1.2.1 Definition

The real symmetric space attached to $V_{\mathbb {R}}$ is the manifold, denoted $X_\infty $ , parametrizing maximal negative-definite subspaces of $V_{\mathbb {R}}$ . Note that $X_\infty $ is also in natural bijection with the set of maximal positive-definite subspaces of $V_{\mathbb {R}}$ , under the map $Z\mapsto Z^\perp $ .

The real orthogonal group $\mathrm {O}_V(\mathbb {R})$ , the special orthogonal group $G(\mathbb {R})={\mathrm {SO}}_V(\mathbb {R})$ , and its connected component $G(\mathbb {R})^+$ , all act transitively on $X_\infty $ . The stabilisers of a point $Z\in X_\infty $ in these three groups are

$$\begin{align*}\mathrm{O}_Z(\mathbb{R}) \times \mathrm{O}_{Z^{\perp}}(\mathbb{R}) \ \supset \ (\mathrm{O}_Z(\mathbb{R}) \times \mathrm{O}_{Z^{\perp}}(\mathbb{R}))_1 \ \supset \ {\mathrm{SO}}_Z(\mathbb{R})\times {\mathrm{SO}}_{Z^\perp}(\mathbb{R}), \end{align*}$$

where the group in the middle consists of elements of determinant $1$ . It follows that

$$ \begin{align*} X_\infty &= \mathrm{O}_V(\mathbb{R})/ (\mathrm{O}_Z(\mathbb{R}) \times \mathrm{O}_{Z^\perp}(\mathbb{R})) \\ &= G(\mathbb{R}) /(\mathrm{O}_Z(\mathbb{R}) \times \mathrm{O}_{Z^\perp}(\mathbb{R}))_1 \\ &= G(\mathbb{R})^+/({\mathrm{SO}}_Z(\mathbb{R})\times {\mathrm{SO}}_{Z^\perp}(\mathbb{R})). \end{align*} $$

Since the orthogonal group of a nondegenerate quadratic form of rank n has dimension $n(n-1)/2$ , the dimension of the real manifold $X_\infty $ is given by

$$\begin{align*}\dim X_\infty = rs. \end{align*}$$

As $\mathrm {O}_{Z^\perp }(\mathbb {R})\times \mathrm {O}_Z(\mathbb {R}) \subseteq O_V(\mathbb {R})$ is a maximal compact subgroup (see, for example, [Reference Hilgert and Neeb28][Proposition 17.2.5]), the polar decomposition [Reference Hilgert and Neeb28][Proposition 16.1.9] implies that the quotient $X_\infty $ is contractible. Since $X_\infty $ is an open subset of the Grassmannian of s-dimensional subspaces in $V_{\mathbb {R}}$ , the tangent space to $X_\infty $ at Z is canonically identified with

(5) $$ \begin{align} \begin{aligned} T_Z(X_\infty) &= \mathrm{Lie}(\mathrm{GL}(V_{\mathbb{R}}))/\mathrm{ Lie}({\mathrm{Stab}_{\mathrm{GL}(V_{\mathbb{R}})}}(Z)) \\ &\cong \operatorname{\mathrm{Hom}}_{\mathbb{R}}(Z,V_{\mathbb{R}}/Z) \\ &\cong \operatorname{\mathrm{Hom}}_{\mathbb{R}}(Z,Z^\perp). \end{aligned} \end{align} $$

The penultimate isomorphism is obtained by restricting an endomorphism of $V_{\mathbb {R}}$ to Z and composing it with the natural surjection $V_{\mathbb {R}}\rightarrow V_{\mathbb {R}}/Z$ .

The spaces Z and $Z^\perp $ are endowed with natural positive inner products obtained from the restrictions of q and $-q$ , respectively. The same is true as well for their duals, and for $\operatorname {\mathrm {Hom}}_{\mathbb {R}}(Z,Z^\perp ) = Z^\vee \otimes Z^{\perp }$ . The resulting euclidean structure on the tangent spaces $T_Z(X_\infty )$ endows $X_\infty $ with a Riemannian structure and a well-defined metric and volume form, which are $\mathrm {O}_{V}(\mathbb {R})$ -invariant. By [Reference Helgason27][Chapter V, Theorem 3.1], $X_\infty $ has non-positive sectional curvature everywhere. In particular, the Cartan–Hadamard theorem implies that, for any point $Z\in X_\infty $ , the exponential map

$$\begin{align*}\exp_Z\colon T_Z(X_\infty)\longrightarrow X_\infty \end{align*}$$

is a diffeomorphism and any two points in $X_\infty $ are connected by a unique geodesic.

Every arithmetic subgroup ${\Gamma _{\!\circ }}\subseteq G(\mathbb {Q})$ acts discretely on $X_\infty $ . The arithmetic quotient ${\Gamma _{\!\circ }}\backslash X_\infty $ has finite volume and is compact when V is anisotropic (i.e., contains no $v\neq 0$ with $q(v)=0$ ) (see, for example, [Reference Bruinier9, (2.25)]).

1.2.2 Functoriality of $X_{\infty }$

More generally, given a finite-dimensional real nondegenerate quadratic space $(W,q_W)$ , one defines $X_\infty (W,q_W)$ to be the space of maximal negative-definite subspaces of W. Sending a maximal negative-definite subspace of W to its orthogonal complement yields an isomorphism between the two symmetric spaces $X_\infty (W,q_W)$ and $X_\infty (W,-q_W)$ . Let $(W',q_W^{\prime })$ be another real nondegenerate quadratic space such that the dimension of a maximal negative subspace of $(W,q_W)$ and $(W',q_W^{\prime })$ agree. For any isometric embedding $f\colon (W,q_W) \hookrightarrow (W',q_W^{\prime })$ , the induced map

$$\begin{align*}X_{\infty}(W,q_W) \hookrightarrow X_{\infty}(W',q_W^{\prime}),\quad Z \longmapsto f(Z), \end{align*}$$

is an isometric embedding of Riemannian manifolds. In particular, if f is an isomorphism of quadratic spaces, then it induces an isometry $f\colon X_{\infty }(W,q_W) \simeq X_{\infty } (W',q_W^{\prime })$ of symmetric spaces.Footnote ² The latter observation implies that the symmetric space $X_{\infty }$ is isometric to $X_{\infty }(V_0,q_0)$ for any particular quadratic space $(V_0,q_0)$ over $\mathbb {R}$ having signature $(r,s)$ . It is sometimes convenient to work with a particular model quadratic space $(V_0,q_0)$ . In the below examples for small values of s, specializing to some particular models allows us to recover classical descriptions of certain well-known symmetric spaces.

1.2.3 Examples of $X_{\infty }$ for small s

Signature $(r,0)$ . In case $V_{\mathbb {R}}$ is positive or negative-definite, the group $G(\mathbb {R})$ is compact and the space $X_\infty $ consists of a single point.

Signature $(r,1)$ . When $s=1$ , the space $X_\infty $ is isometric to the set of lines in the negative cone of $(r+1)$ -dimensional Minkowski space; that is, $X_\infty $ consists of all lines $[x_1:\cdots : x_r: x_{r+1}]\in \mathbb {P}^r(\mathbb {R})$ satisfying the inequality

$$\begin{align*}x_1^2 + \cdots + x_r^2 - x_{r+1}^2 <0.\end{align*}$$

Thus, $X_\infty $ can be identified with the open unit ball in r-dimensional euclidean space, endowed with its natural hyperbolic structure.

Signature $(r,2)$ . The case $s=2$ plays a special role in the theory. First, it represents essentially the only setting where $X_\infty $ has the structure of a Hermitian symmetric domain, possessing a complex structure. Second, the corresponding symmetric space $X_\infty $ admits a convenient description which motivates the definition of the p-adic symmetric space $X_p$ taken up in Section 1.3.

A point in $X_\infty $ corresponds to a negative-definite two-dimensional space $W\subseteq V_{\mathbb {R}}$ . The choice of an orthogonal basis $(v_1,v_2)$ of $ W$ satisfying $q(v_1) = q(v_2)$ determines an isotropic vector in the complexification $V_{\mathbb {C}}$ of V satisfying $\langle \xi _W, \bar \xi _W\rangle <0$ . The complex line spanned by $\xi _W$ depends only on the orientation induced by $(v_1,v_2)$ on W. This suggests identifying $X_\infty $ with an open subset of the quadric of isotropic lines in $V_{\mathbb {C}}$ . More precisely, let Q be the smooth quadric over $\mathbb {Q}$ attached to V by setting

Letting $[v]$ denote the class in $Q(K)$ of the nonzero isotropic vector $v\in V_K$ , we define

(6)

where $v\mapsto \overline v$ denotes the complex conjugation on $V_{\mathbb {C}}$ . The Hermitian symmetric space $\widetilde {X}_\infty $ has two connected components which are interchanged by $[v]\mapsto [\overline {v}]$ . The map sending the line $[v]\in Q(\mathbb {C})$ spanned by $v = v_1 + i v_2$ , with $v_1, v_2 \in V_{\mathbb {R}}$ , to the negative-definite two-dimensional subspace spanned by $v_1$ and $v_2$ gives a two-to-one map $\widetilde {X}_\infty \rightarrow X_\infty $ and identifies each of the connected components with $X_\infty $ . Because $s = 2$ , the orthogonal complement of $\{v_1,v_2\}$ in $V_{\mathbb {R}}$ is positive-definite. In particular, it does not contain any isotropic vectors. It follows that

(7)

This is, in fact, an equality except if $r=2$ . In signature $(2,2)$ , the space $\widetilde {X}_\infty '$ has four connected components: two of them are given by $\widetilde {X}_\infty $ , while the other two correspond to positive-definite planes in $V_{\mathbb {R}}$ . For further discussion of the symmetric space $X_\infty $ in signature $(r,2)$ and its description in terms of what is called the projective model, see [Reference Bruinier9, §2.4].

We now specialise further to discuss indefinite quadratic spaces of dimensions $3$ and $4$ , in which $V_{\mathbb {R}}$ and $X_\infty $ admit useful concrete descriptions:

Signature $(1,2)$ . When $V_{\mathbb {R}}$ is of signature $(1,2)$ (resp. signature $(2,1)$ ), it can be identified with the subspace of trace zero elements $\mathrm {M}_2^0(\mathbb {R})$ of the space of real $2\times 2$ matrices, with quadratic form given by $q=\det $ (resp. $q=-\det $ ). The special orthogonal group $G(\mathbb {R})$ is identified with ${\mathrm {PGL}}_2(\mathbb {R})$ , and $g\in G(\mathbb {R})$ acts on $V_{\mathbb {R}}$ by conjugation:

Isotropic vectors in $V_{\mathbb {C}}$ correspond to nonzero linear endomorphisms of $\mathbb {C}^2$ of trace 0 and determinant 0 – that is, nonzero nilpotent endomorphisms. Such an endomorphism of $\mathbb {C}^2$ is uniquely determined by its kernel, which equals its image, leading to an identification

of the base change of the quadric Q to $\mathbb {R}$ with the projective line, on which $G(\mathbb {R})={\mathrm {PGL}}_2(\mathbb {R})$ acts via Möbius transformations. Under this identification, the hermitian symmetric space $\widetilde {X}_\infty \subseteq Q(\mathbb {C})$ is mapped to the union of the upper and lower complex half space. In coordinates, the identification can be written as follows: the nilpotent endomorphism of $\mathbb {C}^2$ with kernel (and image) spanned by equals

up to scaling. The line spanned by $M_\tau $ belongs to $\widetilde {X}_\infty $ if and only if this vector is not defined over $\mathbb {R}$ . The map $\tau \mapsto M_\tau $ therefore identifies the complex upper half space $\mathcal {H}$ with a connected component of $\widetilde {X}_\infty $ :

(8)

Signature $(2,2)$ . When $V_{\mathbb {R}}$ is of signature $(2,2)$ , it can be identified with the space $\mathrm {M}_2(\mathbb {R})$ of real $2\times 2$ matrices endowed with the determinant quadratic form. The spin group $\mathrm {Spin}_V(\mathbb {R})$ is identified with the group ${\mathrm {SL}}_2(\mathbb {R})\times {\mathrm {SL}}_2(\mathbb {R})$ . The map from ${\mathrm {SL}}_2(\mathbb {R})\times {\mathrm {SL}}_2(\mathbb {R})$ to $G(\mathbb {R})^+$ sends $(\gamma _1,\gamma _2)$ to the orthogonal transformation of $V_{\mathbb {R}}$ given by left and right multiplication:

(9)

This induces an identification of $({\mathrm {SL}}_2(\mathbb {R})\times {\mathrm {SL}}_2(\mathbb {R}))/\langle \pm 1\rangle $ with $G(\mathbb {R})^+$ (cf. the discussion in [Reference Bruinier9, §2.3]). Isotropic vectors in $V_{\mathbb {C}}$ correspond to linear endomorphisms of $\mathbb {C}^2$ of rank one. Such an endomorphism is uniquely determined, up to scaling, by its kernel and image, leading to an identification

of the quadric $Q_{\mathbb {R}}$ with the product of two copies of the projective line. Given $\tau _1, \tau _2 \in \mathbb {C}$ , the endomorphism of $\mathbb {C}^2$ with image and kernel spanned by the vectors respectively equals

up to scaling. Furthermore, the line spanned by $M_{\tau _1,\tau _2}$ belongs to the space $\widetilde {X}_\infty '$ defined in (7) if and only if $\tau _1,\tau _2\notin \mathbb {R}$ (i.e., if both $\tau _1$ and $\tau _2$ belong to the union of the complex upper and lower half planes). The map $(\tau _1,\tau _2) \mapsto [M_{\tau _1,\tau _2}]$ thus identifies $\mathcal {H}\times \mathcal {H}$ with one connected component of $\widetilde {X}^{\prime }_\infty $ , and therefore, one may identify

$$\begin{align*}X_\infty \cong \mathcal{H} \times \mathcal{H}. \end{align*}$$

Under this identification, the action of $G(\mathbb {R})^+$ on $X_\infty $ corresponds to the natural action of $({\mathrm {SL}}_2(\mathbb {R})\times {\mathrm {SL}}_2(\mathbb {R}))/\langle \pm 1\rangle $ on $\mathcal {H}\times \mathcal {H}$ by Möbius transformations on each coordinate.

Signature $(3,1)$ . When $V_{\mathbb {R}}$ is of signature $(3,1)$ , it can be identified with the space

$$\begin{align*}\{ M\in \mathrm{M}_2(\mathbb{C}) \ \vert\ M' = -\overline{M} \}, \qquad q(M) = -{\det}(M), \end{align*}$$

where $M'$ denotes the principal involution on $M_2(\mathbb {C})$ satisfying $MM' = \det (M)$ , and $\overline {M}$ is the complex conjugate of M. The action

of ${\mathrm {PSL}}_2(\mathbb {C})$ on $V_{\mathbb {R}}$ by twisted conjugation identifies this group with the connected component $G(\mathbb {R})^{+}$ of the special orthogonal group of $V_{\mathbb {R}}$ . A point in $X_\infty $ , represented by a negative line in $V_{\mathbb {R}}$ , is spanned by a unique vector of the form with $z\bar z+t<0$ . The map

is a bijection.

The case of signature $(3,1)$ is noteworthy for representing the first scenario in ranks $3$ or $4$ where $X_\infty $ does not admit a complex structure. It will be discussed at greater length in Section 5.2.

1.3.1 Definition

Recall from the introduction that p is assumed to be an odd prime for which $V_{\mathbb {Q}_p}$ admits a self-dual lattice $\Lambda $ . When $n\geq 3$ , the nondegenerate quadratic $\mathbb {F}_p$ -space $\Lambda /p\Lambda $ is isotropic (i.e., contains a nonzero isotropic vector by the Chevalley–Warning theorem). The same is thus true for $\Lambda $ by Hensel’s Lemma. It follows that $V_{\mathbb {Q}_p}$ is isotropic as well: in fact, the Witt indices of $V_{\mathbb {Q}_p}$ and $\Lambda /p\Lambda $ (the dimensions of a maximal isotropic subspace) are essentially maximal: equal to $(n-1)/2$ if n is odd, and greater or equal to $(n-2)/2$ if n is even. This follows from a standard induction argument reducing the statement for $V_{\mathbb {Q}_p}$ to the same statement for the orthogonal complement of a hyperbolic plane in $V_{\mathbb {Q}_p}$ , and likewise for $\Lambda /p\Lambda $ .

The description of $X_\infty $ (or rather, of its double cover $\widetilde {X}_{\infty }$ ) given in (7) in the special case of signature $(r,2)$ motivates the following definition of the p-adic symmetric space $X_p$ , which shall be adopted for all signatures:

(10)

The running assumption on $V_{\mathbb {Q}_p}$ implies that the boundary $Q(\mathbb {Q}_p)$ of $X_p$ is nonempty.

1.3.2 Rigid analytic structure

Like the real symmetric space $X_\infty $ of (6) in signature $(r,2)$ , the space $X_p$ admits a natural analytic structure: in this instance, a rigid analytic structure over $\mathbb {Q}_p$ obtained by expressing $X_p$ as an increasing union of affinoid subsets, as will now be described.

The choice of a self-dual $\mathbb {Z}_p$ -lattice $\Lambda \subseteq V_{\mathbb {Q}_p}$ extends the projective space $\mathbb {P}_{V_{\mathbb {Q}_p}}$ over $\mathbb {Q}_p$ to a smooth proper model $\mathbb {P}_\Lambda $ over $\mathrm {Spec}(\mathbb {Z}_p)$ . Since $\Lambda $ is self-dual, the quadric of isotropic lines in $\Lambda /p\Lambda $ is nonsingular over $\mathbb {F}_p$ . It follows that the zero locus of q in $\mathbb {P}_\Lambda $ extends the quadric $Q_{\mathbb {Q}_p}$ to a smooth integral model $Q_{\Lambda }$ over $\mathrm {Spec}(\mathbb {Z}_p)$ . By the valuative criterion for properness,

$$\begin{align*}Q(\mathbb{Q}_p)=Q_{\Lambda}(\mathbb{Z}_p)\quad \mbox{ and } \quad Q(\mathbb{C}_p)=Q_{\Lambda}(\mathcal{O}_{\mathbb{C}_p}). \end{align*}$$

In more concrete terms, put $\Lambda _{\mathcal {O}_{\mathbb {C}_p}}=\Lambda \otimes _{\mathbb {Z}_p} \mathcal {O}_{\mathbb {C}_p}$ and let

(11)

denote the sets of primitive vectors in $\Lambda $ and $\Lambda _{\mathcal {O}_{\mathbb {C}_p}}$ , respectively. Each $\xi \in Q(\mathbb {Q}_p)$ (resp. $Q(\mathbb {C}_p)$ ) can be represented by an isotropic vector $v_\xi $ in $\Lambda '$ (resp. in $\Lambda _{\mathcal {O}_{\mathbb {C}_p}}'$ ) which is well-defined up to multiplication by $\mathbb {Z}_p^{\times }$ (resp. $\mathcal {O}_{\mathbb {C}_p}^{\times }$ ) and extends $\xi $ to a point of $Q_{\Lambda }$ over $\mathbb {Z}_p$ or $\mathcal {O}_{\mathbb {C}_p}$ , respectively.

For each integer $k\geq 0$ and every self-dual lattice $\Lambda \subseteq V_{\mathbb {Q}_p}$ , the associated basic affinoid subset $X_{p,\Lambda }^{\leq k}\subseteq X_p$ is defined to be

$$\begin{align*}X_{p,\Lambda}^{\leq k} = \left\{\xi \in Q(\mathbb{C}_p), \mbox{ with } {\mathrm{ord}}_p(\langle v_\xi,w\rangle) \leq k \mbox{ for all } w\in (\Lambda')_0\right\}, \end{align*}$$

where the subscript $0$ denotes the set of isotropic vectors. The sets $X_{p,\Lambda }^{\leq k}$ are invariant under the compact group ${\mathrm {SO}}(\Lambda )$ .

Proof. It suffices to prove the inclusion

$$ \begin{align*}\left(\bigcup_{k=0}^{\infty} X_{p,\Lambda}^{\leq k}\right)^c \subseteq X_p^c, \end{align*} $$

where $A^c$ denotes the complement of a subset A inside the projective space $\mathbb {P}_V(\mathbb {C}_p).$ If $\xi $ belongs to $ \bigcap (X_{p,\Lambda }^{\leq k})^c$ , then for each $k \geq 1$ , there is a vector $w_k \in (\Lambda ')_0$ satisfying

$$ \begin{align*}\langle v_\xi, w_k\rangle \equiv 0 \bmod{p^k \mathcal{O}_{\mathbb{C}_p}}.\end{align*} $$

By the compactness of $(\Lambda ')_0$ , the sequence $(w_k)_{k\geq 1}$ contains a convergent subsequence, whose limit is an element $w\in (\Lambda ')_0$ satisfying

$$ \begin{align*}\langle v_\xi,w\rangle = 0.\end{align*} $$

It follows that $\xi $ belongs to $X_p^c$ , as was to be shown.

1.3.3 Low-dimensional examples of $X_p$

There is a related p-adic symmetric space that features prominently in the literature (cf. [Reference Shen42, Appendix A]). Namely, let $Q^{\mathrm {wa}}$ be the connected rigid analytic space over $\mathbb {Q}_p$ for which

$$\begin{align*}Q^{\mathrm{wa}}(\mathbb{C}_p)\ =\ \{\xi \in Q(\mathbb{C}_p)\ \vert\ \xi \cap D_{\mathbb{C}_p}=0 \mbox{ for all totally isotropic } D\subseteq V_{\mathbb{Q}_p} \}. \end{align*}$$

By definition, the p-adic symmetric space $X_p$ is an admissible open subspace of $Q^{\mathrm {wa}}$ . In small dimensions, the two spaces agree and admit familiar descriptions, as will be explained below.

Example 1.4 (Three-dimensional quadratic spaces).

The dimension of a totally isotropic subspace of a three-dimensional quadratic space is at most one, and hence,

$$\begin{align*}X_p=Q^{\mathrm{wa}}= Q(\mathbb{C}_p) \!-\! Q(\mathbb{Q}_p). \end{align*}$$

Let us consider the special case that $V_{\mathbb {Q}_p}$ is given by the space of $2 \times 2$ matrices with trace zero. Just as in the discussion of real quadratic spaces of signature $(2,1)$ , one obtains an identification of $G(\mathbb {Q}_p)$ with ${\mathrm {PGL}}_2(\mathbb {Q}_p)$ and of the quadric $Q_{\mathbb {Q}_p}$ with the projective line over $\mathbb {Q}_p$ . The same reasoning that led to (8) leads to the analogous identification

(12)

of $X_p$ with the Drinfeld p-adic upper half plane $\mathcal {H}_p$ , on which $G(\mathbb {Q}_p) = {\mathrm {PGL}}_2(\mathbb {Q}_p)$ acts via Möbius transformations.

Example 1.5 (Four-dimensional quadratic space of Witt index 2).

An explicit model of the four-dimensional quadratic space over $\mathbb {Q}_p$ of Witt index $2$ is given by

(13) $$ \begin{align} V_{\mathbb{Q}_p}=\mathrm{M}_2(\mathbb{Q}_p) \quad \mbox{with} \quad \langle A,A \rangle=\det(A). \end{align} $$

The group ${\mathrm {SL}}_2(\mathbb {Q}_p) \times {\mathrm {SL}}_2(\mathbb {Q}_p)$ acts on $V_{\mathbb {Q}_p}$ via (9), just as in the real case, inducing an isomorphism

(14) $$ \begin{align} G(\mathbb{Q}_p)^{+} = ({\mathrm{SL}}_{2}(\mathbb{Q}_p) \times {\mathrm{SL}}_{2}(\mathbb{Q}_p))/\{\pm 1\}, \end{align} $$

as well as an identification

$$ \begin{align*}X_p = \mathcal{H}_p \times \mathcal{H}_p \end{align*} $$

of rigid analytic varieties. The action of $G(\mathbb {Q}_p)^{+}$ on $X_p$ is by componentwise Möbius transformations under these identifications.

Let $\xi =[v_\xi ]$ be an element of $Q(\mathbb {C}_p)$ that is not contained in $X_p$ (i.e., for which there is an element $[w]\in Q(\mathbb {Q}_p)$ with $\langle w,v_\xi \rangle =0).$ The two-dimensional regular quadratic space $V'=w^\perp /\operatorname {\mathrm {span}}(w)$ contains an isotropic vector and hence is a hyperbolic plane over $\mathbb {Q}_p$ . Hence, the image of $v_\xi $ in $V^{\prime }_{\mathbb {C}_p}$ is a multiple of a $\mathbb {Q}_p$ -rational isotropic vector. Let $w'\in V_{\mathbb {Q}_p}$ be a preimage of this isotropic vector. By construction, w and $w'$ span a totally isotropic subspace $D\subseteq V_{\mathbb {Q}_p}$ and $v_\xi $ is an element of $D_{\mathbb {C}_p}.$ Thus, v is not an element of $Q^{\mathrm {wa}}$ , and it follows that

$$\begin{align*}X_p=Q^{\mathrm{wa}}. \end{align*}$$

The two examples above are the only cases in which $X_p=Q^{\mathrm {wa}}$ holds. Indeed, assume we are not in one of these situations. Decompose $V_{\mathbb {Q}_p}$ into an orthogonal direct sum

$$\begin{align*}V_{\mathbb{Q}_p}=H\oplus U\end{align*}$$

with H being an hyperbolic plane. If V if four-dimensional, U is anisotropic by assumption. The line generated by any isotropic vector $w_U \in U_{\mathbb {C}_p}$ is clearly contained in $Q^{\mathrm {wa}}$ but not in $X_p$ . If $\dim V \geq 5$ , the quadratic space U fulfils the assumptions we imposed on $V_{\mathbb {Q}_p}$ . Thus, replacing $V_{\mathbb {Q}_p}$ by U, one can construct the p-adic symmetric space $X_p(U)$ as an admissible open subset of the quadric of isotropic lines in U. Take any point of $X_p(U)$ . Then, its image in $Q(\mathbb {C}_p)$ is contained in $Q^{\mathrm {wa}}$ but not in $X_p$ . This also shows that the formation of $X_p$ is – in contrast to the formation of $Q^{\mathrm {wa}}$ – not functorial with respect to isometric embeddings of quadratic spaces.

1.3.4 GIT characterization of $X_p$

We now turn to a description of the points in $X_p$ and $Q^{\mathrm {wa}}$ in the spirit of ‘geometric invariant theory’, characterizing them in terms of their stabilisers and their orbits under the action of split subtori in $G(\mathbb {Q}_p)$ . This in turn relates these spaces to p-adic period spaces appearing in the theory of local Shimura varieties (see Remark 1.9 below). Split subtori of orthogonal groups are closely related to totally isotropic subspaces. More precisely, consider a nondegenerate quadratic space $(W,q_W)$ over field K with $\mathrm {char}(K)\neq 2$ with associated bilinear form $b_W(\cdot ,\cdot )$ , and a nontrivial morphism

$$\begin{align*}\mu\colon T \longrightarrow {\mathrm{SO}}_W \end{align*}$$

of a split torus T over K to the special orthogonal group ${\mathrm {SO}}_W$ . Remember that a split torus over K is just a finite product of copies of $\mathbb {G}_{m,K}$ . Let $X^\ast (T)$ be the character lattice of T – that is, the group of all (K-rational) homomorphisms from T to $\mathbb {G}_{m,K}$ . The action of T on W via $\mu $ induces the decomposition

$$\begin{align*}W=\bigoplus_{\chi \in X^\ast(T)} W_{\chi} \end{align*}$$

into eigenspaces

For $w_1\in W_{\chi _1}$ and $w_2\in W_{\chi _2}$ , one computes

$$\begin{align*}b_W(w_1,w_2)=b_W(\mu(t)w_1,\mu(t)w_2)=\chi_1(t)\chi_2(t)b_W(w_1,w_2)\ \mbox{for all}\ t\in T(K). \end{align*}$$

It follows that $W_{\chi _1}$ and $W_{\chi _2}$ are orthogonal for all characters $\chi _1,\chi _2\in X^\ast (T)$ with $\chi _1\neq \chi _2^{-1}$ . In particular, $W_{\chi }$ is isotropic if $\chi $ is nontrivial. One easily deduces that in that case, $W_{\chi }\oplus W_{\chi ^{-1}}$ is a sum of hyperbolic planes and that the eigenspace $W_{0}$ of the trivial character is a nondegenerate subspace.

In the two examples of Section 1.3.3, the symmetric space $X_p$ consists of exactly the points in $Q(\mathbb {C}_p)$ that are not stabilized by a $\mathbb {Q}_p$ -split torus in $G_{\mathbb {Q}_p}$ . Indeed, a split subtorus of ${\mathrm {SL}}_2(\mathbb {Q}_p)$ has exactly two fixed points in $\mathbb {P}^1(\mathbb {C}_p)$ , both of which already lie in $\mathbb {P}^1(\mathbb {Q}_p)$ . Vice versa, every element in $\mathbb {P}^1(\mathbb {Q}_p)$ is a fixed point of a split subtorus in ${\mathrm {SL}}_2(\mathbb {Q}_p)$ . The following illustrative example shows: in dimension $> 4$ , yet more points of $Q(\mathbb {C}_p)$ beyond those stabilized by a $\mathbb {Q}_p$ -split torus in $G_{\mathbb {Q}_p}$ must be removed in order to obtain $X_p$ from $Q(\mathbb {C}_p)$ .

Example 1.6. Assume that the dimension of $V_{\mathbb {Q}_p}$ is of dimension $5$ . Write $V_{\mathbb {Q}_p}$ as the orthogonal direct sum

$$\begin{align*}V_{\mathbb{Q}_p}=H \oplus U, \end{align*}$$

with H being a hyperbolic plane. Let $0\neq w_H\in H$ be an isotropic vector, $w_U\in U_{\mathbb {C}_p}$ an isotropic vector such that

$$\begin{align*}\langle w_U,u \rangle\notin \mathbb{Q}_p\quad \forall u\in U\!-\!\{0\}, \end{align*}$$

and put . It is easy to check that $v^\perp \cap V_{\mathbb {Q}_p}=\mathbb {Q}_p w.$ In particular, the image $\xi =[v]$ of v in $ Q(\mathbb {C}_p)$ does not belong to $X_p$ . We argue next that $\xi $ is not stabilized by any $\mathbb {Q}_p$ -split torus.

Suppose that $T\subseteq G_{\mathbb {Q}_p}$ is a $\mathbb {Q}_p$ -split torus that stabilizes the line $\xi =[v]$ with corresponding eigenspace decomposition

$$\begin{align*}V_{\mathbb{Q}_p}=\bigoplus_{\chi \in X^\ast(T)} V_{\mathbb{Q}_p,\chi}. \end{align*}$$

The dimension of the subspace $\dim V_{\mathbb {Q}_p,0}$ must be odd, as it is the orthogonal complement of a direct sum of hyperbolic planes. In particular, it is nontrivial. However, since $G_{\mathbb {Q}_p}$ , and hence T, acts faithfully on $V_{\mathbb {Q}_p}$ , we must have that $V_{\mathbb {Q}_p, \chi ^{\pm 1}} \neq \{0\}$ for some nontrivial character $\chi \in X^{\ast }(T)$ . It follows that $\dim v^\perp \cap V_{\mathbb {Q}_p} \geq 2$ . Indeed,

• ○ Since T stabilizes $\xi =[v]$ , the vector v is an eigenvector for the action of $T.$ If $v \in V_{\mathbb {C}_p,0}$ , then it is perpendicular to $V_{\mathbb {Q}_p, \chi } \oplus V_{\mathbb {Q}_p,\chi ^{-1}}$ , which has dimension at least 2. If $v \in V_{\mathbb {C}_p,\chi ^{\pm 1}}$ , however, then v is perpendicular to $V_{\mathbb {Q}_p,0} \oplus V_{\mathbb {Q}_p, \chi ^{\mp 1}}$ , which has dimension at least 2

Since $v^{\perp } \cap V_{\mathbb {Q}_p} = \mathbb {Q}_p w$ is only one-dimensional, the above gives a contradiction. Thus, the stabilizer of $\xi $ contains no $\mathbb {Q}_p$ -split torus.

The point $\xi $ in the example above is not stabilized by a $\mathbb {Q}_p$ -split subtorus of $G_{\mathbb {Q}_p}$ . Nevertheless, it is still unstable with respect to such a subtorus in the sense described next. Recall the definition of (semi)stability with respect to one-parameter subgroups: fix a one-parameter $\mathbb {Q}_p$ -subgroup

$$\begin{align*}\mu\colon\mathbb{G}_{m,\mathbb{Q}_p}\rightarrow G_{\mathbb{Q}_p} \end{align*}$$

(i.e., a nontrivial homomorphism of algebraic groups over $\mathbb {Q}_p$ ). A point $\xi \in Q(\mathbb {C}_p)$ with lift $v_{\xi }\in V_{\mathbb {C}_p}$ is called stable with respect to $\mu $ if

• ○ the stabilizer of $v_{\xi }$ in $\mathbb {C}_p^{\times }$ is finite and

• ○ the $\mathbb {C}_p^{\times }$ -orbit of $v_{\xi }$ is Zariski-closed in $V_{\mathbb {C}_p}$ .

It is called semistable with respect to $\mu $ if $0$ does not lie in the closure of $\mathbb {C}_p^{\times }$ -orbit of $v_\xi $ . Every stable point is clearly also semistable. As the character lattice of $\mathbb {G}_m$ is canonically isomorphic to $\mathbb {Z}$ , we can write the eigenspace decomposition as

$$ \begin{align*} V_{\mathbb{Q}_p}=\bigoplus_{n\in \mathbb{Z}} V_{\mathbb{Q}_p,n}. \end{align*} $$

It is easy to see that $\xi $ is stable with respect to $\mu $ if and only if $v_\xi $ is neither contained in

$$ \begin{align*} V_{\mathbb{C}_p, \geq 0}=\bigoplus_{n\geq 0} V_{\mathbb{C}_p,n}\quad \mbox{nor in}\quad V_{\mathbb{C}_p, \leq 0}=\bigoplus_{n\geq 0} V_{\mathbb{C}_p,n}. \end{align*} $$

In particular, if x is fixed by $\mathbb {G}_{m,\mathbb {C}_p}$ , then x is not stable. Similarly, a point $\xi $ is semistable if $v_\xi $ is neither contained in

$$ \begin{align*} V_{\mathbb{C}_p,> 0}=\bigoplus_{n> 0} V_{\mathbb{C}_p,n}\quad \mbox{nor in}\quad V_{\mathbb{C}_p, < 0}=\bigoplus_{n< 0} V_{\mathbb{C}_p,n}. \end{align*} $$

Proof. (a): Suppose there exists a one-parameter $\mathbb {Q}_p$ -subgroup $\mu \colon \mathbb {G}_{m,\mathbb {Q}_p}\hookrightarrow G_{\mathbb {Q}_p}$ such that $\xi $ is not stable with respect to $\mu $ . After possibly replacing $\mu $ by $-\mu $ , we may assume that $v_{\xi }$ is an element of the nonnegative part $V_{\mathbb {C}_p, \geq 0}$ of the eigenspace decomposition induced by $\mu $ . Let $n_0> 0$ be an integer with $V_{\mathbb {C}_p,n_0}\neq 0$ , which exists as $\mu $ is nontrivial. Then $\langle w,v_{\xi }\rangle =0$ for all $w\in V_{\mathbb {Q}_p,n_0}$ , and hence, $\xi $ is not an element of $X_p$ .

Conversely, suppose that there exists an isotropic vector $w\in V_{\mathbb {Q}_p}$ such that $\langle w,v_{\xi }\rangle =0.$ There exists an isotropic vector $w'\in V_{\mathbb {Q}_p}$ with

$$\begin{align*}\langle w,w'\rangle=1. \end{align*}$$

Let U be the orthogonal complement of $\operatorname {\mathrm {span}}\{w,w'\}$ . This yields a one-parameter $\mathbb {Q}_p$ -subgroup $\mu \colon \mathbb {G}_{m,\mathbb {Q}_p}\rightarrow G_{\mathbb {Q}_p}$ as follows:

$$\begin{align*}\mu(t)(v) = \begin{cases} t\cdot v & \mbox{ if } v \in \mathbb{Q}_p w,\\ t^{-1}\cdot v &\mbox{ if } v \in \mathbb{Q}_p w', \\ v & \mbox{ if } v \in U. \end{cases} \end{align*}$$

In other words, there is an eigenspace decomposition of the form

$$\begin{align*}V_{\mathbb{Q}_p}=V_{\mathbb{Q}_p,1}\oplus V_{\mathbb{Q}_p,-1}\oplus V_{\mathbb{Q}_p,0}\ \mbox{with}\ V_{\mathbb{Q}_p,1}=\mathbb{Q}_p w. \end{align*}$$

Because w and $v_{\xi }$ are perpendicular, we see that $v_{\xi }\in V_{\mathbb {C}_p,1}\oplus V_{\mathbb {C}_p,0}$ , and therefore, $\xi $ is not stable with respect to $\mu $ .

The proof of (b) follows from the observation that every isotropic subspace $D\subseteq V_{\mathbb {Q}_p}$ can be realized as $V_{\mathbb {Q}_p,>0}$ for some one-parameter subgroup.

2.2.1 Definitions

A vector $v\in V_{\mathbb {R}}$ is said to be of positive length if $q(v)> 0$ . Denote by $V_{\mathbb {R},+}$ (resp. $V_{+}$ ) the subset of $V_{\mathbb {R}}$ (resp. V) of vectors of positive length. In [Reference Kudla and Millson33], Kudla and Millson attach to each $v\in V_{\mathbb {R},+}$ a real topological cycle $\Delta _{v,\infty }\subseteq X_\infty $ , consisting of the maximal negative subspaces that are orthogonal to v or, equivalently, the set of maximal positive subspaces that contain v:

$$\begin{align*}\Delta_{v,\infty}=\left\{Z\in X_\infty \ \vert\ v\in Z^\perp \right\}. \end{align*}$$

The cycle $\Delta _{v,\infty }$ is identified with the Archimedean symmetric space attached to $(\mathbb {R} v)^\perp $ , a quadratic space of signature $(r-1,s)$ . Hence, $\Delta _{v,\infty }$ is of dimension $(r-1)s$ and defines a cycle of real codimension s in $X_\infty $ . Alternatively, let

be the tautological rank s vector bundle over $X_\infty $ whose fibre over Z (relative to the second coordinate projection) is the vector space Z itself. A positive vector v defines a section $s_v\colon X_\infty \rightarrow \tau $ sending Z to $(v_Z,Z)$ , where $v_Z$ is the orthogonal projection of v onto Z. The cycle $\Delta _{v,\infty }$ is the zero locus of this section.

Examples for small s

When $s=1$ , the cycle $\Delta _{v,\infty }$ is identified with the set of real lines in the negative cone that intersect the hyperplane $v^\perp \subseteq V_{\mathbb {R}}$ . It is isomorphic to an open unit ball in $(r-1)$ -dimensional euclidean space, embedded as a codimension one submanifold of $X_\infty $ . The complement of $\Delta _{v,\infty }$ is the union of two disjoint open subsets of $X_\infty $ .

When V is of signature $(2,1)$ , this gives a qualitative description of the cycle $\Delta _{v,\infty }$ as a hyperbolic line segment joining two points of the boundary of $X_\infty $ relative to the ‘open unit disc’ model of $X_\infty $ . Alternatively, under the identification of $\mathcal {H}$ with $X_\infty $ sending $\tau $ to given in (8), the cycle $\Delta _{v,\infty }$ attached to the positive vector corresponds to the set of $\tau =x+iy\in \mathcal {H}$ for which v belongs to the real vector space spanned by

$$ \begin{align*}\mathrm{Re}(M_\tau) = \begin{pmatrix} x &-x^2+y^2 \\ 1 & -x\end{pmatrix} \quad\mbox{and}\quad \mathrm{Im}(M_\tau) = \begin{pmatrix} y & -2xy \\ 0 & -y\end{pmatrix} , \end{align*} $$

that is, for which $\tau $ satisfies the equation

$$ \begin{align*}a \cdot |\tau|^2 + b \cdot \mathrm{Re}(\tau) + c = 0.\end{align*} $$

This leads the familiar geodesic cycle in $\mathcal {H}$ joining the real roots of the quadratic equation $ax^2+bx+c$ of positive discriminant $b^2-4ac$ , which plays a key role in defining the rigid meromorphic cocycles of [Reference Darmon and Vonk18].

2.2.2 Orientations

We recall some of the generalities concerning orientations on special cycles $\Delta _{v,\infty }$ as described in [Reference Kudla and Millson33, pp. 130-131].

An orientation on a real vector space W of dimension d is a choice of a connected component $\mathfrak {o}\subseteq \wedge ^d W -\{0\}\simeq \mathbb {R}-\{0\}$ . A basis $(w_1,\ldots ,w_d)$ of W is then said to be positively oriented, or simply positive, if $w_1 \wedge \cdots \wedge w_d$ belongs to $\mathfrak {o}$ . An orientation on W induces one on its dual $W^\ast $ , by declaring that the dual basis of any positive basis for W is positive. Given oriented real vector spaces $W_1$ and $W_2$ with positive bases $\{w_{1,1},\ldots , w_{1,k}\}$ and $\{w_{2,1},\ldots , w_{1,\ell }\}$ , the tensor product $W_1\otimes W_2$ is oriented by the convention that the basis $\{w_{1,i}\otimes w_{2,j}\ \vert \ 1\leq i \leq k, 1\leq j \leq \ell \}$ ordered lexicographically from right to left is positive. The space $\operatorname {\mathrm {Hom}}_{\mathbb {R}}(W_1,W_2)\cong W_1^\ast \otimes _{\mathbb {R}} W_2$ is oriented accordingly. Note that if $\dim _{\mathbb {R}} W_2=1$ , changing the orientation on $W_2$ changes the orientation on $W_1\otimes _{\mathbb {R}} W_2$ if and only if $\dim _{\mathbb {R}} W_1$ is odd.

Fix an orientation on $V_{\mathbb {R}}$ once and for all. Because $X_{\infty }$ is contractible, the real vector bundle $\tau \rightarrow X_{\infty }$ is orientable. An orientation on one of the fibres $\tau _Z$ of $\tau $ thus equips each maximal negative-definite subspace of $V_{\mathbb {R}}$ with a consistent choice of orientation. This choice can be described concretely by writing any maximal negative subspace $Z'\in X_\infty $ as $gZ$ for some $g\in G(\mathbb {R})^+$ and declaring a basis of $Z'$ to be positive if it is the image under g of a positive basis of Z. Similar remarks apply to $Z^\perp $ ; alternately, a basis for a maximal positive-definite subspace $W\subseteq V_{\mathbb {R}}$ is said to be positively oriented if completing it by a positive basis of $W^\perp $ yields a positive basis of $V_{\mathbb {R}}$ .

Let $\Delta _{v,\infty }$ be the special cycle attached to $v\in V$ , and let $Z\in \Delta _{v,\infty }$ . The subspace $U\subseteq V_{\mathbb {R}}$ generated by v is oriented by saying that $\{v\}$ is a positive basis. Following the reasoning in (5), there are canonical isomorphisms

$$ \begin{align*}T_Z(X_\infty) \cong \operatorname{\mathrm{Hom}}_{\mathbb{R}}(Z,Z^\perp), \qquad \nu_Z(X_\infty) \cong \operatorname{\mathrm{Hom}}_{\mathbb{R}}(Z,U), \end{align*} $$

where $\nu _Z(X_\infty )$ denotes the normal space of $\Delta _{v,\infty }$ at Z. These isomorphisms lead to orientations on both the tangent and normal spaces of $X_\infty $ at Z. In other words, the tangent space $T_Z(\Delta _{v,\infty })$ is oriented by the convention that a positive basis of $T_Z(\Delta _{v,\infty })$ completed by a positive basis of $\nu _Z(\Delta _{v,\infty })$ is a positive basis of $T_Z(X_\infty )$ . The cycle $\Delta _{v,\infty }$ is equipped with the induced orientation. The spaces $\Delta _{v,\infty }$ and $\Delta _{-v,\infty }$ carry the same orientation if s is even, while they carry opposite orientations if s is odd. One immediately deduces that the action of $g \in G(\mathbb {R})^{+}$ preserves the chosen orientation on the cycles $\Delta _{v,\infty }$ ; that is, there is an equality of oriented cycles

$$\begin{align*}g\Delta_{v,\infty}=\Delta_{gv,\infty}. \end{align*}$$

2.2.3 Geodesic properties of special cycles on $X_{\infty }$

Let $v \in V_{\mathbb {R}}$ be a positive vector. The reflection

$$\begin{align*}s_v\colon V_{\mathbb{R}} \longrightarrow V_{\mathbb{R}},\quad x \longmapsto x - 2 \frac{\langle x,v \rangle}{\langle v,v \rangle} \cdot v \end{align*}$$

is an element of the orthogonal group $O_{V}(\mathbb {R})$ and thus acts isometrically on $X_\infty $ .

2.2.4 Homology of complements of special cycles

Given a topological space X, denote by the singular chain complex of X with integer coefficients; that is, $C_q(X)$ is the free abelian group generated by all continuous maps from the standard q-simplex to X. If A is a subspace of X, write

for the relative singular chain complex with associated relative homology groups $\mathrm {H}_q(X,A)$ for $q\geq 0$ .

Proposition 2.4. Let v be an element of $V_{\mathbb {R}}$ of positive length. Then

$$ \begin{align*}\mathrm{H}_q(X_\infty,X_\infty\!-\! \Delta_{v,\infty}) \cong\begin{cases} \mathbb{Z} & \mbox{if}\ q=s \\ 0 & \mbox{ otherwise.} \end{cases} \end{align*} $$

Proof. We may assume that $s\geq 1$ , the case $s=0$ being trivial.

The assertion is a consequence of the long exact sequence for relative homology and the following claim: $X_\infty \!-\! \Delta _{v,\infty }$ is homotopy equivalent to an $(s-1)$ -sphere. Indeed, since $\Delta _{v,\infty }$ is a totally geodesic submanifold of the non-positively curved symmetric space $X_\infty $ , the exponential map induces a diffeomorphism

for every $Z\in \Delta _{v,\infty }$ . But for any finite dimensional $\mathbb {R}$ -vector space V, the complement of a proper subspace $W\subsetneq V$ of codimension s is homotopy equivalent to $(V/W) \!-\! \{0\}$ , which in turn is homotopy equivalent to an $(s-1)$ -sphere.

Let Z be an element of $\Delta _{v,\infty }$ . The chosen orientations on $T_Z(\Delta _{v,\infty })$ and $T_Z(X_\infty )$ induce an orientation on the quotient . By the proof of Lemma 2.4, one can identify the relative homology $\mathrm {H}_s(X_\infty ,X_\infty \!-\! \Delta _{v,\infty })$ with the reduced homology $\widetilde {\mathrm {H}}_{s-1}(\mathbb {S}_W)$ of any sphere $\mathbb {S}_W$ in W. By continuously choosing normal vectors on $S_W$ that point outwards, this determines an orientation on $S_W$ and hence an identification

(18) $$ \begin{align} \mathrm{H}_s(X_\infty,X_\infty\!-\! \Delta_{v,\infty})=\widetilde{\mathrm{H}}_{s-1}(S_W)=\mathbb{Z}. \end{align} $$

Lemma 2.5. If $\gamma \in G(\mathbb {R})^{+}$ fixes v, then the induced map

$$ \begin{align*}\gamma_\ast\colon \mathrm{H}_s(X_\infty,X_\infty\!-\! \Delta_{v,\infty}) \longrightarrow \mathrm{H}_s(X_\infty,X_\infty\!-\! \Delta_{v,\infty})\end{align*} $$

is the identity.

2.2.5 Kudla-Millson cycles intersecting a compact region in $X_{\infty }$

Lemma 2.6. Let $C\subseteq {X}_\infty $ be a compact subset and $m>0$ a fixed real number. Then the set

is compact.

Proof. Consider the set

and let $p_1\colon \Omega \rightarrow C$ and $p_2\colon \Omega \rightarrow V_{\mathbb {R}}$ be the two coordinate projections. The map $p_1$ , whose fibres are spheres of radius m in r-dimensional Euclidean space, is proper. Hence, $\Omega $ is compact, and the same follows for $K = p_2(\Omega )$ .

2.3.1 Definition

The quadratic divisor on $X_p$ attached to an anisotropic vector $v\in V_{\mathbb {Q}_p}$ is the subset $\Delta _{v,p}\subseteq X_p$ given by

$$\begin{align*}\Delta_{v,p}=\left\{\xi\in X_p\ \vert\ v \perp \xi \right\}. \end{align*}$$

In case $v\in V_{+}$ , the divisor $\Delta _{v,p}$ is called rational quadratic. By definition,

(19) $$ \begin{align} g \Delta_{v,p}=\Delta_{g v,p} \end{align} $$

for all $g \in G(\mathbb {Q}_p)$ and all anisotropic vectors $v\in V_{\mathbb {Q}_p}$ .

Example 2.7 (Three-dimensional quadratic spaces).

Following the notations and identifications in Example 1.4, the cycle $\Delta _{v,p}\subseteq \mathcal {H}_p$ associated to the vector is given by the equation

$$ \begin{align*}\operatorname{\mathrm{tr}}\left(\begin{pmatrix}b & 2c \\ -2a &-b \end{pmatrix} \begin{pmatrix} \tau & -\tau^2 \\ 1 & -\tau \end{pmatrix}\right) = a\tau^2 + b\tau+c = 0.\end{align*} $$

This set is nonempty precisely when the discriminant $q(v) = b^2-4ac$ is not a square in $\mathbb {Q}_p$ or, equivalently, when the orthogonal complement of v in $V_{\mathbb {Q}_p}$ is anisotropic.

Example 2.8 (Four-dimensional quadratic spaces).

With the notations of Example 1.5, the rational quadratic divisor associated to the vector is given by the equation

$$ \begin{align*}\operatorname{\mathrm{tr}}\left(\begin{pmatrix}b & d \\ -a & -c \end{pmatrix} \begin{pmatrix}\tau_1 & -\tau_1 \tau_2 \\ 1 & -\tau_2 \end{pmatrix}\right) = a\tau_1 \tau_2 + b\tau_1+c \tau_2 + d = 0.\end{align*} $$

This subset of $\mathcal {H}_p\times \mathcal {H}_p$ is always nonempty, and is the p-adic analogue of a Hirzebruch–Zagier divisor on a Hilbert modular surface. In particular, the rational quadratic divisor attached to the identity matrix is simply the diagonal in $\mathcal {H}_p \times \mathcal {H}_p$ .

2.3.2 Quadratic divisors intersecting affinoids

Let $\Lambda \subseteq V_{\mathbb {Q}_p}$ be a self-dual $\mathbb {Z}_p$ -lattice. Recall that $\Lambda '$ denotes the set of primitive vectors of $\Lambda $ in the sense of (11). Given a nonzero vector $v\in V_{\mathbb {Q}_p}$ there exists an integer k and a primitive vector $v_0\in \Lambda '$ such that $v=p^k v_0$ . The order of v with respect to $\Lambda $ is

$$\begin{align*}{\mathrm{ord}}_{\Lambda}(v)=k \in \mathbb{Z}, \end{align*}$$

and the isotropy level of v with respect to $\Lambda $ is given by

Since v determines $v_0$ up to multiplication by $\mathbb {Z}_p^{\times }$ , both ${\mathrm {ord}}_{\Lambda }(v)$ and $\mathrm {iso}_{\Lambda }(v)$ are well-defined.

As explained in Section 1.3.2, the $\mathbb {Z}_p$ -lattice $\Lambda \subseteq V_{\mathbb {Q}_p}$ allows us to express $X_p$ as an increasing union of affinoids:

$$ \begin{align*}X_p = \bigcup_{k=0}^\infty X_{p,\Lambda}^{\le k}.\end{align*} $$

The following p-adic counterpart of Lemma 2.6 relates this filtration to the isotropy level.

Lemma 2.10. Let $\Lambda \subseteq V_{\mathbb {Q}_p}$ be a self-dual $\mathbb {Z}_p$ -lattice and $v\in V_{\mathbb {Q}_p}$ an anisotropic vector. Then

$$\begin{align*}\mathrm{iso}_\Lambda(v)>k \quad \Rightarrow \quad \Delta_{v,p}\cap X_{p,\Lambda}^{\le k}=\emptyset. \end{align*}$$

In particular, for every $m\in \mathbb {Q}_p^{\times }$ and every integer $k\in \mathbb {Z}$ , the set

is contained in $p^{-\ell }\Lambda $ for $\ell \leq \frac {k-{\mathrm {ord}}_p(m)}{2}$ .

Proof. We may assume that $v\in \Lambda '$ and thus that . We note that $q(v)= 0$ modulo $p^{d}$ , and hence, the image of the vector v in $\mathbb {P}({\Lambda }/p^{d}\Lambda )$ lies on the quadric $Q_\Lambda (\mathbb {Z}/p^{d}\mathbb {Z})$ . Since this quadric is smooth, Hensel’s Lemma implies that there is an isotropic vector $\widetilde v\in \Lambda '$ satisfying

$$ \begin{align*}\widetilde v\equiv v \bmod{p^{d}\Lambda}.\end{align*} $$

This implies that if $z\in \Lambda ^{\prime }_{\mathcal {O}_{\mathbb {C}_p}}$ is orthogonal to v, then

$$ \begin{align*}{\mathrm{ord}}_p(\langle z, \widetilde v\rangle ) \geq d> k,\end{align*} $$

and therefore that $[z]\notin X_{p,\Lambda }^{\leq k}$ .

The following is a partial converse to the previous lemma:

Lemma 2.11. Let $\Lambda \subseteq V_{\mathbb {Q}_p}$ be a self-dual $\mathbb {Z}_p$ -lattice and $v \in V_{\mathbb {Q}_p}$ with $q(v)\neq 0$ such that the orthogonal complement of v in $V_{\mathbb {Q}_p}$ is not a hyperbolic plane. Then

$$\begin{align*}\mathrm{iso}_{\Lambda}(v) \leq k \quad \Rightarrow \quad \Delta_{v,p}\cap X_{p,\Lambda}^{\leq \lceil 3k/2\rceil }\neq \emptyset. \end{align*}$$

Proof. As before, we assume that $v\in \Lambda '$ and hence, . Let $V_1$ be the orthogonal complement of v. Write $\pi \colon V_{\mathbb {Q}_p} \rightarrow V_1$ for the orthogonal projection onto $V_1$ and put $\Lambda _1=\Lambda \cap V_1$ . The task at hand is to find an isotropic and primitive vector $z\in \Lambda _1$ such that

$$\begin{align*}{\mathrm{ord}}_p(\langle z,\pi(w)\rangle) = {\mathrm{ord}}_p(\langle z,w\rangle) \leq \lceil 3d/2\rceil \end{align*}$$

for all $w\in (\Lambda ')_0$ . Let us first give a bound on the set $\pi ((\Lambda ')_0)$ . By Lemma 1.1, the projection $\pi (\Lambda )$ is the dual lattice of $\Lambda _1$ , and the quotient $\pi (\Lambda )/\Lambda _1$ is isomorphic to the discriminant module of the $\mathbb {Z}_p$ -lattice $\mathbb {Z}_p v \subseteq \mathbb {Q}_p v$ . In particular, it is cyclic of order $p^d$ . By assumption, v is not isotropic modulo $p^{d+1}$ , and therefore, $v\not \equiv w \bmod p^{d+1}$ for every isotropic vector $w\in (\Lambda ')_0$ . Since $\langle \cdot ,\cdot \rangle $ is a perfect bilinear form on $\Lambda $ , its reduction modulo $p^{d+1}$ of defines a perfect bilinear form on $\Lambda /p^{d+1}\Lambda $ . It follows that for every $w\in (\Lambda ')_0$ , there exists an element $\bar {u}\in \Lambda /p^{d+1}\Lambda $ with

$$\begin{align*}\langle \bar{u}, v\rangle= 0 \bmod{p^{d+1}\mathbb{Z}_p}\quad\mbox{and}\quad\langle \bar{u}, w\rangle\neq 0 \bmod{p^{d+1}\mathbb{Z}_p}. \end{align*}$$

By Hensel’s Lemma, one can lift $\bar {u}$ to an element $u\in \Lambda _1$ . In particular, we see that

(20) $$ \begin{align} \pi(w)\notin p^{d+1}\pi(\Lambda). \end{align} $$

Let $\varpi $ be a square-root of $p^{d}$ in $\mathcal {O}_{\mathbb {C}_p}$ and write $\widetilde {\Lambda }_1\subseteq \pi (\Lambda )_{\mathcal {O}_{\mathbb {C}_p}}$ for the preimage of $\varpi (\pi (\Lambda )/\Lambda _1)_{\mathcal {O}_{\mathbb {C}_p}}\subseteq (\pi (\Lambda )/\Lambda _1)_{\mathcal {O}_{\mathbb {C}_p}}$ under the reduction map. It is the unique self-dual $\mathcal {O}_{\mathbb {C}_p}$ -lattice $\widetilde {\Lambda }_1\subseteq (V_1)_{\mathbb {C}_p}$ such that

$$ \begin{align*}(\Lambda_1)_{\mathcal{O}_{\mathbb{C}_p}}\subseteq \widetilde{\Lambda}_1 \subseteq \pi(\Lambda)_{\mathcal{O}_{\mathbb{C}_p}}.\end{align*} $$

Moreover, each quotient of subsequent lattices in the chain above is annihilated by $p^{\lceil d/2 \rceil }.$ By (20), we may write

$$ \begin{align*}\pi(w)=p^{m_w} \widetilde{w}\end{align*} $$

with $\widetilde {w}$ a primitive vector of $\widetilde {\Lambda }_1$ and $m_w\leq d$ . Since $\Lambda $ is compact, it follows that the set $\{\widetilde {w} \bmod {\mathfrak {m}_{\mathcal {O}_{\mathbb {C}_p}} \widetilde {\Lambda }_1}\ \vert \ w\in (\Lambda ')_0\} \subseteq \widetilde {\Lambda }_1 /\mathfrak {m}_{\mathcal {O}_{\mathbb {C}_p}} \widetilde {\Lambda }_1$ is finite. Assume for the moment that $n\geq 4$ and let $Q_{\widetilde {\Lambda }_1}\subseteq \mathbb {P}(\widetilde {\Lambda }_1)$ be the smooth quadric cut out by the equation $q=0$ . The $\overline {\mathbb {F}}_p$ -valued points of $Q_{\widetilde {\Lambda }_1}$ are not contained in any finite union of hyperplanes. Thus, by Hensel’s Lemma, there exists a primitive vector $\widetilde {z}\in \widetilde {\Lambda }_1$ such that

$$ \begin{align*}q(\widetilde{z})=0\qquad\mbox{and}\qquad {\mathrm{ord}}_p(\langle \widetilde{z},\widetilde{w}\rangle)=0\end{align*} $$

for every $w\in (\Lambda ')_0.$ In case $n=3$ , the statement above follows by a direct calculation using that $\Lambda _1$ is an anisotropic lattice by assumption. There exists a rational number $m_z\leq d/2$ such that $z=p^{m_z}\widetilde {z}$ is primitive in $\Lambda _{\mathcal {O}_{\mathbb {C}_p}}$ . Thus,

$$ \begin{align*}{\mathrm{ord}}_p(\langle z,w\rangle) = m_z+ m_w \leq 3d/2\end{align*} $$

for all $w\in (\Lambda ')_0$ ; that is, the class $[z]$ lies in the intersection $\Delta _{v,p}\cap X_{p,\Lambda }^{\leq \lceil 3d/2\rceil }.$

2.3.3 Locally finite, rational quadratic divisors

Definition 2.13. A formal integer linear combination

of rational quadratic divisors on $X_p$ is said to be locally finite if it satisfies the following equivalent conditions:

1. 1. For each affinoid subset $\mathcal {A}\subseteq X_p$ , the formal sum

   
   is a divisor – that is, a finite linear combination.

2. 2. For one (and thus for each) self-dual lattice $\Lambda \subseteq V_{\mathbb {Q}_p}$ and each $k\geq 0$ , the formal sum $\Delta \cap X_{p,\Lambda }^{\leq k}$ is a divisor.

The equivalence of (1) and (2) follows from Lemma 1.2. Definition 2.13 generalises the locally finite divisors that are studied in [Reference Darmon and Vonk18] and [Reference Gehrmann22], which are infinite sums of real quadratic points in $\mathcal {H}_p$ whose support has finite intersection with each affinoid subset of $\mathcal {H}_p$ . Write $\mathrm {Div^{\dagger }_{rq}}(X_p)$ for the module of locally finite, rational quadratic divisors. The group $G(\mathbb {Q})$ naturally acts on $\mathrm {Div^{\dagger }_{rq}}(X_p)$ .

Let $L\subseteq V$ be the $\mathbb {Z}[1/p]$ -lattice fixed in the introduction. The most common construction of locally finite divisors rests on the following lemma:

Lemma 2.14. Let C be a compact subset of $X_\infty $ , m a positive rational number, and $\mathcal {A}$ an affinoid subset of $X_p$ . Then the set

$$ \begin{align*}X \ = \ \{ v\in L\ \vert\ q(v)=m, \ \Delta_{v,\infty}\cap C \ne \emptyset,\ \mbox{and} \ \Delta_{v,p} \cap \mathcal{A} \ne \emptyset \}\end{align*} $$

is finite.

2.3.4 Compactly supported products

Let $\mathcal {O}\subseteq V_{+}$ be a finite union of $\Gamma $ -orbits, such as the set of all $v\in L$ satisfying $q(v)=m$ for a given m. Lemma 2.14 motivates the following definition:

Definition 2.15. Let $M = (M_v)_{v\in \mathcal {O}}$ be a representation of $\mathcal {O}$ in the sense of Definition 2.1 with values in abelian groups. The compactly supported product of the $M_v$ is the $\Gamma $ -submodule of $ \prod _{v\in \mathcal {O}} M_{v}$ given by

$$ \begin{align*} \sideset{}{'}\prod_{v\in \mathcal{O}} M_{v} = \left\{ (m_v)\in \prod_{v\in \mathcal{O}} M_{v}\ \middle\vert \begin{array}{l}\mbox{there exists}\ C\subseteq X_\infty \text{ compact} \\ \mbox{such that}\ \Delta_{v,\infty}\cap C =\emptyset \Rightarrow m_v=0 \end{array} \right\}. \end{align*} $$

Let $(\mathbb {Z})_{v\in \mathcal {O}}$ be the representation of $\mathcal {O}$ , whose objects are all equal to $\mathbb {Z}$ and whose arrows attached to $\gamma \in \Gamma $ are the identity maps. After multiplying $\mathcal {O}$ with a positive rational number, we may assume that $\mathcal {O}$ is contained in L. Lemma 2.14 implies that the natural $\Gamma $ -equivariant homomorphism

$$\begin{align*}\mathcal{Z}\colon \sideset{}{'}\prod_{v\in \mathcal{O}} \mathbb{Z} \longrightarrow \mathrm{Div^{\dagger}_{rq}}(X_p), \quad (a_v)_{v\in\mathcal{O}} \longmapsto \sum_{v\in \mathcal{O}} a_v \cdot\Delta_{v,p}, \end{align*}$$

is well-defined.

For later purposes, let us discuss an important feature of compactly supported products. The compactly supported product of a chain complex of $\mathcal {O}$ -representations is the subcomplex

defined by taking the compactly supported product in each degree. One easily deduces that taking compactly supported products commutes with taking homology:

Lemma 2.16. For all $q\geq 0$ ,

2.4.1 A subcomplex of the singular chain complex

Every $\gamma \in G(\mathbb {Q})$ induces homomorphisms

as well as homomorphisms

for each $v\in V$ of positive length, which fulfil relation (17). Hence, is a representation of the $\Gamma $ -module $V_+$ . The product of the quotient maps

defines a $G(\mathbb {Q})$ -equivariant homomorphism

of chain complexes.

By Proposition 2.4, the homology of the chain complex

is concentrated in degree s. Therefore, the inclusion

of the canonical s-truncation is a quasi-isomorphism. Since taking homology commutes with products, the inclusion

is a quasi-isomorphism as well.

Define to be the pullback of along . More concretely,

$$ \begin{align*} \mathfrak{C}_q= \begin{cases}C_{q}(X_\infty) & \mbox{ for } q>s,\\ C_{q}(X_\infty \!-\! \bigcup_{v\in V_{+}}\Delta_{v,\infty})& \mbox{ for } q< s, \end{cases} \end{align*} $$

and

$$\begin{align*}\mathfrak{C}_s=\left\{c\in C_{s}(X_\infty)\ \middle\vert\ d_s(c)\in C_{s-1}(X_\infty \!-\! \textstyle\bigcup_{v\in V_{+}}\Delta_{v,\infty})\right\}. \end{align*}$$

Proposition 2.17. The inclusion

is a quasi-isomorphism. In particular, is a resolution of the constant $\mathbb {Z}[\Gamma ]$ -module $\mathbb {Z}$ .

Proof. Since $X_\infty $ is contractible, the homology of is concentrated in degree $0$ . Moreover, its zeroth homology group is generated by the class of any point $Z\in X_\infty $ . Since countable unions of proper closed submanifolds have measure zero, it follows that there exists a point

$$ \begin{align*}Z\in X_\infty \!-\! \bigcup_{v\in V, q(v)>0} \Delta_{v,\infty,}\end{align*} $$

and thus, the inclusion induces a surjective map on homology groups. Injectivity on homology follows from the next general lemma on chain complexes.

Lemma 2.18. Let be a quasi-isomorphism of chain complexes and a homomorphism of chain complexes. Then the pullback

of f along g induces injective maps on homology.

Proof. Let $(x,y)\in A_n \times _{B_n} C_n$ be a cycle; that is, assume that

$$ \begin{align*}d_n (x)=0, \quad d_n (y)=0\quad\mbox{and}\quad f(x)=g(y).\end{align*} $$

Suppose that the image of $(x,y)$ in the homology of $C_n$ is equal to zero. That means that there exists ${z\in C_{n+1}}$ such that $\widetilde {f}(x,y)=y=d_{n+1}(z)$ for some $z\in C_{n+1}$ . Since $f(x)=g(y)=g(d_{n+1}(z))=d_{n+1}(g(z))$ is a boundary and f is a quasi-isomorphism, we see that x is a boundary as well. Thus, there exists an element $u\in A_{n+1}$ such that $x=d_{n+1}(u)$ . It follows that $d_{n+1}(u,z)=(x,y)$ is a boundary, which proves the claim.

2.4.2 Divisor valued cohomology classes attached to $\Gamma $ -orbits

Let $\mathcal {O}\subseteq V_{+}$ be a finite union of $\Gamma $ -orbits. By definition, the image of the $\Gamma $ -equivariant map

is contained in the compactly supported product. Thus, restricting this map to gives a homomorphism

The natural quotient map to homology induces a $\Gamma $ -equivariant quasi-isomorphism

Moreover, the identification (18) induces a $\Gamma $ -equivariant isomorphism

The composition

defines a morphism of $\mathbb {Z}[\Gamma ]$ -chain complexes and, therefore, by (15) a class

$$ \begin{align*}{\mathscr{D}}_{\mathcal{O}}^{\Gamma}\in \mathrm{H}^s(\Gamma, \mathrm{Div^{\dagger}_{rq}}(X_p)).\end{align*} $$

Let us unravel this construction. It is convenient to introduce the following signed intersection number for $c\in \mathfrak {C}_s$ and $\Delta _v$ , $v\in V_{\mathbb {R},+}$ : by definition,

$$ \begin{align*}d_s(c)\in {\mbox{Ker}}(d_{s-1}\colon C_{s-1}(X_\infty\!-\! \Delta_{v,\infty})\longrightarrow C_{s-2}(X_\infty\!-\! \Delta_{v,\infty})).\end{align*} $$

The intersection number $\Delta _{v,\infty }\cap c$ is the image of $d_s(c)$ in the reduced homology group $\tilde {H}_{s-1}(X_\infty \!-\! \Delta _{v,\infty })$ that we identify with $\mathbb {Z}$ by (18). If c is smooth and does intersect $\Delta _{v,\infty }$ transversely, this is simply the signed count of intersection points of c with $\Delta _{v,\infty }$ , where the latter is oriented according to the definitions spelled out in Section 2.2.2. Now the class ${\mathscr {D}}_{\mathcal {O}}^{\Gamma }$ is in fact given by the s-cocycle

$$\begin{align*}\mathfrak{C}_s\longrightarrow \mathrm{Div^{\dagger}_{rq}}(X_p),\qquad c\longmapsto \sum_{v\in\mathcal{O}}(\Delta_{v,\infty} \cap c) \cdot \Delta_{v,p}. \end{align*}$$

2.4.3 Restriction to a subgroup

Let $\Gamma '\subseteq \Gamma $ be a finite index subgroup and

$$\begin{align*}\mathrm{res}\colon \mathrm{H}^s(\Gamma, \mathrm{Div^{\dagger}_{rq}}(X_p))\longrightarrow \mathrm{H}^s(\Gamma', \mathrm{Div^{\dagger}_{rq}}(X_p))\end{align*}$$

the restriction map on cohomology. The group $\Gamma '$ fulfils all the conditions that were imposed on $\Gamma $ . Moreover, the set $\mathcal {O}$ is also a finite union of $\Gamma '$ -orbits. Thus, we can define the class

$$ \begin{align*}{\mathscr{D}}_{\mathcal{O}}^{\Gamma'}\in \mathrm{H}^s(\Gamma', \mathrm{Div^{\dagger}_{rq}}(X_p)).\end{align*} $$

The following proposition follows directly from the construction.

Proposition 2.19. Let $\Gamma '\subseteq \Gamma $ be a finite index subgroup. Then

$$\begin{align*}\mathrm{res}({\mathscr{D}}_{\mathcal{O}}^{\Gamma})={\mathscr{D}}_{\mathcal{O}}^{\Gamma'}. \end{align*}$$

Because of Proposition 2.19, the class ${\mathscr {D}}_{\mathcal {O}}^{\Gamma }$ does not depend on the group $\Gamma $ preserving $\mathcal {O}$ in an essential way, and we adopt the abbreviation

when the underlying group $\Gamma $ is clear from the context.

Remark 2.20. If $\mathcal {O}$ decomposes into the union of $\Gamma $ -orbits $\mathcal {O}_1,\ldots ,\mathcal {O}_h$ , then

$$\begin{align*}{\mathscr{D}}_{\mathcal{O}} =\sum_{i=1}^{h}{\mathscr{D}}_{\mathcal{O}_i}. \end{align*}$$

Moreover, ${\mathscr {D}}_{a\mathcal {O}}=\mathrm {sign}(a)^{s}\cdot {\mathscr {D}}_{\mathcal {O}}$ for every $a\in \mathbb {Q}^{\times }$ . If V is three-dimensional and $\mathcal {O}$ consists of a single $\Gamma $ -orbit, the class ${\mathscr {D}}_{\mathcal {O}}$ is obviously zero if $\Delta _{v,p}=\emptyset $ for one (and thus any) $v\in \mathcal {O}$ . It is natural to wonder whether the latter relations generate all relations between the divisors attached to $\Gamma $ -orbits in $V_{+}$ . This is known in signature $(3,0)$ and $(2,1)$ . The case that V is the three-dimensional quadratic space of rational $2\times 2$ matrices of trace $0$ and $\Gamma ={\mathrm {SL}}_2(\mathbb {Z}[1/p])$ is implicitly treated in [Reference Darmon and Vonk18, Proof of Thm. 1.24], where it is shown that the parabolic part of is generated by the classes ${\mathscr {D}}_{\mathcal {O}_i}$ as the $\mathcal {O}_i$ range over all $\Gamma $ -orbits of vectors. A more general case is treated in [Reference Gehrmann22]. It is unclear whether such a strong result can reasonably be expected to hold in higher rank.

2.4.4 Kudla–Millson divisors attached to Schwartz functions

The class ${\mathscr {D}}_{\mathcal {O}}$ attached to a $\Gamma $ -orbit $\mathcal {O}\subseteq V_+$ should be viewed as an analogue of the connected cycles on orthogonal Shimura varieties introduced by Kudla in [Reference Kudla31, Section 3]. In the following, we introduce the analogue of weighted cycles in the sense of [Reference Kudla31, Section 5]. Let $\mathbb {A}^{p}_{f}$ denote the ring of finite adeles away from p – that is, the restricted product over all completions $\mathbb {Q}_\ell $ for all rational primes $\ell \neq p$ . Put and write

for the space of $\mathbb {Z}$ -valued Schwartz functions on $V_{\mathbb {A}^{p}_{f}}$ with its natural $G(\mathbb {A}^{p}_{f})$ -action. The group $G(\mathbb {Q})$ acts on $S(V_{\mathbb {A}^{p}_{f}})$ via the diagonal embedding $G(\mathbb {Q})\hookrightarrow G(\mathbb {A}^{p}_{f})$ . Let $\Phi \in S(V_{\mathbb {A}^{p}_{f}})$ be a $\Gamma $ -invariant Schwartz function and m a positive rational number. For every nonzero integer r, the set is a finite union of $\Gamma $ -orbits. Moreover, it is empty for all but finitely many r. Thus, the cohomology class

is well-defined. Its definition is inspired by the formula for the restriction of a weighted cycle to a connected component of an orthogonal Shimura variety (cf. [Reference Kudla31, Proposition 5.4]).

Definition 2.21. The space of Kudla–Millson divisors of level $\Gamma $ is the subspace

$$\begin{align*}\mathcal{KM}(\Gamma)\subseteq \mathrm{H}^s(\Gamma, \mathrm{Div^{\dagger}_{rq}}(X_p))\end{align*}$$

generated by the classes ${\mathscr {D}}_{m,\Phi }$ with $m\in \mathbb {Q}_{>0}$ and $\Phi \in S(V_{\mathbb {A}^{p}_{f}})^\Gamma $ .

2.4.5 Kudla–Millson divisors attached to cosets

The following class of Kudla–Millson divisors features prominently in the construction of p-adic Borcherds products in Section 3.3. Assume for the moment that $\Gamma $ acts trivially on the discriminant module $\mathbb {D}_{L}$ . Let $\widehat {L}$ be the completion of the lattice L inside $V_{\mathbb {A}^{p}_{f}}$ . Then for any $\beta \in \mathbb {D}_{L}$ , the coset $\beta +\widehat {L}$ is a $\Gamma $ -invariant compact open subset of $V_{\mathbb {A}^{p}_{f}}$ , and thus, the characteristic function $\mathbf {1}_{\beta +\widehat {L}}$ is a $\Gamma $ -invariant Schwartz function.

Definition 2.22. The Kudla–Millson divisor ${\mathscr {D}}_{m,\beta }\in \mathcal {KM}(\Gamma )$ is defined as

Denote by $\mathcal {O}_L(m,\beta )$ the set of vectors $v\in \beta +L$ such that $q(v)=m$ . The equality

$$\begin{align*}{\mathscr{D}}_{m,\beta}={\mathscr{D}}_{\mathcal{O}_L(m,\beta)}. \end{align*}$$

follows directly from the definition. If $v\in \beta $ , then clearly, $-v\in -\beta $ . Since $q(v)=q(-v)$ , the equality

$$\begin{align*}{\mathscr{D}}_{m,\beta}=(-1)^{s}{\mathscr{D}}_{m,-\beta} \end{align*}$$

follows from the discussion of orientations in Section 2.2.2. Moreover, the fact that $q(pv) = p^2 q(v)$ implies that Kudla–Millson divisors are ‘p-ordinary’ in the following sense:

Proposition 2.23. For all $\beta \in {\mathbb D}_{L}$ and all positive rational numbers m,

$$\begin{align*}{\mathscr{D}}_{p^2m,p\beta}={\mathscr{D}}_{m,\beta}. \end{align*}$$

2.5.1 Transversal base points

When $s\geq 1$ , the complex is rather big. For both computational and theoretical purposes, it is desirable to replace it by something more manageable.

Consider the bar resolution defined in (16). Every point $Z\in X_\infty $ gives rise to a map of chain complexes of $\Gamma $ -modules

sending $(\gamma _0, \ldots , \gamma _q)\in B(\Gamma )_q$ to the barycentric simplex $[\gamma _0 Z,\ldots , \gamma _q Z]$ with corners $\gamma _0 Z, \ldots , \gamma _q Z$ . Let us briefly recall the construction of barycentric simplices: let

denote the standard q-dimensional simplex in $\mathbb {R}^{q+1}$ . The choice of a $(q+1)$ -tuple of points $(Z_0, Z_1,\ldots , Z_q)$ in $X_\infty $ determines a map

$$ \begin{align*}\Phi_{Z_0,\ldots,Z_q}\colon \Delta_q\longrightarrow X_\infty\end{align*} $$

sending $(t_0,t_1,\ldots , t_q)$ to the unique minimumFootnote ³ of the real-valued function $g\colon X_\infty \rightarrow \mathbb {R}$ defined by

The map $\Phi _{Z_0,\ldots ,Z_s}$ maps the vertices of $\Delta _q$ to the points $Z_0,\ldots , Z_q$ , and we set

The boundary of this q-dimensional simplex is given by

$$ \begin{align*}\partial([Z_0,\ldots,Z_q]) = \sum_{i=0}^q (-1)^i [Z_0,\ldots, \hat{Z_i},\ldots Z_q],\end{align*} $$

where $[Z_0,\ldots , \hat {Z_i},\ldots Z_q]$ denotes the $(q-1)$ -dimensional simplex in $X_\infty $ obtained by removing the i-th vertex from $[Z_0,\ldots , Z_q]$ .

Let $Z\in X_\infty $ be a transversal base point. For $c\in B(\Gamma )_s$ and $v\in V_{\mathbb {R},+}$ , define the intersection product

$$\begin{align*}\Delta_{v,\infty} \cap_{Z} c= \Delta_{v,\infty} \cap b^{Z}_s(c). \end{align*}$$

Then a cocycle representing the class $\mathcal {D}_{\mathcal {O}}$ is given by the map

$$\begin{align*}B(\Gamma)_s \longrightarrow \mathrm{Div^{\dagger}_{rq}}(X_p),\quad c \longmapsto \sum_{v\in\mathcal{O}} (\Delta_{v,\infty} \cap_{Z} c) \cdot \Delta_{v,p}. \end{align*}$$

2.5.2 Transversal base points in signature $(r,1)$

It is natural to ask whether transversal base points exist. This question can be answered affirmatively if $s=1$ . Indeed, in that case, the only condition required of Z is that it not be an element of the union of all $\Delta _{v,\infty }$ , $v\in \mathcal {O}$ , and it was already observed in the proof of Proposition 2.17 that such points exist. So in case $s=1$ , we may chose Z as above and get the following description of the class attached to the orbit of v: the space $X_\infty \!-\! \Delta _{v,\infty }$ decomposes into two connected components $\mathrm {H}_v^+$ and $\mathrm {H}_v^-$ for all $v\in \mathcal {O}$ , where the choice of $\mathrm {H}_v^+$ is determined by the chosen orientations. Then the intersection product is given by

(22) $$ \begin{align} \Delta_{v,\infty} \cap_{Z} [\gamma_0,\gamma_1]= \begin{cases} \phantom{-}0 & \mbox{if } \ \gamma_0 Z \mbox{ and } \gamma_1 Z \mbox{ both belong to } \mathrm{H}_v^+ \mbox{ or } \mathrm{H}_v^-;\\ \phantom{-}1 & \mbox{if }\ \gamma_0 Z\in \mathrm{H}_v^+ \mbox{ and } \gamma_1 Z \in \mathrm{H}_v^-;\\ -1 & \mbox{if }\ \gamma_0 Z\in \mathrm{H}_v^- \mbox{ and } \gamma_1 Z \in \mathrm{H}_v^+. \end{cases} \end{align} $$

2.6.1 Abstract modular symbols

The maximal isotropic subspaces of V are rational lines on the boundary of the negative cone in $V_{\mathbb {R}}$ . In particular, they are one-dimensional (i.e., for any two distinct elements $\ell _1=[w_1],\ \ell _2=[w_2]\in Q(\mathbb {Q})$ the inequality

$$ \begin{align*}\langle w_1, w_2 \rangle \neq 0\end{align*} $$

holds).

The p-arithmetic group $\Gamma $ acts on the set $Q(\mathbb {Q})$ via the inclusion $\Gamma \subseteq G(\mathbb {Q})$ . Therefore, it acts naturally on the free abelian group $\mathbb {Z}[Q(\mathbb {Q})]$ generated by $Q(\mathbb {Q})$ and its subgroup

Definition 2.25. Let M be any $\Gamma $ -module. The space of M-valued modular symbols is the $\Gamma $ -module

Given a modular symbol $m\in \mathrm {MS}(M)$ , it is customary to set for any $r,s\in Q(\mathbb {Q})$ . The resulting function

$$ \begin{align*}m\colon Q(\mathbb{Q})\times Q(\mathbb{Q}) \longrightarrow M\end{align*} $$

then satisfies the familiar two and three term relations

$$\begin{align*}m(r,s)+m(s,r)=0\quad\mbox{and}\quad m(r,s)+m(s,t)= m(r,t) \quad \mbox{ for all } r,s,t\in Q(\mathbb{Q}). \end{align*}$$

The short exact sequence of $\Gamma $ -modules