundle), denoted by $\mathscr{L}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D6E4})$ , is a fundamental fractional (orbifold) line bundle on $\mathscr{F}_{\unicode[STIX]{x1D6EC}}(\unicode[STIX]{x1D6E4})$ ; it is defined as the quotient of $\mathscr{O}_{\mathscr{D}_{\unicode[STIX]{x1D6EC}}^{+}}(-1)$ by $\unicode[STIX]{x1D6E4}$ , where $\mathscr{O}_{\mathscr{D}_{\unicode[STIX]{x1D6EC}}^{+}}(-1)$ is the restriction to $\mathscr{D}_{\unicode[STIX]{x1D6EC}}^{+}$ of the tautological line bundle on $\mathbb{P}(\unicode[STIX]{x1D6EC}_{\mathbb{C}})$ . We recall that $\mathscr{L}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D6E4})$ extends to an ample fractional line bundle $\mathscr{L}^{\ast }(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D6E4})$ on the Baily–Borel compactification $\mathscr{F}_{\unicode[STIX]{x1D6EC}}(\unicode[STIX]{x1D6E4})^{\ast }$ , and that the sections of $m\mathscr{L}^{\ast }(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D6E4})$ are precisely the weight- $m$   $\unicode[STIX]{x1D6E4}$ -automorphic forms. We let $\unicode[STIX]{x1D706}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D6E4}):=c_{1}(\mathscr{L}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D6E4}))$ ; thus, $\unicode[STIX]{x1D706}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D6E4})$ is a $\mathbb{Q}$ -Cartier divisor class.

As is well known, any Weil divisor $D$ on the quasi-projective variety $X$ is $\mathbb{Q}$ -Cartier. Thus, we may identify $\operatorname{CH}^{1}(X)_{\mathbb{Q}}$ and $\operatorname{Pic}(X)_{\mathbb{Q}}$ .

Heegner divisors on $\mathscr{F}_{\unicode[STIX]{x1D6EC}}(\unicode[STIX]{x1D6E4})$ are defined as follows. Let $\unicode[STIX]{x1D70B}:\mathscr{D}_{\unicode[STIX]{x1D6EC}}^{+}\rightarrow \mathscr{F}_{\unicode[STIX]{x1D6EC}}(\unicode[STIX]{x1D6E4})$ be the quotient map. Given a non-zero $v\in \unicode[STIX]{x1D6EC}$ , we let

$$\begin{eqnarray}\mathscr{H}_{v,\unicode[STIX]{x1D6EC}}(\unicode[STIX]{x1D6E4}):=\mathop{\bigcup }_{g\in \unicode[STIX]{x1D6E4}}g(v)^{\bot }\cap \mathscr{D}_{\unicode[STIX]{x1D6EC}}^{+},\quad H_{v,\unicode[STIX]{x1D6EC}}(\unicode[STIX]{x1D6E4}):=\unicode[STIX]{x1D70B}(\mathscr{H}_{v,\unicode[STIX]{x1D6EC}}(\unicode[STIX]{x1D6E4})).\end{eqnarray}$$

Notice that $g(v)^{\bot }\cap \mathscr{D}_{\unicode[STIX]{x1D6EC}}^{+}$ is empty if $v^{2}\geqslant 0$ ; hence, we will always assume that $v^{2}<0$ . Then $\mathscr{H}_{v,\unicode[STIX]{x1D6EC}}(\unicode[STIX]{x1D6E4})$ is a (particular) hyperplane arrangement (see [Reference LooijengaLoo03a, Reference LooijengaLoo03b]): we call it the pre-Heegner divisor associated to $v$ . One checks that $H_{v,\unicode[STIX]{x1D6EC}}(\unicode[STIX]{x1D6E4})$ is a prime divisor in the quasi-projective variety $\mathscr{F}_{\unicode[STIX]{x1D6EC}}(\unicode[STIX]{x1D6E4})$ . We say that $H_{v,\unicode[STIX]{x1D6EC}}(\unicode[STIX]{x1D6E4})$ is the Heegner divisor associated to $v$ . Notice that $\mathscr{H}_{v,\unicode[STIX]{x1D6EC}}(\unicode[STIX]{x1D6E4})$ and $H_{v,\unicode[STIX]{x1D6EC}}(\unicode[STIX]{x1D6E4})$ depend only on the $\unicode[STIX]{x1D6E4}$ -orbit of $v$ . We say that $\mathscr{H}_{v,\unicode[STIX]{x1D6EC}}(\unicode[STIX]{x1D6E4})$ and $H_{v,\unicode[STIX]{x1D6EC}}(\unicode[STIX]{x1D6E4})$ are reflective if the reflection $\unicode[STIX]{x1D70C}_{v}$ (see Example 1.2.1) belongs to $\unicode[STIX]{x1D6E4}$ (in particular, $\unicode[STIX]{x1D70C}_{v}(\unicode[STIX]{x1D6EC})=\unicode[STIX]{x1D6EC}$ ). If this is the case, we say that $v$ is a reflective vector of $(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D6E4})$ . If $(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})$ is a decorated $D$ -lattice, and $\unicode[STIX]{x1D6E4}=\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D709}}$ , then we say that $v$ is a reflective vector of $(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})$ .

1.3.2 Relevant Heegner divisors for $D$ locally symmetric varieties

Let $(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})$ be a decorated $D$ -lattice. The Heegner divisors which are relevant for the present work are associated to vectors $v\in \unicode[STIX]{x1D6EC}$ (of negative square) which minimize $|v^{2}|$ among vectors such that $v^{\ast }$ equals a given element of $A_{\unicode[STIX]{x1D6EC}}$ . It will be convenient to write

(1.3.1) $$\begin{eqnarray}A_{\unicode[STIX]{x1D6EC}}=\{0,\unicode[STIX]{x1D701},\unicode[STIX]{x1D709},\unicode[STIX]{x1D701}^{\prime }\}.\end{eqnarray}$$

Thus,

(1.3.2) $$\begin{eqnarray}q_{\unicode[STIX]{x1D6EC}}(\unicode[STIX]{x1D701})=q_{\unicode[STIX]{x1D6EC}}(\unicode[STIX]{x1D701}^{\prime })\equiv -(N-2)/4\hspace{0.6em}({\rm mod}\hspace{0.2em}2\mathbb{Z}),\end{eqnarray}$$

always, and

(1.3.3) $$\begin{eqnarray}\displaystyle 2\unicode[STIX]{x1D701}=2\unicode[STIX]{x1D709}=2\unicode[STIX]{x1D701}^{\prime }=0 & \quad \text{if }N\text{ is even,} & \displaystyle\end{eqnarray}$$

(1.3.4) $$\begin{eqnarray}\displaystyle 2\unicode[STIX]{x1D701}=2\unicode[STIX]{x1D701}^{\prime }=\unicode[STIX]{x1D709} & \quad \text{if }N\text{ is odd.} & \displaystyle\end{eqnarray}$$

Remark 1.3.1. With notation as above, there exists $g\in \unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D709}}$ such that $g(\unicode[STIX]{x1D701})=\unicode[STIX]{x1D701}^{\prime }$ . In fact, it suffices to let $g=(\operatorname{Id}_{U^{2}},\unicode[STIX]{x1D70C})$ , where $\unicode[STIX]{x1D70C}\in O^{+}(D_{N-2})$ is the reflection in the vector $(2,0,\ldots ,0)$ , i.e.  $\unicode[STIX]{x1D70C}(x_{1},\ldots ,x_{N-2})=(-x_{1},x_{2},\ldots ,x_{N-2})$ .

The result below will be used throughout the paper in order to classify $\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D709}}$ -orbits of vectors of $\unicode[STIX]{x1D6EC}$ .

Proposition 1.3.2 (Eichler’s criterion, [Reference Gritsenko, Hulek and SankaranGHS09, Proposition 3.3]).

Let $\unicode[STIX]{x1D6EC}$ be an even lattice which contains $U\oplus U$ . Let $v,w\in \unicode[STIX]{x1D6EC}$ be non-zero and primitive. There exists $g\in \widetilde{O}^{+}(\unicode[STIX]{x1D6EC})$ such that $g(v)=w$ if and only if $v^{2}=w^{2}$ and $v^{\ast }=w^{\ast }$ .

Proposition 1.3.3. Let $(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})$ be a decorated $D$ -lattice of dimension $N$ . The following hold.

1. (1) There exists $v\in \unicode[STIX]{x1D6EC}$ such that $v^{2}=-2$ and $\operatorname{div}(v)=1$ , and it is unique up to $\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D709}}$ (notice that $v^{\ast }=0$ ).

2. (2) There exists $v\in \unicode[STIX]{x1D6EC}$ such that $v^{2}=-4$ , $\operatorname{div}(v)=2$ , and $v^{\ast }=\unicode[STIX]{x1D709}$ . Such a $v$ is unique up to $\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D709}}$ .

3. (3) Let $a\in \{0,\ldots ,7\}$ be the residue mod $8$ of $N-2$ . There exists $v\in \unicode[STIX]{x1D6EC}$ such that:

   1. (3a) $v^{2}=-4a$ , $\operatorname{div}(v)=4$ , and $v^{\ast }=\unicode[STIX]{x1D701}$ (or $v^{\ast }=\unicode[STIX]{x1D701}^{\prime }$ ) if $N$ is odd;

   2. (3b) $v^{2}=-a$ , $\operatorname{div}(v)=2$ , and $v^{\ast }=\unicode[STIX]{x1D701}$ (or $v^{\ast }=\unicode[STIX]{x1D701}^{\prime }$ ) if $N$ is even.

   Such a $v$ is unique up to $\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D709}}$ (recall that $\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D709}}$ exchanges $\unicode[STIX]{x1D701}$ and $\unicode[STIX]{x1D701}^{\prime }$ ).

Proof. (1) and (2): Existence is obvious. Such a $v$ is unique up to $\widetilde{O}^{+}(\unicode[STIX]{x1D6EC})$ by Eichler’s criterion, and hence also up to $\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D709}}$ , because $\widetilde{O}^{+}(\unicode[STIX]{x1D6EC})<\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D709}}$ . (3): Let us prove existence. Assume first that $N-2\not \equiv 0\hspace{0.6em}({\rm mod}\hspace{0.2em}8)$ and hence $a\in \{1,\ldots ,7\}$ . Let $N-2=8k+a$ ; by Remark 1.1.4, we may identify $\unicode[STIX]{x1D6EC}$ with $\text{II}_{2,2+8k}\oplus D_{a}$ . If $N$ is odd, the vector $v:=(0_{4+8k},(2,\ldots ,2))\in \unicode[STIX]{x1D6EC}_{N}$ satisfies (3a) and, if $N$ is even, the vector $v:=(0_{4+8k},(1,\ldots ,1))\in \unicode[STIX]{x1D6EC}_{N}$ satisfies (3b). If $N-2\equiv 0\hspace{0.6em}({\rm mod}\hspace{0.2em}8)$ , let $N=8k+2$ . Then $\unicode[STIX]{x1D6EC}\cong \text{II}_{1,1+8k}\oplus U(2)$ by Theorem 1.13.2 of [Reference NikulinNik80]. With this identification understood, Item (3b) holds for $v:=(0_{2+8k},e)\in \unicode[STIX]{x1D6EC}$ , where $e\in U(2)$ is a primitive isotropic vector. Lastly, we prove unicity of $v$ up to $\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D709}}$ . Let $v_{1},v_{2}$ be two vectors such that (3a) holds for both, or (3b) holds for both. If $v_{1}^{\ast }=v_{2}^{\ast }$ , then $v_{1},v_{2}$ are $\widetilde{O}^{+}(\unicode[STIX]{x1D6EC})$ -equivalent by Eichler’s criterion and hence they are $\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D709}}$ -equivalent because $\widetilde{O}^{+}(\unicode[STIX]{x1D6EC})<\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D709}}$ . If $v_{1}^{\ast }\not =v_{2}^{\ast }$ , then $\{v_{1}^{\ast },v_{2}^{\ast }\}=\{\unicode[STIX]{x1D701},\unicode[STIX]{x1D701}^{\prime }\}$ and hence by Remark 1.3.1 there exists $g\in \unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D709}}$ such that $g(v_{2})^{\ast }=v_{1}^{\ast }$ ; thus, $v_{1},v_{2}$ are $\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D709}}$ -equivalent by Eichler’s criterion.◻

Below is a key definition for all that follows.

Definition 1.3.4. Let $(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})$ be a decorated $D$ -lattice.

1. (1) A vector $v\in \unicode[STIX]{x1D6EC}$ is nodal if Item (1) of Proposition 1.3.3 holds. The nodal pre-Heegner divisor and the nodal Heegner divisor are $\mathscr{H}_{v,\unicode[STIX]{x1D6EC}}(\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D709}})$ and $H_{v,\unicode[STIX]{x1D6EC}}(\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D709}})$ , respectively, where $v\in \unicode[STIX]{x1D6EC}$ is a nodal vector. We will denote them by $\mathscr{H}_{n}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})$ and $H_{n}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})$ , respectively.

2. (2) A vector $v\in \unicode[STIX]{x1D6EC}$ is hyperelliptic if Item (2) of Proposition 1.3.3 holds. The hyperelliptic pre-Heegner divisor and the hyperelliptic Heegner divisor are $\mathscr{H}_{v,\unicode[STIX]{x1D6EC}}(\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D709}})$ and $H_{v,\unicode[STIX]{x1D6EC}}(\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D709}})$ , respectively, where $v\in \unicode[STIX]{x1D6EC}$ is a hyperelliptic vector. We will denote them by $\mathscr{H}_{h}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})$ and $H_{h}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})$ , respectively.

3. (3) A vector $v\in \unicode[STIX]{x1D6EC}$ is unigonal if Item (3) of Proposition 1.3.3 holds. The unigonal pre-Heegner divisor and the unigonal Heegner divisor are $\mathscr{H}_{v,\unicode[STIX]{x1D6EC}}(\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D709}})$ and $H_{v,\unicode[STIX]{x1D6EC}}(\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D709}})$ , respectively, where $v\in \unicode[STIX]{x1D6EC}$ is a unigonal vector. We will denote them by $\mathscr{H}_{u}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})$ and $H_{u}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})$ , respectively. Notice that if $N\equiv 2\hspace{0.6em}({\rm mod}\hspace{0.2em}8)$ , then $\mathscr{H}_{u}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})=0$ and $H_{u}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})=0$ because $v^{2}=0$ .

(The definition makes sense because by Proposition 1.3.3 there is a single $\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D709}}$ -orbit of nodal, hyperelliptic, or unigonal vectors.)

1.3.3 Reflective Heegner divisors and the boundary of $D$ period spaces

Proposition 1.3.5. Let $(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})$ be a dimension- $N$ decorated $D$ -lattice. Let $v\in \unicode[STIX]{x1D6EC}$ be primitive, of negative square, i.e.  $v^{2}<0$ . The reflection $\unicode[STIX]{x1D70C}_{v}$ (see (1.2.1)) belongs to $\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D709}}$ (i.e.  $v$ is a reflective vector of $(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})$ ) if and only if $v$ is either nodal or hyperelliptic, or $N\equiv 3,4\hspace{0.6em}({\rm mod}\hspace{0.2em}8)$ and $v$ is unigonal.

Proof. Assume that $\unicode[STIX]{x1D70C}_{v}$ belongs to $\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D709}}$ . Since $\unicode[STIX]{x1D70C}_{v}(\unicode[STIX]{x1D6EC})=\unicode[STIX]{x1D6EC}$ , it follows that $v^{2}|2\operatorname{div}(v)$ (see Example 1.2.1). On the other hand, $A_{\unicode[STIX]{x1D6EC}}$ is isomorphic to $\mathbb{Z}/(4)$ if $N$ is odd and to the Klein group if $N$ is even. Thus, one of the following holds:

1. (a) $v^{2}=-2$ and $\operatorname{div}(v)\in \{1,2,4\}$ ;

2. (b) $v^{2}=-4$ and $\operatorname{div}(v)\in \{2,4\}$ ;

3. (c) $v^{2}=-8$ and $\operatorname{div}(v)=4$ .

Suppose that Item (a) holds. If $\operatorname{div}(v)=1$ , then $v$ is nodal. Thus, $\unicode[STIX]{x1D70C}_{v}\in \widetilde{O}^{+}(\unicode[STIX]{x1D6EC})$ by Example 1.2.1 and hence $\unicode[STIX]{x1D70C}_{v}\in \unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D709}}$ . If $\operatorname{div}(v)=2$ , then $q_{\unicode[STIX]{x1D6EC}}(v^{\ast })\equiv -1/2\hspace{0.6em}({\rm mod}\hspace{0.2em}2\mathbb{Z})$ . By (1.1.4), Claim 1.1.1, and (1.1.2), it follows that $N\equiv 4\hspace{0.6em}({\rm mod}\hspace{0.2em}8)$ , and $v$ is unigonal. In particular, $\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D709}}=O^{+}(\unicode[STIX]{x1D6EC})$ by Proposition 1.2.3 and hence $\unicode[STIX]{x1D70C}_{v}\in \unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D709}}$ because $\unicode[STIX]{x1D70C}_{v}\in O^{+}(\unicode[STIX]{x1D6EC})$ . If $\operatorname{div}(v)=4$ , then $q_{\unicode[STIX]{x1D6EC}}(v^{\ast })\equiv -1/8\hspace{0.6em}({\rm mod}\hspace{0.2em}2\mathbb{Z})$ . On the other hand, $N$ is odd because $\operatorname{div}(v)=4$ and hence by (1.1.4), Claim 1.1.1, and (1.1.2), it follows that $-(N-2)/4\equiv -1/8\hspace{0.6em}({\rm mod}\hspace{0.2em}2\mathbb{Z})$ , which is absurd. Thus, the case $v^{2}=-2$ and $\operatorname{div}(v)=4$ does not occur.

Now suppose that Item (b) holds. If $\operatorname{div}(v)=2$ , then either $v$ is hyperelliptic or $N\equiv 6\hspace{0.6em}({\rm mod}\hspace{0.2em}8)$ . If $v$ is hyperelliptic, then $\unicode[STIX]{x1D70C}_{v}\in O^{+}(\unicode[STIX]{x1D6EC})$ by Example 1.2.1. Since $\unicode[STIX]{x1D70C}_{v}(v)=-v$ and $\unicode[STIX]{x1D709}=v^{\ast }$ , the reflection $\unicode[STIX]{x1D70C}_{v}$ fixes the $2$ -torsion element $\unicode[STIX]{x1D709}$ and hence belongs to $\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D709}}$ . If, on the other hand, $N\equiv 6\hspace{0.6em}({\rm mod}\hspace{0.2em}8)$ and $v$ is not hyperelliptic, we may assume that $v^{\ast }=\unicode[STIX]{x1D701}$ . Then $\unicode[STIX]{x1D70C}_{v}(\unicode[STIX]{x1D709})=\unicode[STIX]{x1D701}^{\prime }$ (let $N=8k+6$ , so that $\unicode[STIX]{x1D6EC}\cong \text{II}_{2,2+8k}\oplus D_{4}$ , and compute) and hence $\unicode[STIX]{x1D70C}_{v}\notin \unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D709}}$ . If $\operatorname{div}(v)=4$ , then $q_{\unicode[STIX]{x1D6EC}}(v^{\ast })\equiv -1/4\hspace{0.6em}({\rm mod}\hspace{0.2em}2\mathbb{Z})$ . By (1.1.4), Claim 1.1.1, and (1.1.2), it follows that $N\equiv 3\hspace{0.6em}({\rm mod}\hspace{0.2em}8)$ , and $v$ is unigonal. Then $\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D709}}=O^{+}(\unicode[STIX]{x1D6EC})$ by Proposition 1.2.3 and hence $\unicode[STIX]{x1D70C}_{v}\in \unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D709}}$ because $\unicode[STIX]{x1D70C}_{v}\in O^{+}(\unicode[STIX]{x1D6EC})$ .

Lastly, let us show that Item (c) cannot hold. In fact, $q_{\unicode[STIX]{x1D6EC}}(v^{\ast })\equiv -1/2\hspace{0.6em}({\rm mod}\hspace{0.2em}2\mathbb{Z})$ , and $N$ is odd because $\operatorname{div}(v)=4$ ; by  (1.1.4), Claim 1.1.1, and (1.1.2), it follows that $-(N-2)/4\equiv -1/2\hspace{0.6em}({\rm mod}\hspace{0.2em}2\mathbb{Z})$ , which is absurd.◻

The following definition of the boundary divisor $\unicode[STIX]{x1D6E5}$ is motivated by the discussion of § 4.

Definition 1.3.6. Let $(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})$ be a decorated $D$ -lattice of dimension $N$ . Let $\unicode[STIX]{x1D6E5}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})$ be the $\mathbb{Q}$ -Cartier divisor on $\mathscr{F}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})$ defined as

(1.3.5) $$\begin{eqnarray}\unicode[STIX]{x1D6E5}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709}):=\left\{\begin{array}{@{}ll@{}}H_{h}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})/2\quad & \text{if }N\not \equiv 3,4\hspace{0.6em}({\rm mod}\hspace{0.2em}8),\\ (H_{h}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})+H_{u}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709}))/2\quad & \text{if }N\equiv 3,4\hspace{0.6em}({\rm mod}\hspace{0.2em}8).\end{array}\right.\end{eqnarray}$$

1.3.4 Heegner divisors for the stable orthogonal group

Let $(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})$ be a dimension- $N$ decorated $D$ -lattice. The main object of study in this paper is the geometry of the locally symmetric variety $\mathscr{F}_{\unicode[STIX]{x1D6EC}}(\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D709}})$ . (The reason for this is the inductive behavior described in § 1.7.) It is more standard to discard the decoration $\unicode[STIX]{x1D709}$ and consider $\mathscr{F}_{\unicode[STIX]{x1D6EC}}(\widetilde{O}^{+}(\unicode[STIX]{x1D6EC}))$ , the variety associated to the stable orthogonal group. If $N$ is odd, $\mathscr{F}_{\unicode[STIX]{x1D6EC}}(\widetilde{O}^{+}(\unicode[STIX]{x1D6EC}))\cong \mathscr{F}_{\unicode[STIX]{x1D6EC}}(\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D709}})$ and there is nothing to be said. On the other hand, if $N$ is even, then $\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D709}}$ is an index- $2$ subgroup of $\widetilde{O}^{+}(\unicode[STIX]{x1D6EC})$ (see Proposition 1.2.3) and hence we have a double cover map

(1.3.6) $$\begin{eqnarray}\mathscr{F}_{\unicode[STIX]{x1D6EC}}(\widetilde{O}^{+}(\unicode[STIX]{x1D6EC}))\overset{\unicode[STIX]{x1D70C}}{\longrightarrow }\mathscr{F}_{\unicode[STIX]{x1D6EC}}(\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D709}}).\end{eqnarray}$$

We will describe the inverse image by $\unicode[STIX]{x1D70C}$ of the Heegner divisors $H_{n}$ , $H_{h}$ , and $H_{u}$ .

Definition 1.3.7. Let $\unicode[STIX]{x1D6EC}$ be a dimension- $N$   $D$ -lattice. A minimal norm vector of $\unicode[STIX]{x1D6EC}$ is a $v\in \unicode[STIX]{x1D6EC}$ such that one of the following holds:

1. (1) $v^{2}=-2$ and $\operatorname{div}(v)=1$ ; or

2. (2) $v^{2}=-4$ and $\operatorname{div}(v)=2$ ; or

3. (3) $(N-2)\not \equiv 0\hspace{0.6em}({\rm mod}\hspace{0.2em}8)$ and, letting $a$ be the residue of $(N-2)$ modulo $8$ :

   1. (3a) $v^{2}=-4a$ and $\operatorname{div}(v)=4$ if $N$ is odd; or else

   2. (3b) $v^{2}=-a$ and $\operatorname{div}(v)=2$ if $N$ is even.

Remark 1.3.8. Given $\unicode[STIX]{x1D702}\in A_{\unicode[STIX]{x1D6EC}}$ , there exists a minimal norm vector $v\in \unicode[STIX]{x1D6EC}$ such that $v^{\ast }=\unicode[STIX]{x1D702}$ and, by Eichler’s criterion (Proposition 1.3.2) the set of such minimal norm vectors is a single $\widetilde{O}^{+}(\unicode[STIX]{x1D6EC})$ -orbit. If $u\in \unicode[STIX]{x1D6EC}$ is another vector such that $u^{2}<0$ and $u^{\ast }=\unicode[STIX]{x1D702}$ , then $u^{2}\leqslant v^{2}$ ; this is the reason for our choice of terminology.

Definition 1.3.9. Let $\unicode[STIX]{x1D6EC}$ be a dimension- $N$   $D$ -lattice and $\unicode[STIX]{x1D702}\in A_{\unicode[STIX]{x1D6EC}}$ . We let

$$\begin{eqnarray}\mathscr{H}_{\unicode[STIX]{x1D702}}(\unicode[STIX]{x1D6EC}):=\mathscr{H}_{v,\unicode[STIX]{x1D6EC}}(\widetilde{O}^{+}(\unicode[STIX]{x1D6EC})),\quad H_{\unicode[STIX]{x1D702}}(\unicode[STIX]{x1D6EC}):=H_{v,\unicode[STIX]{x1D6EC}}(\widetilde{O}^{+}(\unicode[STIX]{x1D6EC})),\end{eqnarray}$$

where $v\in \unicode[STIX]{x1D6EC}$ is any minimal norm vector such that $v^{\ast }=\unicode[STIX]{x1D702}$ . (The definition makes sense by Remark 1.3.8.)

Remark 1.3.10. Let $\unicode[STIX]{x1D6EC}$ be a dimension- $N$   $D$ -lattice with $N$ odd. Choose a decoration $\unicode[STIX]{x1D709}$ of $\unicode[STIX]{x1D6EC}$ and let $A_{\unicode[STIX]{x1D6EC}}=\{\unicode[STIX]{x1D709},\unicode[STIX]{x1D701},\unicode[STIX]{x1D701}^{\prime }\}$ , as usual. Then, under the identification $\mathscr{F}_{\unicode[STIX]{x1D6EC}}(\widetilde{O}^{+}(\unicode[STIX]{x1D6EC}))\cong \mathscr{F}_{\unicode[STIX]{x1D6EC}}(\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D709}})$ , we have $H_{0}(\unicode[STIX]{x1D6EC})=H_{n}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})$ , $H_{\unicode[STIX]{x1D709}}(\unicode[STIX]{x1D6EC})=H_{h}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})$ , and $H_{\unicode[STIX]{x1D701}}(\unicode[STIX]{x1D6EC})=H_{\unicode[STIX]{x1D701}^{\prime }}(\unicode[STIX]{x1D6EC})=H_{u}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})$ (notice that $\unicode[STIX]{x1D701}^{\prime }=-\unicode[STIX]{x1D701}$ because $N$ is odd).

Proposition 1.3.11. Let $(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})$ be a dimension- $N$ decorated $D$ -lattice and assume that $N$ is even. Let $\unicode[STIX]{x1D70C}$ be the double covering in (1.3.6). Then (notation as in Remark 1.3.10)

(1.3.7) $$\begin{eqnarray}\unicode[STIX]{x1D70C}^{\ast }H_{n}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})=H_{0}(\unicode[STIX]{x1D6EC}),\quad \unicode[STIX]{x1D70C}^{\ast }H_{h}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})=2H_{\unicode[STIX]{x1D709}}(\unicode[STIX]{x1D6EC}),\quad \unicode[STIX]{x1D70C}^{\ast }H_{u}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})=H_{\unicode[STIX]{x1D701}}(\unicode[STIX]{x1D6EC})+H_{\unicode[STIX]{x1D701}^{\prime }}(\unicode[STIX]{x1D6EC}).\end{eqnarray}$$

Proof. Let $v$ be a hyperelliptic vector of $(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})$ and let $\unicode[STIX]{x1D70C}_{v}$ be the associated reflection. Then $\unicode[STIX]{x1D70C}_{v}\in \unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D709}}$ , but $\unicode[STIX]{x1D70C}_{v}\not \in \widetilde{O}^{+}(\unicode[STIX]{x1D6EC})$ (see Remark 1.3.1). Thus, the class of $\unicode[STIX]{x1D70C}_{v}$ in $\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D709}}/\widetilde{O}^{+}(\unicode[STIX]{x1D6EC})$ is the covering involution of $\unicode[STIX]{x1D70C}$ . One may choose a nodal vector $w$ of $(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})$ which is orthogonal to $v$ . Then $\unicode[STIX]{x1D70C}_{v}$ acts non-trivially on $w^{\bot }\cap \mathscr{D}_{\unicode[STIX]{x1D6EC}}^{+}$ ; it follows that the covering involution of $\unicode[STIX]{x1D70C}$ maps $H_{0}(\unicode[STIX]{x1D6EC})$ to itself and is not the identity. The first equality of (1.3.7) follows from this. On the other hand, $\unicode[STIX]{x1D70C}_{v}$ acts trivially on $v^{\bot }\cap \mathscr{D}_{\unicode[STIX]{x1D6EC}}^{+}$ and the second equality of (1.3.7) follows. Lastly, $\unicode[STIX]{x1D70C}_{v}$ switches the vectors $\unicode[STIX]{x1D701}$ and $\unicode[STIX]{x1D701}^{\prime }$ in $A_{\unicode[STIX]{x1D6EC}}$ (cf. Remark 1.3.1) and this proves the third equality of (1.3.7).◻

Applying $\unicode[STIX]{x1D70C}_{\ast }$ to the equalities in (1.3.7), we get the following result.

Corollary 1.3.12. Keep assumptions as in Proposition 1.3.11. Then

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70C}_{\ast }H_{0}(\unicode[STIX]{x1D6EC})=2H_{n}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709}),\quad \unicode[STIX]{x1D70C}_{\ast }H_{\unicode[STIX]{x1D709}}(\unicode[STIX]{x1D6EC})=H_{h}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709}),\quad \unicode[STIX]{x1D70C}_{\ast }(H_{\unicode[STIX]{x1D701}}(\unicode[STIX]{x1D6EC})+H_{\unicode[STIX]{x1D701}^{\prime }}(\unicode[STIX]{x1D6EC}))=2H_{u}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709}). & & \displaystyle \nonumber\end{eqnarray}$$

Below is the last result of the present subsection.

Claim 1.3.13. Let $\unicode[STIX]{x1D6EC}$ be a $D$ -lattice. Choose a decoration $\unicode[STIX]{x1D709}$ of $\unicode[STIX]{x1D6EC}$ and let $A_{\unicode[STIX]{x1D6EC}}=\{\unicode[STIX]{x1D709},\unicode[STIX]{x1D701},\unicode[STIX]{x1D701}^{\prime }\}$ , as usual. Then, with respect to the group $\widetilde{O}^{+}(\unicode[STIX]{x1D6EC}_{N})$ , the following hold:

1. (1) $H_{0}(\unicode[STIX]{x1D6EC})$ is a reflective Heegner divisor;

2. (2) $H_{\unicode[STIX]{x1D709}}(\unicode[STIX]{x1D6EC})$ is reflective if and only if $N$ is odd;

3. (3) $H_{\unicode[STIX]{x1D701}}(\unicode[STIX]{x1D6EC})$ is reflective if and only if $N\equiv 3,4\hspace{0.6em}({\rm mod}\hspace{0.2em}8)$ , and similarly for $H_{\unicode[STIX]{x1D701}^{\prime }}(\unicode[STIX]{x1D6EC})$ .

Proof. If $N$ is odd, the result follows from Proposition 1.3.5 because $\mathscr{F}_{\unicode[STIX]{x1D6EC}}(\widetilde{O}^{+}(\unicode[STIX]{x1D6EC}))=\mathscr{F}_{\unicode[STIX]{x1D6EC}}(\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D709}})$ . Suppose that $N$ is even. Item (1) holds because the reflection associated to $v\in \unicode[STIX]{x1D6EC}$ with $v^{2}=-2$ and $\operatorname{div}(v)=1$ belongs to $\widetilde{O}^{+}(\unicode[STIX]{x1D6EC})$ . On the other hand, as noted above, if $v$ is a hyperelliptic vector of $(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})$ , then $\pm \unicode[STIX]{x1D70C}_{v}\not \in \widetilde{O}^{+}(\unicode[STIX]{x1D6EC})$ and Item (2) follows. In order to prove Item (3), let $\unicode[STIX]{x1D70C}$ be the double covering in (1.3.6). If $H_{\unicode[STIX]{x1D701}}(\unicode[STIX]{x1D6EC})$ is reflective, then $\unicode[STIX]{x1D70C}_{\ast }$ is reflective as well and hence $N\equiv 3,4\hspace{0.6em}({\rm mod}\hspace{0.2em}8)$ by Proposition 1.3.5. On the other hand, one easily checks that if $N\equiv 3,4\hspace{0.6em}({\rm mod}\hspace{0.2em}8)$ , the reflection associated to a unigonal vector is in $\widetilde{O}^{+}(\unicode[STIX]{x1D6EC})$ .◻

1.4 Hyperelliptic Heegner divisors

We will prove that the hyperelliptic Heegner divisor $H_{h}(\unicode[STIX]{x1D6EC}_{N},\unicode[STIX]{x1D709}_{N})$ is isomorphic to $\mathscr{F}(\unicode[STIX]{x1D6EC}_{N-1},\unicode[STIX]{x1D709}_{N-1})$ .

Remark 1.4.2. Let $N\geqslant 4$ , let $(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})$ be a dimension- $N$ decorated $D$ -lattice, and let $v\in \unicode[STIX]{x1D6EC}$ be hyperelliptic. The $D$ -lattice $v^{\bot }$ comes with a decoration. In fact, since $v^{2}=-4$ and $\operatorname{div}(v)=2$ , the sublattice $\langle v\rangle \oplus v^{\bot }$ has index $2$ in $\unicode[STIX]{x1D6EC}$ . Thus, there exists $w\in v^{\bot }$ , well determined modulo $2v^{\bot }$ , such that $(v+w)/2$ is contained in $\unicode[STIX]{x1D6EC}$ . It follows that $(w/2,v^{\bot })\subset \mathbb{Z}$ and hence $w/2$ represents an element $\unicode[STIX]{x1D702}\in A_{v^{\bot }}$ , independent of the choice of $w$ . Moreover, $q_{v^{\bot }}(\unicode[STIX]{x1D702})\equiv 1\hspace{0.6em}({\rm mod}\hspace{0.2em}2)$ because

$$\begin{eqnarray}-1+(w/2)^{2}=(v/2)^{2}+(w/2)^{2}=((v+w)/2)^{2}\equiv 0\hspace{0.6em}({\rm mod}\hspace{0.2em}2\mathbb{Z}).\end{eqnarray}$$

Thus, $(v^{\bot },\unicode[STIX]{x1D702})$ is a dimension- $(N-1)$ decorated $D$ -lattice.

By Proposition 1.4.3, we have an injection of groups

(1.4.1) $$\begin{eqnarray}\begin{array}{@{}ccc@{}}\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D702}} & {\hookrightarrow} & \unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D709}}\\ g & \mapsto & \widetilde{g}.\end{array}\end{eqnarray}$$

The following is immediate.

Let $(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})$ be a decorated $D$ -lattice and $v\in \unicode[STIX]{x1D6EC}$ be a hyperelliptic vector. Let $\unicode[STIX]{x1D6EC}^{\prime }:=v^{\bot }$ (a $D$ -lattice) and $\unicode[STIX]{x1D709}^{\prime }$ be the associated decoration of $\unicode[STIX]{x1D6EC}^{\prime }$ . We have defined an injection $\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D709}^{\prime }}{\hookrightarrow}\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D709}}$ , see (1.4.1), and hence there is a well-defined regular map of quasi-projective varieties

(1.4.2) $$\begin{eqnarray}\begin{array}{@{}ccc@{}}\mathscr{F}(\unicode[STIX]{x1D6EC}^{\prime },\unicode[STIX]{x1D709}^{\prime }) & \overset{f}{\longrightarrow } & H_{h}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})\\ \unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D709}^{\prime }}[\unicode[STIX]{x1D70E}] & \mapsto & \unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D709}}[\unicode[STIX]{x1D70E}].\end{array}\end{eqnarray}$$

Below is the main result of the present subsection.

We will prove Proposition 1.4.5 at the end of the present subsubsection. First, we will go through a series of preliminary results.

Proof. (1): Trivial. (2): We may assume that $\unicode[STIX]{x1D6EC}=U^{2}\oplus D_{N-2}$ and

$$\begin{eqnarray}v=(0_{4},(0,\ldots ,0,2)).\end{eqnarray}$$

By Propositions 1.4.3 and 1.3.3, we may assume that $w=(0_{4},(0,\ldots ,0,2,0))$ , and (2) follows. (3): Let $N=4+8k$ , where $k\geqslant 0$ . We may assume that $\unicode[STIX]{x1D6EC}=\text{II}_{2,2+8k}\oplus D_{2}$ and $v=(0_{4+8k},(0,2))$ . By Propositions 1.4.3 and 1.3.3, we may assume that $w=(0_{4+8k},(2,0))$ , and (3) follows. (4): Let $N=5+8k$ , where $k\geqslant 0$ . We may assume that $\unicode[STIX]{x1D6EC}=\text{II}_{2,2+8k}\oplus D_{3}$ and $v=(0_{4+8k},(0,0,2))$ . By Propositions 1.4.3 and 1.3.3, we may assume that $w=(0_{4+8k},(1,-1,0))$ , and (4) follows.◻

Let $\unicode[STIX]{x1D6EC}$ be a lattice (any lattice, not necessarily a $D$ -lattice) and $\unicode[STIX]{x1D6FA}\subset \unicode[STIX]{x1D6EC}$ a subgroup. We let $\overline{\unicode[STIX]{x1D6FA}}\subset \unicode[STIX]{x1D6EC}$ be the saturation of $\unicode[STIX]{x1D6FA}$ , i.e. the subgroup of vectors $v\in \unicode[STIX]{x1D6EC}$ such that $mv\in \unicode[STIX]{x1D6FA}$ for some $0\not =m\in \mathbb{Z}$ .

Proof. By definition, $H_{h}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})=\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D709}}\backslash \mathscr{H}_{h}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})$ . Let $p\in H_{h}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})$ and $[\unicode[STIX]{x1D70E}]\in \mathscr{H}_{h}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})$ be a representative of $p$ . Let $\text{Stab}([\unicode[STIX]{x1D70E}])<\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D709}}$ be the stabilizer of the line $[\unicode[STIX]{x1D70E}]$ . By construction, we have the following isomorphisms of analytic germs:

(1.4.4) $$\begin{eqnarray}\displaystyle (\mathscr{F}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709}),p) & \cong & \displaystyle (\text{Stab}([\unicode[STIX]{x1D70E}])\backslash \mathscr{D}_{\unicode[STIX]{x1D6EC}}^{+},\overline{[\unicode[STIX]{x1D70E}]}),\end{eqnarray}$$

(1.4.5) $$\begin{eqnarray}\displaystyle (H_{h}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709}),p) & \cong & \displaystyle (\text{Stab}([\unicode[STIX]{x1D70E}])\backslash \mathscr{H}_{h}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709}),\overline{[\unicode[STIX]{x1D70E}]}).\end{eqnarray}$$

Since $\unicode[STIX]{x1D70E}^{\bot }\cap \unicode[STIX]{x1D6EC}_{\mathbb{R}}$ is negative definite, the set of hyperelliptic vectors $v\in \unicode[STIX]{x1D70E}^{\bot }\cap \unicode[STIX]{x1D6EC}$ is finite (and non-empty). Let $\{v_{1},\ldots ,v_{k}\}$ be a maximal collection of such vectors with the property that $v_{i}\not =\pm v_{j}$ for $i\not =j$ . Let $\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D70E}}\subset \unicode[STIX]{x1D6EC}$ be the subgroup generated by $v_{1},\ldots ,v_{k}$ . As noticed above, the restriction of the quadratic form to $\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D70E}}$ is negative definite. Thus, the hypotheses of Lemma 1.4.7 are satisfied and hence the saturation $\overline{\unicode[STIX]{x1D6FA}}_{\unicode[STIX]{x1D70E}}$ is isomorphic (as a lattice) to $D_{k}$ . (Notice that by Lemma 1.4.7, $\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D70E}}$ is independent of the choice of a maximal collection as above.) Let $R_{\unicode[STIX]{x1D70E}}^{\prime }$ be the set of $r\in \overline{\unicode[STIX]{x1D6FA}}_{\unicode[STIX]{x1D70E}}$ such that $r^{2}=-2$ and let $R_{\unicode[STIX]{x1D70E}}^{\prime \prime }$ be the set of hyperelliptic vectors of $\overline{\unicode[STIX]{x1D6FA}}_{\unicode[STIX]{x1D70E}}$ . Then

$$\begin{eqnarray}R_{\unicode[STIX]{x1D70E}}^{\prime }=\{((-1)^{m_{i}}v_{i}+(-1)^{m_{j}}v_{j})/2\mid 1\leqslant i<j\leqslant k\},\quad R_{\unicode[STIX]{x1D70E}}^{\prime \prime }=\{(-1)^{m_{i}}v_{i}\mid 1\leqslant i\leqslant k\}.\end{eqnarray}$$

(The last equality holds by Lemma 1.4.7.) Let $R_{\unicode[STIX]{x1D70E}}:=R_{\unicode[STIX]{x1D70E}}^{\prime }\cup R_{\unicode[STIX]{x1D70E}}^{\prime \prime }$ . Notice that $\unicode[STIX]{x1D70C}_{r}\in \unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D709}}$ for all $r\in R_{\unicode[STIX]{x1D70E}}$ . Let $W_{\unicode[STIX]{x1D70E}},W_{\unicode[STIX]{x1D70E}}^{\prime },W_{\unicode[STIX]{x1D70E}}^{\prime \prime }<\text{Stab}([\unicode[STIX]{x1D70E}])$ be the subgroups generated by the reflections $\unicode[STIX]{x1D70C}_{r}$ for $r\in R_{\unicode[STIX]{x1D70E}}$ , $r\in R_{\unicode[STIX]{x1D70E}}^{\prime }$ , and $r\in R_{\unicode[STIX]{x1D70E}}^{\prime \prime }$ , respectively. Clearly, $W_{\unicode[STIX]{x1D70E}}$ is a normal subgroup of $\text{Stab}([\unicode[STIX]{x1D70E}])$ ; let $G_{\unicode[STIX]{x1D70E}}:=\text{Stab}([\unicode[STIX]{x1D70E}])/W_{\unicode[STIX]{x1D70E}}$ . Then $G_{\unicode[STIX]{x1D70E}}$ acts on $W_{\unicode[STIX]{x1D70E}}\backslash \mathscr{D}_{\unicode[STIX]{x1D6EC}}^{+}$ and on $W_{\unicode[STIX]{x1D70E}}\backslash \mathscr{H}_{h}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})$ . The analytic germ $(\text{Stab}([\unicode[STIX]{x1D70E}])\backslash \mathscr{H}_{h}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709}),\overline{[\unicode[STIX]{x1D70E}]})$ is isomorphic to the germ at $\overline{[\unicode[STIX]{x1D70E}]}$ of $W_{\unicode[STIX]{x1D70E}}\backslash \mathscr{H}_{h}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})$ modulo the action of $G_{\unicode[STIX]{x1D70E}}$ . Hence, it suffices to prove that $W_{\unicode[STIX]{x1D70E}}\backslash \mathscr{H}_{h}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})$ is smooth in a neighborhood of $\overline{[\unicode[STIX]{x1D70E}]}$ .

We identify each $r\in R_{\unicode[STIX]{x1D70E}}$ with $F^{-1}(r)\in \mathbb{C}^{k}$ , where $F$ is the isometry of (1.4.3), and we denote it by $r$ . Thus, $r\in R_{\unicode[STIX]{x1D70E}}^{\prime }$ , $r\in R_{\unicode[STIX]{x1D70E}}^{\prime \prime }$ are of the form

$$\begin{eqnarray}(0,\ldots ,0,\pm 1,0,\ldots ,0,\pm 1,0,\ldots ,0),\quad (0,\ldots ,0,\pm 2,0,\ldots ,0),\end{eqnarray}$$

respectively. The action of $W_{\unicode[STIX]{x1D70E}}$ is trivial on $\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D70E}}^{\bot }$ . It follows that there exist local analytic coordinates $(\mathbf{x},\mathbf{t})=((x_{1},\ldots ,x_{k}),\mathbf{t})$ on $\mathscr{D}_{\unicode[STIX]{x1D6EC}}^{+}$ , centered at $[\unicode[STIX]{x1D70E}]$ , such that $x_{i}=0$ is a local equation of $v_{i}^{\bot }\cap \mathscr{D}_{\unicode[STIX]{x1D6EC}}^{+}$ , and

$$\begin{eqnarray}\unicode[STIX]{x1D70C}_{r}(\mathbf{x},\mathbf{t})=\bigg(\mathbf{x}-\frac{2(\mathbf{x},r)}{r^{2}}r,\mathbf{t}\bigg),\end{eqnarray}$$

where $(\mathbf{x},r)$ is the opposite of the standard Euclidean product of $\mathbf{x}$ and $r$ .

In order to describe $W_{\unicode[STIX]{x1D70E}}\backslash \mathscr{D}_{\unicode[STIX]{x1D6EC}}^{+}$ and $W_{\unicode[STIX]{x1D70E}}\backslash \mathscr{H}_{h}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})$ , we first take the quotient by the normal subgroup $W_{\unicode[STIX]{x1D70E}}^{\prime \prime }$ and then we act by $W_{\unicode[STIX]{x1D70E}}^{\prime }/W_{\unicode[STIX]{x1D70E}}^{\prime \prime }$ on the quotient. Local analytic coordinates on $W_{\unicode[STIX]{x1D70E}}^{\prime \prime }\backslash \mathscr{D}_{\unicode[STIX]{x1D6EC}}^{+}$ are $(y_{1},\ldots ,y_{k},\mathbf{t})$ , where $y_{i}=x_{i}^{2}$ . The action of $W_{\unicode[STIX]{x1D70E}}^{\prime }/W_{\unicode[STIX]{x1D70E}}^{\prime \prime }$ on $(y_{1},\ldots ,y_{k})$ is the standard representation of the symmetric group $\mathscr{S}_{k}$ . Thus, local analytic coordinates on $W_{\unicode[STIX]{x1D70E}}\backslash \mathscr{D}_{\unicode[STIX]{x1D6EC}}^{+}$ are $(\unicode[STIX]{x1D70F}_{1},\ldots ,\unicode[STIX]{x1D70F}_{k},\mathbf{t})$ , where $\unicode[STIX]{x1D70F}_{i}$ is the degree- $i$ elementary symmetric function in $y_{1},\ldots ,y_{k}$ . In particular, $W_{\unicode[STIX]{x1D70E}}\backslash \mathscr{D}_{\unicode[STIX]{x1D6EC}}^{+}$ is smooth in a neighborhood of $\overline{[\unicode[STIX]{x1D70E}]}$ . Since a local equation of $\mathscr{H}_{h}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})$ is given by $x_{1}\cdot \cdots \cdot x_{k}=0$ , a local equation of $W_{\unicode[STIX]{x1D70E}}\backslash \mathscr{H}_{h}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})$ in $W_{\unicode[STIX]{x1D70E}}\backslash \mathscr{D}_{\unicode[STIX]{x1D6EC}}^{+}$ is $\unicode[STIX]{x1D70F}_{k}=0$ . We have proved that $W_{\unicode[STIX]{x1D70E}}\backslash \mathscr{H}_{h}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})$ is smooth in a neighborhood of $\overline{[\unicode[STIX]{x1D70E}]}$ , as claimed.◻

Proof of Proposition 1.4.5.

We adopt the notation introduced in the proof of Proposition 1.4.8. We start by noting that $f$ is surjective by definition. The composition of $f$ and the inclusion $H_{h}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709}){\hookrightarrow}\mathscr{F}^{\ast }(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})$ extends to a regular map

$$\begin{eqnarray}\unicode[STIX]{x1D711}:\mathscr{F}^{\ast }(\unicode[STIX]{x1D6EC}^{\prime },\unicode[STIX]{x1D709}^{\prime })\longrightarrow \mathscr{F}^{\ast }(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709}),\end{eqnarray}$$

compatible with the Baily–Borel boundaries. Since the Baily–Borel compactifications are projective, it follows that $f$ is a projective map. By Claim 1.4.4, the fiber of $f$ at the equivalence class represented by $\unicode[STIX]{x1D70E}\in \mathscr{H}_{h}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})$ is identified with the set of hyperelliptic vectors $v\in \unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D70E}}$ , modulo the action of $\text{Stab}([\unicode[STIX]{x1D70E}])$ (notice that $-\text{Id}_{\unicode[STIX]{x1D6EC}}\in \text{Stab}([\unicode[STIX]{x1D70E}])$ ). Hence, the proof of Proposition 1.4.8 shows that the fiber of $f$ is a singleton; in particular, $f$ is birational. Now $H_{h}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})$ is normal by Proposition 1.4.8; since $f$ is birational and projective, it follows that it is an isomorphism.

It remains to prove that the intersection of $H_{h}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})$ and the singular locus of $\mathscr{F}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})$ has codimension at least $2$ in $H_{h}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})$ . Suppose the contrary: we will reach a contradiction. Since $\mathscr{F}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})$ is normal, there is an irreducible component $Z$ of $H_{h}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})\cap \operatorname{sing}\mathscr{F}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})$ which has codimension $1$ in $H_{h}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})$ . Let $v\in \unicode[STIX]{x1D6EC}$ be a hyperelliptic vector and $\unicode[STIX]{x1D702}$ the decoration of the $D$ -lattice $v^{\bot }$ associated to $\unicode[STIX]{x1D709}$ . Let $\unicode[STIX]{x1D70B}:\mathscr{D}_{v^{\bot }}^{+}\longrightarrow H_{h}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})$ be the natural map; we have proved that $\unicode[STIX]{x1D70B}$ is the quotient map for the action of $\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D702}}$ on $\mathscr{D}_{v^{\bot }}^{+}$ . Now let $\widetilde{Z}\subset \mathscr{D}_{v^{\bot }}^{+}$ be an irreducible component of $\unicode[STIX]{x1D70B}^{-1}Z$ . If $[\unicode[STIX]{x1D70E}]\in \widetilde{Z}$ , then

$$\begin{eqnarray}\text{Stab}([\unicode[STIX]{x1D70E}])\supsetneq \langle \unicode[STIX]{x1D70C}_{v},-1_{\unicode[STIX]{x1D6EC}}\rangle\end{eqnarray}$$

because $\unicode[STIX]{x1D70B}([\unicode[STIX]{x1D70E}])$ is a singular point of $\mathscr{F}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})$ . We distinguish between the two cases.

1. (1) For very general $[\unicode[STIX]{x1D70E}]\in \widetilde{Z}$ , the set of hyperelliptic vectors in $\unicode[STIX]{x1D70E}^{\bot }$ is $\{v,-v\}$ .

2. (2) For very general $[\unicode[STIX]{x1D70E}]\in \widetilde{Z}$ , the set of hyperelliptic vectors in $\unicode[STIX]{x1D70E}^{\bot }$ has cardinality strictly greater than $2$ .

Assume that (1) holds and let $[\unicode[STIX]{x1D70E}]\in \widetilde{Z}$ be very general. Let $g\in (\text{Stab}([\unicode[STIX]{x1D70E}])\setminus \langle \unicode[STIX]{x1D70C}_{v},-1_{\unicode[STIX]{x1D6EC}}\rangle )$ . Then $g(v)=\pm v$ and hence, multiplying $g$ by $\unicode[STIX]{x1D70C}_{v}$ if necessary, we may assume that $g(v)=v$ , i.e.  $g|_{v^{\bot }}\in \unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D702}}$ . Now $g|_{v^{\bot }}$ fixes every point of $\widetilde{Z}$ , which has codimension $1$ in $\mathscr{D}_{v^{\bot }}^{+}$ . It follows that $g|_{v^{\bot }}=\pm \unicode[STIX]{x1D70C}_{w}^{v^{\bot }}$ , where $w$ is a reflective vector of $(v^{\bot },\unicode[STIX]{x1D702})$ and $\unicode[STIX]{x1D70C}_{w}^{v^{\bot }}$ denotes the associated reflection of $v^{\bot }$ . By Proposition 1.4.6, the vector $w$ is a reflective vector of $(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})$ as well; it follows that $g=\unicode[STIX]{x1D70C}_{w}$ . Thus, $\widetilde{Z}=v^{\bot }\cap w^{\bot }\cap \mathscr{D}_{\unicode[STIX]{x1D6EC}}^{+}$ and hence $\text{Stab}([\unicode[STIX]{x1D70E}])=\langle \unicode[STIX]{x1D70C}_{v},\unicode[STIX]{x1D70C}_{w},-1_{\unicode[STIX]{x1D6EC}}\rangle$ for a very general $[\unicode[STIX]{x1D70E}]\in \widetilde{Z}$ . It follows that if $[\unicode[STIX]{x1D70E}]\in \widetilde{Z}$ is very general, then $\mathscr{F}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})$ is smooth at $\unicode[STIX]{x1D70B}([\unicode[STIX]{x1D70E}])$ , and that contradicts our assumption.

Lastly, assume that (2) holds. Let $[\unicode[STIX]{x1D70E}]\in \widetilde{Z}$ be very general. Since $\widetilde{Z}$ has codimension $2$ in $\mathscr{D}_{\unicode[STIX]{x1D6EC}}^{+}$ , the set of hyperelliptic vectors in $\unicode[STIX]{x1D70E}^{\bot }$ is equal to $\{\pm v_{1},\pm v_{2}\}$ , where $v_{1},v_{2}$ are orthogonal hyperelliptic vectors, and moreover $\text{Stab}([\unicode[STIX]{x1D70E}])=\langle \unicode[STIX]{x1D70C}_{v_{1}},\unicode[STIX]{x1D70C}_{v_{2}},-1_{\unicode[STIX]{x1D6EC}}\rangle$ . As shown in the proof of Proposition 1.4.8, it follows that $\mathscr{F}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})$ is smooth at $\unicode[STIX]{x1D70B}([\unicode[STIX]{x1D70E}])$ , and that contradicts our assumption.◻

1.5 Reflective unigonal divisors

1.5.1 $N\equiv 3\hspace{0.6em}({\rm mod}\hspace{0.2em}8)$

Let $(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})$ be a decorated $D$ -lattice of dimension $N\equiv 3\hspace{0.6em}({\rm mod}\hspace{0.2em}8)$ . Let $N=8k+3$ , where $k\geqslant 0$ . Let $v\in \unicode[STIX]{x1D6EC}$ be a unigonal vector, i.e.  $v^{2}=-4$ and $\operatorname{div}(v)=4$ . Then $\unicode[STIX]{x1D6EC}=\langle v\rangle \oplus v^{\bot }$ . Since $\det \unicode[STIX]{x1D6EC}=-4$ , it follows that $v^{\bot }$ is unimodular. Thus,

(1.5.1) $$\begin{eqnarray}v^{\bot }\cong \text{II}_{2,2+8k}.\end{eqnarray}$$

Given $g\in O^{+}(\text{II}_{2,2+8k})$ , let $\widetilde{g}\in O(\unicode[STIX]{x1D6EC})$ be the isometry such that

(1.5.2) $$\begin{eqnarray}\widetilde{g}(v)=v,\quad \widetilde{g}|_{v^{\bot }}=g.\end{eqnarray}$$

Then $\widetilde{g}\in O^{+}(\unicode[STIX]{x1D6EC})$ because $g\in O^{+}(\text{II}_{2,2+8k})$ , and $\widetilde{g}\in \widetilde{O}(\unicode[STIX]{x1D6EC})$ because $A_{\unicode[STIX]{x1D6EC}}$ is generated by $v^{\ast }$ . Thus, we have an injection of groups

(1.5.3) $$\begin{eqnarray}\begin{array}{@{}ccc@{}}O^{+}(\text{II}_{2,2+8k}) & {\hookrightarrow} & \unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D709}}\\ g & \mapsto & \widetilde{g}.\end{array}\end{eqnarray}$$

It follows that the injection $\mathscr{D}_{v^{\bot }}^{+}{\hookrightarrow}\mathscr{D}_{\unicode[STIX]{x1D6EC}}^{+}$ descends to a regular map

(1.5.4) $$\begin{eqnarray}\mathscr{F}_{\text{II}_{2,2+8k}}(O^{+}(\text{II}_{2,2+8k}))\overset{l}{\longrightarrow }H_{u}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709}).\end{eqnarray}$$

Proposition 1.5.1. Let $(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})$ be a decorated $D$ -lattice of dimension $N\equiv 3\hspace{0.6em}({\rm mod}\hspace{0.2em}8)$ and let $N=8k+3$ , where $k\geqslant 0$ . The map $l$ in (1.5.4) is an isomorphism onto the unigonal Heegner divisor $H_{u}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})$ . Moreover, the intersection of $H_{u}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})$ and the singular locus of $\mathscr{F}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})$ has codimension at least $2$ in $H_{u}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})$ .

Proof. The map $l$ is finite (see the proof of Proposition 1.4.5) and it has degree $1$ because $-\text{Id}_{\unicode[STIX]{x1D6EC}}\in \unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D709}}$ . Thus, in order to prove that $l$ is an isomorphism, it suffices to prove that $H_{u}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})$ is normal. We claim that the pre-Heegner divisor $\mathscr{H}_{u}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})$ is smooth, i.e. that if $v_{1},v_{2}$ are non-proportional unigonal vectors, then $v_{1}^{\bot }\cap v_{2}^{\bot }\cap \mathscr{D}_{\unicode[STIX]{x1D6EC}}^{+}=\varnothing$ . In fact, suppose the contrary. Then $v_{1},v_{2}$ span a negative-definite rank- $2$ sublattice of $\unicode[STIX]{x1D6EC}$ ; since $v_{i}^{2}=-4$ and $(v_{1},v_{2})\in 4\mathbb{Z}$ , it follows that $v_{1}\bot v_{2}$ . On the other hand, $v_{1}^{\bot }\cong \text{II}_{2,2+8k}$ by (1.5.1) and hence $v_{1}^{\bot }$ does not contain a primitive vector of divisibility greater than $1$ . This proves that $\mathscr{H}_{u}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})$ is smooth and hence $H_{u}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})=\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D709}}\backslash \mathscr{H}_{u}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})$ is normal.

The proof that $H_{u}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})\cap \operatorname{sing}\mathscr{F}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})$ has codimension at least $2$ in $H_{u}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})$ is similar to the analogous statement for $H_{h}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})$ ; see Proposition 1.4.5. We leave details to the reader.◻

In order to simplify notation, from now on we let

(1.5.5) $$\begin{eqnarray}\mathscr{F}(\text{II}_{2,2+8k}):=\mathscr{F}_{\text{II}_{2,2+8k}}(O^{+}(\text{II}_{2,2+8k})).\end{eqnarray}$$

A vector $v\in \text{II}_{2,2+8k}$ is nodal if it has square $-2$ . By Eichler’s criterion, i.e. Proposition 1.3.2, any two nodal vectors of $\text{II}_{2,2+8k}$ are $O^{+}(\text{II}_{2,2+8k})$ -equivalent. We let $H_{n}(\text{II}_{2,2+8k})$ be the Heegner divisor of $\mathscr{F}(\text{II}_{2,2+8k})$ corresponding to a nodal $v\in \text{II}_{2,2+8k}$ .

1.5.2 $N\equiv 4\hspace{0.6em}({\rm mod}\hspace{0.2em}8)$

Let $(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})$ be a decorated $D$ -lattice of dimension $N\equiv 4\hspace{0.6em}({\rm mod}\hspace{0.2em}8)$ . Let $N=8k+4$ , where $k\geqslant 0$ . Let $v\in \unicode[STIX]{x1D6EC}$ be a unigonal vector, i.e.  $v^{2}=-2$ and $\operatorname{div}(v)=2$ . Then $\unicode[STIX]{x1D6EC}=\langle v\rangle \oplus v^{\bot }$ . Moreover,

(1.5.6) $$\begin{eqnarray}v^{\bot }\cong \text{II}_{2,2+8k}\oplus A_{1}.\end{eqnarray}$$

In fact, $A_{\unicode[STIX]{x1D6EC}}=\mathbb{Z}/(2)\oplus A_{v^{\bot }}$ , where a generator of the first summand has square $-1/2$ modulo $2\mathbb{Z}$ . It follows that the discriminant group of $v^{\bot }$ is $\mathbb{Z}/(2)$ and a generator has square $-1/2$ modulo $2\mathbb{Z}$ . Thus, $v^{\bot }$ and $U^{2}\oplus E_{8}^{k}\oplus A_{1}$ have the same signature and isomorphic discriminant groups; thus, (1.5.6) follows from Theorem 1.13.2 of [Reference NikulinNik80]. Given $g\in O^{+}(v^{\bot })$ , let $\widetilde{g}\in O(\unicode[STIX]{x1D6EC})$ be the isometry such that

(1.5.7) $$\begin{eqnarray}\widetilde{g}(v)=v,\quad \widetilde{g}|_{v^{\bot }}=g.\end{eqnarray}$$

Then $\widetilde{g}\in O^{+}(\unicode[STIX]{x1D6EC})$ because $g\in O^{+}(v^{\bot })$ , and $\widetilde{g}\in \widetilde{O}(\unicode[STIX]{x1D6EC})$ because $O(v^{\bot })=\widetilde{O}(v^{\bot })$ , and $A_{\unicode[STIX]{x1D6EC}}=A_{v^{\bot }}\oplus \langle v^{\ast }\rangle$ . Thus, we have an injection of groups

(1.5.8) $$\begin{eqnarray}\begin{array}{@{}ccc@{}}O^{+}(\text{II}_{2,2+8k}\oplus A_{1}) & {\hookrightarrow} & \unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D709}}\\ g & \mapsto & \widetilde{g}.\end{array}\end{eqnarray}$$

It follows that the injection $\mathscr{D}_{v^{\bot }}^{+}{\hookrightarrow}\mathscr{D}_{\unicode[STIX]{x1D6EC}}^{+}$ descends to a regular map

(1.5.9) $$\begin{eqnarray}\mathscr{F}_{\text{II}_{2,2+8k}\oplus A_{1}}(O^{+}(\text{II}_{2,2+8k}\oplus A_{1}))\overset{m}{\longrightarrow }H_{u}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709}).\end{eqnarray}$$

Proposition 1.5.2. Let $(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})$ be a decorated $D$ -lattice of dimension $N\equiv 4\hspace{0.6em}({\rm mod}\hspace{0.2em}8)$ and let $N=8k+4$ , where $k\geqslant 0$ . The map $m$ in (1.5.9) is an isomorphism onto the unigonal Heegner divisor $H_{u}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})$ . Moreover, the intersection of $H_{u}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})$ and the singular locus of $\mathscr{F}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})$ has codimension at least $2$ in $H_{u}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})$ .

Proof. The proof is similar to the proof of Proposition 1.5.1 (see also Proposition 1.4.5). We leave details to the reader. ◻

In order to simplify notation, from now on we let

(1.5.10) $$\begin{eqnarray}\mathscr{F}(\text{II}_{2,2+8k}\oplus A_{1}):=\mathscr{F}_{\text{II}_{2,2+8k}\oplus A_{1}}(O^{+}(\text{II}_{2,2+8k}\oplus A_{1})).\end{eqnarray}$$

A vector $v\in \text{II}_{2,2+8k}\oplus A_{1}$ is nodal if it has square $-2$ and divisibility $1$ ; it is unigonal if it has square $-2$ and divisibility $2$ . By Eichler’s criterion, i.e. Proposition 1.3.2, any two nodal vectors of $\text{II}_{2,2+8k}\oplus A_{1}$ are $O^{+}(\text{II}_{2,2+8k}\oplus A_{1})$ -equivalent, and similarly for any two unigonal vectors. We let $H_{n}(\text{II}_{2,2+8k}\oplus A_{1})$ and $H_{u}(\text{II}_{2,2+8k}\oplus A_{1})$ be the Heegner divisors of $\mathscr{F}(\text{II}_{2,2+8k}\oplus A_{1})$ corresponding to a nodal or a unigonal $v\in \text{II}_{2,2+8k}\oplus A_{1}$ , respectively.

One describes the Heegner divisor $H_{u}(\text{II}_{2,2+8k}\oplus A_{1})$ proceeding as in § 1.5.1. In fact, let $v\in (\text{II}_{2,2+8k}\oplus A_{1})$ be a unigonal vector. Then $(\text{II}_{2,2+8k}\oplus A_{1})=\langle v\rangle \oplus v^{\bot }$ , and $v^{\bot }\cong \text{II}_{2,2+8k}$ is unimodular. It follows that the injection $\mathscr{D}_{v^{\bot }}^{+}{\hookrightarrow}\mathscr{D}_{\unicode[STIX]{x1D6EC}}^{+}$ descends to a regular map

(1.5.11) $$\begin{eqnarray}\mathscr{F}(\text{II}_{2,2+8k})\overset{p}{\longrightarrow }H_{u}(\text{II}_{2,2+8k}\oplus A_{1}),\end{eqnarray}$$

which satisfies the following proposition.

Proposition 1.5.3. Let $k\geqslant 0$ . The map $p$ of (1.5.11) is an isomorphism onto the unigonal Heegner divisor $H_{u}(\text{II}_{2,2+8k}\oplus A_{1})$ . Moreover, the intersection of $H_{u}(\text{II}_{2,2+8k}\oplus A_{1})$ and the singular locus of $\mathscr{F}(\text{II}_{2,2+8k}\oplus A_{1})$ has codimension at least $2$ in $H_{u}(\text{II}_{2,2+8k}\oplus A_{1})$ .

1.6 The (non-reflective) unigonal divisors for $N\equiv 5\hspace{0.6em}({\rm mod}\hspace{0.2em}8)$

Let $(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709})$ be a decorated $D$ -lattice of dimension $N\equiv 5\hspace{0.6em}({\rm mod}\hspace{0.2em}8)$ . Let $N=8k+5$ , where $k\geqslant 0$ ; thus, $\unicode[STIX]{x1D6EC}\cong \text{II}_{2,2+8k}\oplus D_{3}$ . Let $v\in \unicode[STIX]{x1D6EC}$ be a unigonal vector, i.e.  $v^{2}=-12$ and $\operatorname{div}(v)=4$ . Then

(1.6.1) $$\begin{eqnarray}v^{\bot }\cong \text{II}_{2,2+8k}\oplus A_{2}.\end{eqnarray}$$

Given $g\in \widetilde{O}^{+}(v^{\bot })$ , let $\widetilde{g}\in O(\unicode[STIX]{x1D6EC})$ be the isometry such that $\widetilde{g}(v)=v$ and $\widetilde{g}|_{v^{\bot }}=g$ . Then $\widetilde{g}\in O^{+}(\unicode[STIX]{x1D6EC})$ . Thus, we have an injection of groups

(1.6.2) $$\begin{eqnarray}\begin{array}{@{}ccc@{}}O^{+}(\text{II}_{2,2+8k}\oplus A_{2}) & {\hookrightarrow} & \widetilde{O}^{+}(\unicode[STIX]{x1D6EC})\\ g & \mapsto & \widetilde{g}.\end{array}\end{eqnarray}$$

It follows that the injection $\mathscr{D}_{v^{\bot }}^{+}{\hookrightarrow}\mathscr{D}_{\unicode[STIX]{x1D6EC}}^{+}$ descends to a regular map

(1.6.3) $$\begin{eqnarray}\mathscr{F}_{\text{II}_{2,2+8k}\oplus A_{2}}(\widetilde{O}^{+}(\text{II}_{2,2+8k}\oplus A_{2}))\overset{q}{\longrightarrow }H_{u}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D709}).\end{eqnarray}$$

The following proposition follows by similar arguments to those used above; we omit the proof.

In order to simplify notation, from now on we let

(1.6.4) $$\begin{eqnarray}\mathscr{F}(\text{II}_{2,2+8k}\oplus A_{2}):=\mathscr{F}_{\text{II}_{2,2+8k}\oplus A_{2}}(\widetilde{O}^{+}(\text{II}_{2,2+8k}\oplus A_{2})).\end{eqnarray}$$

A vector $v\in \text{II}_{2,2+8k}\oplus A_{2}$ is nodal if it has square $-2$ and divisibility $1$ ; it is unigonal if it has square $-12$ and divisibility $3$ . Notice that the set of nodal vectors of $\text{II}_{2,2+8k}\oplus A_{2}$ is a single $\widetilde{O}^{+}(\text{II}_{2,2+8k}\oplus A_{2})$ -orbit, and similarly the set of unigonal vectors up to $\pm 1$ is a single $\widetilde{O}^{+}(\text{II}_{2,2+8k}\oplus A_{2})$ -orbit. We let $H_{n}(\text{II}_{2,2+8k}\oplus A_{2})$ and $H_{u}(\text{II}_{2,2+8k}\oplus A_{2})$ be the Heegner divisors of $\mathscr{F}(\text{II}_{2,2+8k}\oplus A_{2})$ corresponding to a nodal or a unigonal $v\in \text{II}_{2,2+8k}\oplus A_{2}$ , respectively (both are irreducible).

Let $v\in (\text{II}_{2,2+8k}\oplus A_{2})$ be a unigonal vector. Then $v^{\bot }\cong \text{II}_{2,2+8k}\oplus A_{1}$ . As in § 1.5.2, we get a regular map

(1.6.5) $$\begin{eqnarray}\mathscr{F}(\text{II}_{2,2+8k}\oplus A_{1})\overset{r}{\longrightarrow }H_{u}(\text{II}_{2,2+8k}\oplus A_{2}).\end{eqnarray}$$

1.7 Nested locally symmetric varieties

1.7.1 The infinite tower of $D$ locally symmetric varieties

For $3\leqslant M$ , we let $\unicode[STIX]{x1D709}_{M}$ be a decoration of $\unicode[STIX]{x1D6EC}_{M}$ . Choose $N\geqslant 3$ . By Proposition 1.4.5, we have a sequence of inclusions of $D$ period spaces:

(1.7.1) $$\begin{eqnarray}\mathscr{F}(\unicode[STIX]{x1D6EC}_{3},\unicode[STIX]{x1D709}_{3})\overset{f_{4}}{{\hookrightarrow}}\mathscr{F}(\unicode[STIX]{x1D6EC}_{4},\unicode[STIX]{x1D709}_{4})\overset{f_{5}}{{\hookrightarrow}}\cdots \overset{f_{N-1}}{{\hookrightarrow}}\mathscr{F}(\unicode[STIX]{x1D6EC}_{N-1},\unicode[STIX]{x1D709}_{N-1})\overset{f_{N}}{{\hookrightarrow}}\mathscr{F}(\unicode[STIX]{x1D6EC}_{N},\unicode[STIX]{x1D709}_{N}).\end{eqnarray}$$

Thus, $\operatorname{Im}f_{M}=H_{h}(\unicode[STIX]{x1D6EC}_{M},\unicode[STIX]{x1D709}_{M})$ for $4\leqslant M\leqslant N$ . There is a unique continuation of the above sequence. In fact, let $w\in \unicode[STIX]{x1D6EC}_{N}$ be a vector of divisibility $2$ and such that $[v/2]=\unicode[STIX]{x1D709}_{N}$ (e.g. a hyperelliptic vector). Let $L\subset (\unicode[STIX]{x1D6EC}_{N}\oplus (-4))_{\mathbb{Q}}$ be the sublattice generated by $(\unicode[STIX]{x1D6EC}_{N}\oplus (-4))$ and $(w/2,1/2)$ . Then $L$ and $\unicode[STIX]{x1D6EC}_{N+1}$ are even lattices of signature $(2,N+1)$ and their discriminant groups (equipped with the discriminant quadratic forms) are isomorphic. By Theorem 1.13.2 of [Reference NikulinNik80], it follows that $L$ is isomorphic to $\unicode[STIX]{x1D6EC}_{N+1}$ ; we choose an identification of $L$ with $\unicode[STIX]{x1D6EC}_{N+1}$ . Let $\unicode[STIX]{x1D709}_{N+1}$ be the decoration of $\unicode[STIX]{x1D6EC}_{N+1}$ (i.e.  $L$ ) defined by $(0,1/2)$ . Then $v:=(0_{\unicode[STIX]{x1D6EC}_{N}},1)$ is a hyperelliptic vector of $\unicode[STIX]{x1D6EC}_{N+1}$ , and $v^{\bot }=\unicode[STIX]{x1D6EC}_{N}$ . Furthermore, $\unicode[STIX]{x1D709}_{N}$ is the decoration of $v^{\bot }$ associated to $\unicode[STIX]{x1D709}_{N+1}$ . In conclusion, there is an infinite prolongation of (1.7.1), unique up to isomorphism:

(1.7.2) $$\begin{eqnarray}\mathscr{F}(\unicode[STIX]{x1D6EC}_{3},\unicode[STIX]{x1D709}_{3})\overset{f_{4}}{{\hookrightarrow}}\cdots \overset{f_{N}}{{\hookrightarrow}}\mathscr{F}(\unicode[STIX]{x1D6EC}_{N},\unicode[STIX]{x1D709}_{N})\overset{f_{N+1}}{{\hookrightarrow}}\mathscr{F}(\unicode[STIX]{x1D6EC}_{N+1},\unicode[STIX]{x1D709}_{N+1})\overset{f_{N+2}}{{\hookrightarrow}}\cdots \,.\end{eqnarray}$$

For $3\leqslant M<N$ , we let $f_{M,N}:=f_{N}\circ f_{N-1}\circ \cdots \circ f_{M+1}$ . Thus,

(1.7.3) $$\begin{eqnarray}f_{M,N}:\mathscr{F}(\unicode[STIX]{x1D6EC}_{M},\unicode[STIX]{x1D709}_{M}){\hookrightarrow}\mathscr{F}(\unicode[STIX]{x1D6EC}_{N},\unicode[STIX]{x1D709}_{N}).\end{eqnarray}$$