Rational equivalence of cusps

TL;DR

This paper proves that cusps of the same dimension in certain modular varieties are rationally linearly dependent, extending the Manin-Drinfeld theorem to higher dimensions and generalizing modular units as higher Chow cycles.

Contribution

It establishes rational equivalence of cusps in higher dimensions and introduces higher Chow cycles as a generalization of modular units.

Findings

Cusps of the same dimension are linearly dependent in the rational Chow group.

Higher dimensional analogue of the Manin-Drinfeld theorem is proven.

Higher Chow cycles generalize modular units in this context.

Abstract

We prove that two cusps of the same dimension in the Baily-Borel compactification of some classical series of modular varieties are linearly dependent in the rational Chow group of the compactification. This gives a higher dimensional analogue of the Manin-Drinfeld theorem. As a consequence, we obtain a higher dimensional generalization of modular units as higher Chow cycles on the modular variety.

Equations

$$Q_{\Lambda }=\{[\mathbb{C}\omega ]\in \mathbb{P}{\Lambda }_{\mathbb{C}}|(\omega ,\omega )=0\}$$

$${\mathcal{D}}_{\Lambda }^{\ast }={\mathcal{D}}_{\Lambda }\bigsqcup \underset{\dim J=2}{⨆}{\mathbb{H}}_J\bigsqcup \underset{\dim I=1}{⨆}{p}_I$$

$$N_1{\Lambda }^{\prime }\subset N\Lambda \subset \Lambda \subset N_2^{-1}{\Lambda }^{\prime }.$$

$${\mathrm{O}}^{+}({\Lambda }^{\prime },N^{\prime })\subset {\mathrm{O}}^{+}(\Lambda ,N)\subset \Gamma$$

$${\mathrm{SL}}_2(\mathbb{Z})\simeq \mathrm{SL}(J_{\mathbb{Z}})\hookrightarrow {\mathrm{O}}^{+}({\Lambda }^{\prime }),\gamma \mapsto (\gamma {|}_{J_{\mathbb{Z}}})\oplus ({\gamma }^{\vee }{|}_{J_{\mathbb{Z}}^{\prime }})\oplus {\mathrm{id}}_{{\Lambda }_2},$$

$$[p_1]=f_{\ast }[q_1]=f_{\ast }[q_2]=[p_2]$$

$$\iota :{\mathrm{O}}^{+}({\Lambda }_{\mathbb{Q}}^{\prime })\hookrightarrow {\mathrm{O}}^{+}({\Lambda }_{\mathbb{Q}}),\gamma \mapsto \gamma \oplus {\mathrm{id}}_{({\Lambda }_{\mathbb{Q}}^{\prime }{)}^{\perp }}.$$

$$\phi :{\mathrm{SL}}_2(\mathbb{Q})\to {\mathrm{O}}^{+}(V)={\mathrm{O}}^{+}(({\Lambda }_1{)}_{\mathbb{Q}})$$

$${\mathbb{H}}^{\ast }\to {\mathcal{D}}_{{\Lambda }_1}^{\ast },\tau \mapsto e+\tau v_0-{\tau }^{2}f,$$

$$\mathbb{H}\times \mathbb{H}\to {\mathcal{D}}_{2U},({\tau }_1,{\tau }_2)\mapsto e_1-{\tau }_1{\tau }_2{f}_1+{\tau }_1{e}_2+{\tau }_2{f}_2.$$

$${\mathrm{SL}}_2(\mathbb{Q})\times {\mathrm{SL}}_2(\mathbb{Q})=\mathrm{SL}(J_1^{\prime })\times \mathrm{SL}(J_3^{\prime })\hookrightarrow {\mathrm{O}}^{+}(2U_{\mathbb{Q}})$$

$$C_1^{\prime }=X(N)\times ({\tau }_2=0),C_2^{\prime }=X(N)\times ({\tau }_2=i\infty )$$

$${\mathrm{O}}^{+}(2U_{\mathbb{Q}})\simeq {\mathrm{O}}^{+}({\Lambda }_{\mathbb{Q}}^{\prime })\hookrightarrow {\mathrm{O}}^{+}({\Lambda }_{\mathbb{Q}}),\gamma \mapsto \gamma \oplus {\mathrm{id}}_{({\Lambda }_{\mathbb{Q}}^{\prime }{)}^{\perp }},$$

$$J_1\cap J_3=\{0\},J_3\cap J_4=\{0\},J_4\cap J_2=\{0\}.$$

$$\mathbb{Q}[Z_1]=\mathbb{Q}[Z_3]=\mathbb{Q}[Z_4]=\mathbb{Q}[Z_2]$$

$$L{G}_{\Lambda }=\{[V]\in G(g,{\Lambda }_{\mathbb{C}})|(,){|}_V\equiv 0\}$$

$${\mathcal{D}}_{\Lambda }^{\ast }={\mathcal{D}}_{\Lambda }\bigsqcup \underset{I\subset {\Lambda }_{\mathbb{Q}}}{⨆}{\mathcal{D}}_I$$

$${\mathcal{D}}_{{\Lambda }^{\prime }}\times {\mathcal{D}}_{{\Lambda }^{\prime \prime }}\hookrightarrow {\mathcal{D}}_{\Lambda },(V^{\prime },V^{\prime \prime })\mapsto V^{\prime }\oplus V^{\prime \prime }.$$

$$\mathrm{Sp}({\Lambda }_{\mathbb{Q}}^{\prime })\times \mathrm{Sp}({\Lambda }_{\mathbb{Q}}^{\prime \prime })\hookrightarrow \mathrm{Sp}({\Lambda }_{\mathbb{Q}}),({\gamma }^{\prime },{\gamma }^{\prime \prime })\mapsto {\gamma }^{\prime }\oplus {\gamma }^{\prime \prime }.$$

$$Z_i^{\prime }=p_i\times X^{\prime \prime }\subset X^{\prime }\times X^{\prime \prime },$$

$$\mathbb{Q}[Z_1]=\mathbb{Q}[Z_3]=\mathbb{Q}[Z_4]=\mathbb{Q}[Z_2]$$

$${\mathcal{D}}_{\Lambda }=\{[V]\in G_{\Lambda }|(,){|}_V>0\}$$

$${\mathcal{D}}_{\Lambda }^{\ast }={\mathcal{D}}_{\Lambda }\bigsqcup \underset{I\subset {\Lambda }_K}{⨆}{\mathcal{D}}_I$$

$$(A,B)=\mathrm{tr}(A{B}^{\ast })+{\sqrt{-D}}^{-1}\mathrm{tr}(J_{D}A{B}^{\ast }),A,B\in M_2(\mathbb{Q}),$$

$${\mathrm{SL}}_2(\mathbb{Q})\to \mathrm{SU}({\Lambda }_K)$$

$$\mathcal{O}=\{X\in M_2(\mathbb{Q})|\Lambda X\subset \Lambda \}.$$

$$I+N{N}_0{\mathcal{O}}_{max}\subset I+N\mathcal{O}\subset \mathcal{O}\subset {\mathcal{O}}_{max}.$$

$$(I+N{N}_0{\mathcal{O}}_{max}{)}^1\subset \mathrm{SU}(\Lambda ,N)\subset \mathrm{SU}(\Lambda )\subset {\mathcal{O}}_{max}^1.$$

$$\Gamma (N{N}_0)\subset {\mathrm{Ad}}_g(\mathrm{SU}(\Lambda ,N))\subset {\mathrm{Ad}}_g(\mathrm{SU}(\Lambda ))\subset {\mathrm{SL}}_2(\mathbb{Z}).$$

$$Y_{\Gamma }\stackrel{j}{\hookrightarrow }{X}_{\Gamma }\stackrel{i}{\hookleftarrow }\partial X_{\Gamma }.$$

Full text

Supported by JSPS KAKENHI 15H05738 and 17K14158.

1. Introduction

The classical theorem of Manin and Drinfeld ([13], [6]) asserts that the difference of two cusps is torsion in the Picard group of the modular curve for a congruence subgroup of ${{\rm SL}_{2}(\mathbb{Z})}$. This had stimulated the development of the theory of modular units and cuspidal class groups (see [10]). The original proof used modular symbols and Hecke operators on the cohomology of the modular curve ([13], [6]). Later an interpretation in terms of the mixed Hodge structure of the modular curve minus the cusps was also found ([7]).

Our purpose in this paper is to prove a generalization of the Manin-Drinfeld theorem for cusps in the Baily-Borel compactification of some higher dimensional classical modular varieties. In higher dimension cusps are no longer divisors, but algebraic cycles of various codimension. We wish to clarify their contribution to the Chow group of the Baily-Borel compactification.

The modular varieties of our object of study are of the following three types:

1. (1)

modular varieties of orthogonal type attached to rational quadratic forms of signature $(2,n)$, which have only [math]-dimensional and $1$-dimensional cusps; 2. (2)

Siegel modular varieties attached to rational symplectic forms; and 3. (3)

modular varieties of unitary type, including the Picard modular varieties, attached to Hermitian forms over imaginary quadratic fields.

In Cartan’s classification of irreducible Hermitian symmetric domains, these correspond to the domains $\mathcal{D}$ of type IV, III, I respectively. The Baily-Borel compactification ([1]) of the modular variety ${\Gamma}\backslash\mathcal{D}$ for an arithmetic group ${\Gamma}$ is obtained by adjoining rational boundary components to $\mathcal{D}$ and then taking quotient by ${\Gamma}$. Below, by a cusp we mean the closure of the image of a rational boundary component in the Baily-Borel compactification.

Our main results are the following.

Theorem 1.1** (orthogonal case).**

Let $\Lambda$ be an integral quadratic lattice of signature $(2,n)$, ${\Gamma}$ a congruence subgroup of the orthogonal group ${\rm O}^{+}(\Lambda)$, and ${X_{\Gamma}}$ the Baily-Borel compactification of the modular variety defined by ${\Gamma}$. Let $Z_{1},Z_{2}$ be two cusps of ${X_{\Gamma}}$ of the same dimension, say $k\in\{0,1\}$. Assume that $n\geq 3$ if $k=1$. Then we have ${\mathbb{Q}}[Z_{1}]={\mathbb{Q}}[Z_{2}]$ in the rational Chow group $CH_{k}({X_{\Gamma}})_{{\mathbb{Q}}}=CH_{k}({X_{\Gamma}})\otimes_{{\mathbb{Z}}}{\mathbb{Q}}$ of ${X_{\Gamma}}$.

Theorem 1.2** (symplectic case).**

Let $\Lambda$ be an integral symplectic lattice, ${\Gamma}$ a congruence subgroup of the symplectic group ${\rm Sp}(\Lambda)$, and ${X_{\Gamma}}$ the Satake-Baily-Borel compactification of the Siegel modular variety defined by ${\Gamma}$. If $Z_{1},Z_{2}$ are two cusps of ${X_{\Gamma}}$ of the same dimension, say $k$, then ${\mathbb{Q}}[Z_{1}]={\mathbb{Q}}[Z_{2}]$ in $CH_{k}({X_{\Gamma}})_{{\mathbb{Q}}}$.

Theorem 1.3** (unitary case).**

Let $K$ be an imaginary quadratic field, $\Lambda$ a Hermitian lattice over $\mathcal{O}_{K}$, ${\Gamma}$ a congruence subgroup of the unitary group ${\rm U}(\Lambda)$, and ${X_{\Gamma}}$ the Baily-Borel compactification of the modular variety defined by ${\Gamma}$. If $Z_{1},Z_{2}$ are two cusps of ${X_{\Gamma}}$ of the same dimension, say $k$, then ${\mathbb{Q}}[Z_{1}]={\mathbb{Q}}[Z_{2}]$ in $CH_{k}({X_{\Gamma}})_{{\mathbb{Q}}}$.

Note that the equality ${\mathbb{Q}}[Z_{1}]={\mathbb{Q}}[Z_{2}]$ in $CH_{k}({X_{\Gamma}})_{{\mathbb{Q}}}$ is the same as the equality $N_{1}[Z_{1}]=N_{2}[Z_{2}]$ in the integral Chow group $CH_{k}({X_{\Gamma}})$ for some natural numbers $N_{1},N_{2}$. When $k=0$, we must have $N_{1}=N_{2}$, so $[Z_{1}]-[Z_{2}]$ is torsion in $CH_{0}({X_{\Gamma}})$.

In the symplectic case, when $\Lambda$ has rank $\geq 4$, every finite-index subgroup of ${\rm Sp}(\Lambda)$ is a congruence subgroup by Mennicke [14] and Bass-Lazard-Serre [2]. The case ${\rm rk}(\Lambda)=2$ is just the case of modular curves.

The case $(n,k)=(2,1)$ in the orthogonal case is indeed an exception. We have self products of modular curves as typical examples of ${X_{\Gamma}}$ in $n=2$, for which two transversal boundary curves are not homologically equivalent. On the other hand, we should note that some consideration in the case $n=2$ is necessary for our proof for the case $n\geq 3$.

The proof of Theorems 1.1 – 1.3 is based on the same simple idea. We connect $Z_{1}$ and $Z_{2}$ by a chain of sub modular varieties or their products, through the interior or the boundary, and use induction on the dimension of modular varieties. This eventually reduces the problem to the Manin-Drinfeld theorem for modular curves. The actual argument requires case-by-case construction depending on the combinatorics of rational boundary components. We need to argue the three cases separately, though the symplectic and the unitary cases are similar. Theorem 1.1 is proved in §2; Theorem 1.2 in §3; and Theorem 1.3 in §4.

In §5, as a consequence of these results, we associate an explicit nonzero element of the higher Chow group $CH_{k}({\Gamma}\backslash\mathcal{D},1)_{{\mathbb{Q}}}$ of the modular variety ${\Gamma}\backslash\mathcal{D}$ (before compactification) to each pair $(Z_{1},Z_{2})$ of cusps of maximal dimension $k$. This gives a higher dimensional analogue of modular units from the viewpoint of algebraic cycles. If the span of all such higher Chow cycles on ${\Gamma}\backslash\mathcal{D}$ has dimension no less than the number of maximal cusps, we would then obtain a nontrivial subspace of $CH_{k}({X_{\Gamma}},1)_{{\mathbb{Q}}}$ for the Baily-Borel compactification ${X_{\Gamma}}$.

Throughout the paper $\Gamma(N)$ stands for the principal congruence subgroup of ${{\rm SL}_{2}(\mathbb{Z})}$ of level $N$, and $X(N)=\Gamma(N)\backslash{\mathbb{H}}^{\ast}$ the (compactified) modular curve for $\Gamma(N)$. In §2 and §3, for a free ${\mathbb{Z}}$-module $\Lambda$ of finite rank, we denote by $\Lambda^{\vee}={\operatorname{Hom}}_{{\mathbb{Z}}}(\Lambda,{\mathbb{Z}})$ its dual ${\mathbb{Z}}$-module and denote $\Lambda_{F}=\Lambda\otimes_{{\mathbb{Z}}}F$ for $F={\mathbb{Q}},{\mathbb{R}},{\mathbb{C}}$. For a ${\mathbb{Q}}$-vector space $V$ we also write $V^{\vee}={\operatorname{Hom}}_{{\mathbb{Q}}}(V,{\mathbb{Q}})$ and $V_{F}=V\otimes_{{\mathbb{Q}}}F$ when no confusion is likely to occur.

2. The orthogonal case

In this section we prove Theorem 1.1. We first recall orthogonal modular varieties (cf. [15], [11]). Let $\Lambda$ be a free ${\mathbb{Z}}$-module of rank $2+n$ equipped with a nondegenerate symmetric bilinear form $(\>,\>):\Lambda\times\Lambda\to{\mathbb{Z}}$ of signature $(2,n)$. Let

[TABLE]

be the isotropic quadric in ${{\mathbb{P}}}{\Lambda_{\mathbb{C}}}$. The open set of $Q_{\Lambda}$ defined by the condition $(\omega,\bar{\omega})>0$ consists of two connected components, and the Hermitian symmetric domain ${\mathcal{D}_{\Lambda}}$ attached to $\Lambda$ is defined as one of them. This choice is equivalent to the choice of an orientation of a positive definite plane in $\Lambda_{{\mathbb{R}}}$.

Let ${\rm O}(\Lambda)$ be the orthogonal group of $\Lambda$, namely the group of isomorphisms $\Lambda\to\Lambda$ preserving the quadratic form. We write ${\rm O}^{+}(\Lambda)$ for the subgroup of ${\rm O}(\Lambda)$ preserving the component ${\mathcal{D}_{\Lambda}}$. For a natural number $N$ let ${\rm O}^{+}(\Lambda,N)<{\rm O}^{+}(\Lambda)$ be the kernel of the reduction map ${\rm O}^{+}(\Lambda)\to{\rm GL}(\Lambda/N\Lambda)$. A subgroup ${\Gamma}$ of ${\rm O}^{+}(\Lambda)$ is called a congruence subgroup if it contains ${\rm O}^{+}(\Lambda,N)$ for some level $N$. A typical example is the kernel of the reduction map ${\rm O}^{+}(\Lambda)\to{\rm GL}(\Lambda^{\vee}/\Lambda)$ for the discriminant group $\Lambda^{\vee}/\Lambda$.

There are two types of rational boundary components of ${\mathcal{D}_{\Lambda}}$, [math]-dimensional and $1$-dimensional components. [math]-dimensional components correspond to isotropic ${\mathbb{Q}}$-lines $I$ in ${\Lambda_{\mathbb{Q}}}$: we take the point $p_{I}=[I_{{\mathbb{C}}}]\in Q_{\Lambda}$, which is in the closure of ${\mathcal{D}_{\Lambda}}$, for each such $I$. $1$-dimensional components correspond to isotropic ${\mathbb{Q}}$-planes $J$ in ${\Lambda_{\mathbb{Q}}}$: we take the connected component of ${{\mathbb{P}}}J_{{\mathbb{C}}}-{{\mathbb{P}}}J_{{\mathbb{R}}}\simeq{\mathbb{H}}\sqcup{\mathbb{H}}$, say ${\mathbb{H}}_{J}$, that is in the closure of ${\mathcal{D}_{\Lambda}}$. The union

[TABLE]

is equipped with the Satake topology ([1], [5]). By Baily-Borel [1], the quotient space ${X_{\Gamma}}={\Gamma}\backslash{\mathcal{D}_{\Lambda}^{\ast}}$ has the structure of a normal projective variety and contains ${\Gamma}\backslash{\mathcal{D}_{\Lambda}}$ as a Zariski open set.

In §2.1 we prove Theorem 1.1 for [math]-dimensional cusps, and in §2.2 for $1$-dimensional cusps. Throughout this section $U$ stands for the rank $2$ unimodular hyperbolic lattice with Gram matrix $\begin{pmatrix}0&1\\ 1&0\end{pmatrix}$. The symbol $\Lambda_{1}\perp\Lambda_{2}$ stands for the orthogonal direct sum of two quadratic lattices (or spaces) $\Lambda_{1},\Lambda_{2}$, while $\Lambda_{1}\oplus\Lambda_{2}$ just stands for the direct sum of $\Lambda_{1},\Lambda_{2}$ as ${\mathbb{Z}}$-module (or linear space) and does not necessarily mean that $(\Lambda_{1},\Lambda_{2})\equiv 0$.

2.1. [math]-dimensional cusps

In this subsection we prove Theorem 1.1 for [math]-dimensional cusps. Let $I_{1}\neq I_{2}$ be two isotropic lines in ${\Lambda_{\mathbb{Q}}}$ and $p_{1},p_{2}\in{X_{\Gamma}}$ the corresponding [math]-dimensional cusps. We consider separately the cases where $(I_{1},I_{2})\equiv 0$ or $(I_{1},I_{2})\not\equiv 0$. In the former case $p_{1}$ and $p_{2}$ are joined by a boundary curve, while in the latter case they are joined by a modular curve through the interior of ${X_{\Gamma}}$.

2.1.1. The case $(I_{1},I_{2})\equiv 0$

We first assume that $(I_{1},I_{2})\equiv 0$. The direct sum $J=I_{1}\oplus I_{2}$ is an isotropic plane in ${\Lambda_{\mathbb{Q}}}$. Let ${\mathbb{H}}_{J}^{\ast}={\mathbb{H}}_{J}\sqcup\bigsqcup_{I\subset J}p_{I}$ and $\Gamma_{J}\subset{\rm SL}(J)$ be the image of the stabilizer of $J$ in ${\Gamma}$. We have a generically injective morphism $f:X_{J}\to{X_{\Gamma}}$ from the modular curve $X_{J}=\Gamma_{J}\backslash{\mathbb{H}}_{J}^{\ast}$ whose image is the $1$-dimensional cusp associated to $J$.

Claim 2.1*.*

$\Gamma_{J}$ is a congruence subgroup of ${\rm SL}(J_{{\mathbb{Z}}})$ where $J_{{\mathbb{Z}}}=J\cap\Lambda$.

Proof.

There exists a rank $2$ isotropic sublattice $J^{\prime}_{{\mathbb{Z}}}$ in ${\Lambda_{\mathbb{Q}}}$ such that $J^{\prime}_{{\mathbb{Z}}}\simeq(J_{{\mathbb{Z}}})^{\vee}$ by the pairing. The lattice $\Lambda_{1}=J_{{\mathbb{Z}}}\oplus J_{{\mathbb{Z}}}^{\prime}$ is isometric to $U\perp U$. We set $\Lambda_{2}=(\Lambda_{1})^{\perp}\cap\Lambda$ and $\Lambda^{\prime}=\Lambda_{1}\perp\Lambda_{2}$. Recall that ${\Gamma}$ contains ${\rm O}^{+}(\Lambda,N)$ for some level $N$. Since both $\Lambda$ and $\Lambda^{\prime}$ are full lattices in ${\Lambda_{\mathbb{Q}}}$, we can find natural numbers $N_{1},N_{2}$ such that

[TABLE]

If we set $N^{\prime}=N_{1}N_{2}$, this tells us that

[TABLE]

inside ${\rm O}(\Lambda_{{\mathbb{Q}}})={\rm O}(\Lambda^{\prime}_{{\mathbb{Q}}})$. Now we have the embedding

[TABLE]

whose image is contained in the stabilizer of $J$. Since this maps $\Gamma(N^{\prime})$ into ${\rm O}^{+}(\Lambda^{\prime},N^{\prime})\subset{\Gamma}$, we see that $\Gamma_{J}$ contains $\Gamma(N^{\prime})$. ∎

Let $q_{1},q_{2}$ be the cusps of $X_{J}$ corresponding to $I_{1},I_{2}$ respectively. By this claim we can apply the Manin-Drinfeld theorem to $X_{J}$. Therefore $[q_{1}]=[q_{2}]$ in $CH_{0}(X_{J})_{{\mathbb{Q}}}$. Since $f(q_{1})=p_{1}$ and $f(q_{2})=p_{2}$, we obtain

[TABLE]

in $CH_{0}({X_{\Gamma}})_{{\mathbb{Q}}}$.

2.1.2. The case $(I_{1},I_{2})\not\equiv 0$

Next we assume that $(I_{1},I_{2})\not\equiv 0$. In this case $I_{1}\oplus I_{2}$ is isometric to $U_{{\mathbb{Q}}}$. Its orthogonal complement $(I_{1}\oplus I_{2})^{\perp}$ has signature $(1,n-1)$. We choose a vector $v$ of positive norm from $(I_{1}\oplus I_{2})^{\perp}$ and put $\Lambda^{\prime}_{{\mathbb{Q}}}=I_{1}\oplus I_{2}\oplus{\mathbb{Q}}v$. Then $\Lambda^{\prime}_{{\mathbb{Q}}}$ has signature $(2,1)$. Let $\mathcal{D}_{\Lambda^{\prime}}$ be the Hermitian symmetric domain attached to $\Lambda^{\prime}_{{\mathbb{Q}}}$. We have the natural inclusion $\mathcal{D}_{\Lambda^{\prime}}^{\ast}\subset{\mathcal{D}_{\Lambda}^{\ast}}$ which is compatible with the embedding of orthogonal groups

[TABLE]

Claim 2.2*.*

There is a subgroup $\Gamma^{\prime}<{\rm O}^{+}(\Lambda^{\prime}_{{\mathbb{Q}}})$ such that $\iota(\Gamma^{\prime})\subset{\Gamma}$ and that $X^{\prime}=\Gamma^{\prime}\backslash\mathcal{D}_{\Lambda^{\prime}}^{\ast}$ is naturally isomorphic to $X(N)$ for some level $N$.

Proof.

Let $\Lambda_{1}=U\perp\langle 2\rangle$. Then $\Lambda_{{\mathbb{Q}}}^{\prime}$ is isometric to the scaling of $(\Lambda_{1})_{{\mathbb{Q}}}$ by some positive rational number. This gives natural isomorphisms $\mathcal{D}_{\Lambda^{\prime}}^{\ast}\simeq\mathcal{D}_{\Lambda_{1}}^{\ast}$ and ${\rm O}^{+}(\Lambda_{{\mathbb{Q}}}^{\prime})\simeq{\rm O}^{+}((\Lambda_{1})_{{\mathbb{Q}}})$. The group ${\rm O}^{+}((\Lambda_{1})_{{\mathbb{Q}}})$ is related to ${\rm SL}_{2}({\mathbb{Q}})$ by the following well-known construction (cf. [12] §2.4). Let $V\subset M_{2}({\mathbb{Q}})$ be the space of $2\times 2$ matrices with trace [math], equipped with the symmetric form $(A,B)={\operatorname{tr}}(AB)$. Then $V\cap M_{2}({\mathbb{Z}})$ is isometric to $\Lambda_{1}$. By conjugation ${\rm SL}_{2}({\mathbb{Q}})$ acts on $V$. This defines a homomorphism

[TABLE]

with ${\operatorname{Ker}}(\varphi)=\{\pm I\}$. (We have ${\rm Im}(\varphi)={\rm SO}^{+}(V)$, but we do not need this fact.) It is readily checked that $\varphi(\Gamma(N))\subset{\rm O}^{+}(\Lambda_{1},N)$ for every level $N$. Furthermore, $\varphi$ is compatible with the Veronese isomorphism

[TABLE]

where $e,f$ are the standard basis of $U$ and $v_{0}$ is a generator of $\langle 2\rangle$. Now by the same argument as (2.1), there exists a level $N$ such that the embedding $\iota$ maps ${\rm O}^{+}(\Lambda_{1},N)$ into ${\Gamma}$. This proves our claim. ∎

Let $q_{1},q_{2}$ be the cusps of $X^{\prime}$ corresponding to the isotropic lines $I_{1},I_{2}$ of $\Lambda^{\prime}_{{\mathbb{Q}}}$. By this claim we have a finite morphism $f:X^{\prime}\to{X_{\Gamma}}$ which sends $q_{1}$ to $p_{1}$ and $q_{2}$ to $p_{2}$. By the Manin-Drinfeld theorem for $X^{\prime}$ we have $[q_{1}]=[q_{2}]$ in $CH_{0}(X^{\prime})_{{\mathbb{Q}}}$. Applying $f_{\ast}$, we obtain $[p_{1}]=[p_{2}]$ in $CH_{0}({X_{\Gamma}})_{{\mathbb{Q}}}$. This finishes the proof of Theorem 1.1 for [math]-dimensional cusps.

Remark 2.3*.*

If $\Lambda$ has Witt index $2$, $(I_{1}\oplus I_{2})^{\perp}$ contains an isotropic line, say $I_{3}$. Then we could also apply the result of §2.1.1 to $I_{1}$ vs $I_{3}$ and to $I_{3}$ vs $I_{2}$, thus obtaining $[p_{1}]=[p_{2}]$ via $I_{3}$. Together with the case of §2.1.1, this shows that when $X_{\Gamma}$ contains at least one $1$-dimensional cusp, then any two [math]-dimensional cusps can be connected by a chain of $1$-dimensional cusps of length $\leq 2$ which provides their rational equivalence.

2.2. $1$-dimensional cusps

In this subsection we prove Theorem 1.1 for $1$-dimensional cusps.

2.2.1. Preliminaries in $n=2$

Although the case $n=2$ is not included in Theorem 1.1 for $1$-dimensional cusps, we need to study a specific example in $n=2$ as preliminaries for the proof for the case $n\geq 3$. We consider the lattice $2U=U\perp U$. Let $e_{1},f_{1}$ be the standard basis of the first copy of $U$, and $e_{2},f_{2}$ be that of the second $U$. Let $J_{1}^{\prime}={\mathbb{Q}}e_{2}\oplus{\mathbb{Q}}e_{1}$ and $J_{2}^{\prime}={\mathbb{Q}}f_{2}\oplus{\mathbb{Q}}f_{1}$, which are isotropic planes in $2U_{{\mathbb{Q}}}$. We take an arbitrary natural number $N$ and consider the modular surface $S(N)={\rm O}^{+}(2U,N)\backslash\mathcal{D}_{2U}^{\ast}$. Let $C_{1},C_{2}$ be the boundary curves of $S(N)$ associated to $J_{1}^{\prime},J_{2}^{\prime}$ respectively.

Lemma 2.4**.**

We have ${\mathbb{Q}}[C_{1}]={\mathbb{Q}}[C_{2}]$ in $CH_{1}(S(N))_{{\mathbb{Q}}}$.

Proof.

Recall that we have the Segre isomorphism

[TABLE]

This extends to ${\mathbb{H}}^{\ast}\times{\mathbb{H}}^{\ast}\to\mathcal{D}_{2U}^{\ast}$, and maps the boundary components ${\mathbb{H}}\times(\tau_{2}=0),{\mathbb{H}}\times(\tau_{2}=i\infty)$ of ${\mathbb{H}}^{\ast}\times{\mathbb{H}}^{\ast}$ to the boundary components ${\mathbb{H}}_{J_{1}^{\prime}},{\mathbb{H}}_{J_{2}^{\prime}}$ of $\mathcal{D}_{2U}^{\ast}$ respectively.

Let $J_{3}^{\prime}={\mathbb{Q}}f_{2}\oplus{\mathbb{Q}}e_{1}$ and $J_{4}^{\prime}={\mathbb{Q}}e_{2}\oplus{\mathbb{Q}}f_{1}$. By the pairing we identify $J_{2}^{\prime}\simeq(J_{1}^{\prime})^{\vee}$ and $J_{4}^{\prime}\simeq(J_{3}^{\prime})^{\vee}$. Then we define an embedding

[TABLE]

by sending $\gamma_{1}\in{\rm SL}(J_{1}^{\prime})$ to $(\gamma_{1}|_{J_{1}^{\prime}})\oplus(\gamma_{1}^{\vee}|_{J_{2}^{\prime}})$ and $\gamma_{3}\in{\rm SL}(J_{3}^{\prime})$ to $(\gamma_{3}|_{J_{3}^{\prime}})\oplus(\gamma_{3}^{\vee}|_{J_{4}^{\prime}})$. This embedding of groups is compatible with the isomorphism (2.2) of domains, and it maps $\Gamma(N)\times\Gamma(N)$ into ${\rm O}^{+}(2U,N)$. We thus obtain a finite morphism $f:X(N)\times X(N)\to S(N)$ which maps the boundary curves

[TABLE]

of $X(N)\times X(N)$ onto $C_{1},C_{2}$ respectively. By the Manin-Drinfeld theorem for the second copy of $X(N)$, we have $[C_{1}^{\prime}]=[C_{2}^{\prime}]$ in $CH_{1}(X(N)\times X(N))_{{\mathbb{Q}}}$. Applying $f_{\ast}$, we obtain the assertion. ∎

2.2.2. The case $J_{1}\cap J_{2}=\{0\}$

We go back to the proof of Theorem 1.1. Let $\Lambda$ have signature $(2,n)$ with $n\geq 3$. Let $J_{1}\neq J_{2}$ be two isotropic planes in ${\Lambda_{\mathbb{Q}}}$ and $Z_{1},Z_{2}\subset{X_{\Gamma}}$ the corresponding $1$-dimensional cusps. We first consider the case where $J_{1}\cap J_{2}=\{0\}$. In this case the pairing between $J_{1}$ and $J_{2}$ is perfect because $J_{i}^{\perp}/J_{i}$ is negative definite. The direct sum $\Lambda_{{\mathbb{Q}}}^{\prime}=J_{1}\oplus J_{2}$ is isometric to $2U_{{\mathbb{Q}}}$. We can take an isometry $2U_{{\mathbb{Q}}}\to\Lambda_{{\mathbb{Q}}}^{\prime}$ which maps $J_{1}^{\prime},J_{2}^{\prime}$ to $J_{1},J_{2}$ respectively. This gives an embedding of orthogonal groups

[TABLE]

which is compatible with the embedding $\mathcal{D}_{2U}\simeq\mathcal{D}_{\Lambda^{\prime}}\subset{\mathcal{D}_{\Lambda}}$ of domains. By the same argument as (2.1), we can find a level $N$ such that the embedding (2.3) maps ${\rm O}^{+}(2U,N)$ into ${\Gamma}$. We thus obtain a finite morphism $f:S(N)\to{X_{\Gamma}}$ with $f(C_{1})=Z_{1}$ and $f(C_{2})=Z_{2}$. Sending the equality ${\mathbb{Q}}[C_{1}]={\mathbb{Q}}[C_{2}]$ of Lemma 2.4 by $f_{\ast}$, we obtain ${\mathbb{Q}}[Z_{1}]={\mathbb{Q}}[Z_{2}]$ in $CH_{1}({X_{\Gamma}})_{{\mathbb{Q}}}$.

2.2.3. The case $J_{1}\cap J_{2}\neq\{0\}$

We next consider the case where $J_{1}\cap J_{2}\neq\{0\}$. Let $I=J_{1}\cap J_{2}$ and choose splittings $J_{1}=I\oplus I_{1}$ and $J_{2}=I\oplus I_{2}$. Since $(I_{1},I_{2})\not\equiv 0$, we have $I_{1}\oplus I_{2}\simeq U_{{\mathbb{Q}}}$. Let ${\Lambda^{\prime}_{\mathbb{Q}}}=I_{1}\oplus I_{2}$ and ${\Lambda^{\prime\prime}_{\mathbb{Q}}}=({\Lambda^{\prime}_{\mathbb{Q}}})^{\perp}$. Then ${\Lambda^{\prime\prime}_{\mathbb{Q}}}$ has signature $(1,n-1)$. Since $n-1\geq 2$ and ${\Lambda^{\prime\prime}_{\mathbb{Q}}}$ contains at least one isotropic line $I$, then ${\Lambda^{\prime\prime}_{\mathbb{Q}}}$ contains infinitely many isotropic lines. We can choose isotropic lines $I_{3},I_{4}$ in ${\Lambda^{\prime\prime}_{\mathbb{Q}}}$ such that $I,I_{3},I_{4}$ are linearly independent. Put $J_{3}=I_{4}\oplus I_{2}$ and $J_{4}=I_{3}\oplus I_{1}$. Then $J_{3},J_{4}$ are isotropic of dimension $2$ and we have

[TABLE]

If $Z_{i}\subset{X_{\Gamma}}$ is the $1$-dimensional cusp associated to $J_{i}$, we can apply the result of §2.2.2 successively and obtain

[TABLE]

in $CH_{1}({X_{\Gamma}})_{{\mathbb{Q}}}$. This finishes the proof of Theorem 1.1 for $1$-dimensional cusps.

3. The symplectic case

In this section we prove Theorem 1.2. We first recall Siegel modular varieties (cf. [9], [11]). Let $\Lambda$ be a free ${\mathbb{Z}}$-module of rank $2g$ equipped with a nondegenerate symplectic form $(\>,\>):\Lambda\times\Lambda\to{\mathbb{Z}}$. Let ${{\rm Sp}(\Lambda)}$ be the symplectic group of $\Lambda$, namely the group of isomorphisms $\Lambda\to\Lambda$ preserving the symplectic form. For a natural number $N$ we write ${{\rm Sp}(\Lambda,N)}$ for the kernel of the reduction map ${{\rm Sp}(\Lambda)}\to{\rm GL}(\Lambda/N\Lambda)$. A subgroup ${\Gamma}$ of ${{\rm Sp}(\Lambda)}$ is called a congruence subgroup if it contains ${{\rm Sp}(\Lambda,N)}$ for some level $N$. When $g\geq 2$, every finite-index subgroup of ${{\rm Sp}(\Lambda)}$ is a congruence subgroup ([14], [2]).

Let

[TABLE]

be the Lagrangian Grassmannian parametrizing $g$-dimensional (= maximal) isotropic ${\mathbb{C}}$-subspaces of ${\Lambda_{\mathbb{C}}}$. The Hermitian symmetric domain attached to $\Lambda$ is defined as the open locus ${\mathcal{D}_{\Lambda}}\subset LG_{\Lambda}$ of those $[V]$ such that the Hermitian form $i(\cdot,\bar{\cdot})|_{V}$ on $V$ is positive definite.

Rational boundary components of ${\mathcal{D}_{\Lambda}}$ correspond to isotropic ${\mathbb{Q}}$-subspaces $I$ of ${\Lambda_{\mathbb{Q}}}$. To each such $I$ we associate the locus $\mathcal{D}_{I}\subset LG_{\Lambda}$ of those $[V]$ which contains $I$ and for which $i(\cdot,\bar{\cdot})|_{V}$ is positive semidefinite with kernel $I_{{\mathbb{C}}}$. If we consider the rational symplectic space ${\Lambda^{\prime}_{\mathbb{Q}}}=I^{\perp}/I$, then $\mathcal{D}_{I}$ is canonically isomorphic to the Hermitian symmetric domain $\mathcal{D}_{\Lambda^{\prime}}$ attached to ${\Lambda^{\prime}_{\mathbb{Q}}}$ by mapping $[V]\in\mathcal{D}_{I}$ to $[V/I_{{\mathbb{C}}}]\in\mathcal{D}_{\Lambda^{\prime}}$. The union

[TABLE]

is equipped with the Satake topology ([1], [5], [9]). By Baily-Borel [1], the quotient space ${X_{\Gamma}}={\Gamma}\backslash{\mathcal{D}_{\Lambda}^{\ast}}$ has the structure of a normal projective variety and contains ${\Gamma}\backslash{\mathcal{D}_{\Lambda}}$ as a Zariski open set.

Theorem 1.2 is proved by induction on $g$. The case $g=1$ follows from the Manin-Drinfeld theorem. Let $g\geq 2$. Assume that the theorem is proved for every congruence subgroup of ${\rm Sp}(\Lambda^{\prime})$ for every symplectic lattice $\Lambda^{\prime}$ of rank $<2g$. We then prove the theorem for ${\Gamma}<{{\rm Sp}(\Lambda)}$ with $\Lambda$ rank $2g$.

Let $I_{1}\neq I_{2}$ be two isotropic ${\mathbb{Q}}$-subspaces of ${\Lambda_{\mathbb{Q}}}$ of the same dimension, say $g^{\prime}$, and $Z_{1},Z_{2}\subset{X_{\Gamma}}$ the corresponding cusps. If we write $g^{\prime\prime}=g-g^{\prime}$, then $Z_{i}$ has dimension $k=g^{\prime\prime}(g^{\prime\prime}+1)/2$. We consider the following three cases separately:

1. (1)

$I_{1}\cap I_{2}\neq\{0\}$; 2. (2)

the pairing between $I_{1}$ and $I_{2}$ is perfect; 3. (3)

$I_{1}\cap I_{2}=\{0\}$ but the pairing between $I_{1}$ and $I_{2}$ is not perfect.

The case (1) is studied in §3.1 where $Z_{1}$ and $Z_{2}$ are joined by a modular variety in the boundary. The case (2) is studied in §3.2 where $Z_{1}$ and $Z_{2}$ are joined by a product of two modular varieties (when $g^{\prime}=1$) or by a chain of boundary modular varieties (when $g^{\prime}>1$). The remaining case (3) is considered in §3.3 where we combine the results of (1) and (2).

3.1. The case $I_{1}\cap I_{2}\neq\{0\}$

Assume that $I_{1}\cap I_{2}\neq\{0\}$. Let $I=I_{1}\cap I_{2}$. In this case $\mathcal{D}_{I_{1}},\mathcal{D}_{I_{2}}$ are in the boundary of $\mathcal{D}_{I}$. We set ${\Lambda^{\prime}_{\mathbb{Q}}}=I^{\perp}/I$, $I_{1}^{\prime}=I_{1}/I$ and $I_{2}^{\prime}=I_{2}/I$. Then $I_{1}^{\prime},I_{2}^{\prime}$ are isotropic subspaces of ${\Lambda^{\prime}_{\mathbb{Q}}}$. The isomorphism $\mathcal{D}_{I}\to\mathcal{D}_{\Lambda^{\prime}}$ extends to $\mathcal{D}_{I}^{\ast}\to\mathcal{D}_{\Lambda^{\prime}}^{\ast}$ and maps $\mathcal{D}_{I_{i}}$ to $\mathcal{D}_{I_{i}^{\prime}}$. The stabilizer of $I$ in ${\Gamma}$ acts on ${\Lambda^{\prime}_{\mathbb{Q}}}$ naturally. Let $\Gamma_{I}<{{\rm Sp}(\Lambda^{\prime}_{\mathbb{Q}})}$ be its image in ${{\rm Sp}(\Lambda^{\prime}_{\mathbb{Q}})}$. By a similar argument as Claim 2.1, $\Gamma_{I}$ is a congruence subgroup of ${\rm Sp}(\Lambda^{\prime})$ for some lattice $\Lambda^{\prime}\subset{\Lambda^{\prime}_{\mathbb{Q}}}$. If we put $X_{I}=\Gamma_{I}\backslash\mathcal{D}_{\Lambda^{\prime}}^{\ast}$, we have a generically injective morphism $f:X_{I}\to{X_{\Gamma}}$ onto the $I$-cusp.

Let $Z_{1}^{\prime},Z_{2}^{\prime}\subset X_{I}$ be the cusps of $X_{I}$ corresponding to $I_{1}^{\prime},I_{2}^{\prime}\subset{\Lambda^{\prime}_{\mathbb{Q}}}$ respectively. By our hypothesis of induction, we have ${\mathbb{Q}}[Z_{1}^{\prime}]={\mathbb{Q}}[Z_{2}^{\prime}]$ in $CH_{k}(X_{I})_{{\mathbb{Q}}}$. Since $f(Z_{i}^{\prime})=Z_{i}$, applying $f_{\ast}$ gives ${\mathbb{Q}}[Z_{1}]={\mathbb{Q}}[Z_{2}]$ in $CH_{k}({X_{\Gamma}})_{{\mathbb{Q}}}$.

3.2. The case $(I_{1},I_{2})$ perfect

Next we consider the case where the pairing between $I_{1}$ and $I_{2}$ is perfect. We shall distinguish the case $g^{\prime}>1$ and the case $g^{\prime}=1$ (i.e., top dimensional cusps).

3.2.1. The case $g^{\prime}>1$

First let $g^{\prime}>1$. We can choose a proper subspace $J_{1}\neq\{0\}$ of $I_{1}$. We put $J_{2}=J_{1}^{\perp}\cap I_{2}$ and $I_{3}=J_{1}\oplus J_{2}$. Then $I_{3}$ is isotropic of dimension $g^{\prime}$. By construction we have $I_{1}\cap I_{3}\neq\{0\}$ and $I_{3}\cap I_{2}\neq\{0\}$. Therefore we can apply the result of §3.1 to $I_{1}$ vs $I_{3}$ and to $I_{3}$ vs $I_{2}$. If $Z_{3}$ is the cusp of ${X_{\Gamma}}$ associated to $I_{3}$, this gives ${\mathbb{Q}}[Z_{1}]={\mathbb{Q}}[Z_{3}]={\mathbb{Q}}[Z_{2}]$ in $CH_{k}({X_{\Gamma}})_{{\mathbb{Q}}}$.

3.2.2. The case $g^{\prime}=1$

Next let $g^{\prime}=1$. We set ${\Lambda^{\prime}_{\mathbb{Q}}}=I_{1}\oplus I_{2}$, which is a nondegenerate symplectic space of dimension $2$. Then ${\Lambda^{\prime\prime}_{\mathbb{Q}}}:=({\Lambda^{\prime}_{\mathbb{Q}}})^{\perp}$ is also nondegenerate of dimension $2g-2$ and we have ${\Lambda_{\mathbb{Q}}}={\Lambda^{\prime}_{\mathbb{Q}}}\perp{\Lambda^{\prime\prime}_{\mathbb{Q}}}$. Let $\mathcal{D}_{\Lambda^{\prime}}$, $\mathcal{D}_{\Lambda^{\prime\prime}}$ be the Hermitian symmetric domains attached to ${\Lambda^{\prime}_{\mathbb{Q}}},{\Lambda^{\prime\prime}_{\mathbb{Q}}}$ respectively. We have the embedding of domains

[TABLE]

This is compatible with the embedding of groups

[TABLE]

The isotropic lines $I_{1},I_{2}$ in ${\Lambda^{\prime}_{\mathbb{Q}}}$ correspond to the rational boundary points $[(I_{1})_{{\mathbb{C}}}],[(I_{2})_{{\mathbb{C}}}]$ of $\mathcal{D}_{\Lambda^{\prime}}\simeq{\mathbb{H}}$. Then (3.1) extends to $\mathcal{D}_{\Lambda^{\prime}}^{\ast}\times\mathcal{D}_{\Lambda^{\prime\prime}}^{\ast}\hookrightarrow{\mathcal{D}_{\Lambda}^{\ast}}$ and maps $[(I_{i})_{{\mathbb{C}}}]\times\mathcal{D}_{\Lambda^{\prime\prime}}$ to $\mathcal{D}_{I_{i}}$.

We take some full lattices $\Lambda^{\prime}\subset{\Lambda^{\prime}_{\mathbb{Q}}}$ and $\Lambda^{\prime\prime}\subset{\Lambda^{\prime\prime}_{\mathbb{Q}}}$. By the same argument as (2.1), we can find a level $N$ such that (3.2) maps ${\rm Sp}(\Lambda^{\prime},N)\times{\rm Sp}(\Lambda^{\prime\prime},N)$ into ${\Gamma}$. If we put $X^{\prime}={\rm Sp}(\Lambda^{\prime},N)\backslash\mathcal{D}_{\Lambda^{\prime}}^{\ast}$ and $X^{\prime\prime}={\rm Sp}(\Lambda^{\prime\prime},N)\backslash\mathcal{D}_{\Lambda^{\prime\prime}}^{\ast}$, we thus obtain a finite morphism $f:X^{\prime}\times X^{\prime\prime}\to{X_{\Gamma}}$. Let $p_{1},p_{2}$ be the cusps of the modular curve $X^{\prime}$ corresponding $I_{1},I_{2}\subset{\Lambda^{\prime}_{\mathbb{Q}}}$ respectively. If we set

[TABLE]

the above consideration shows that $f(Z_{i}^{\prime})=Z_{i}$.

We have $[p_{1}]=[p_{2}]$ in $CH_{0}(X^{\prime})_{{\mathbb{Q}}}$ by the Manin-Drinfeld theorem. Taking pullback by $X^{\prime}\times X^{\prime\prime}\to X^{\prime}$, we obtain $[Z_{1}^{\prime}]=[Z_{2}^{\prime}]$ in $CH_{k}(X^{\prime}\times X^{\prime\prime})_{{\mathbb{Q}}}$. Then, taking pushforward by $f$, we obtain ${\mathbb{Q}}[Z_{1}]={\mathbb{Q}}[Z_{2}]$ in $CH_{k}({X_{\Gamma}})_{{\mathbb{Q}}}$.

3.3. The remaining case

Finally we consider the remaining case, namely $I_{1}\cap I_{2}=\{0\}$ but the pairing between $I_{1}$ and $I_{2}$ is not perfect. Let $J_{1}\subset I_{1}$ and $J_{2}\subset I_{2}$ be the kernels of the pairing between $I_{1}$ and $I_{2}$. We choose splittings $I_{1}=J_{1}\oplus K_{1}$ and $I_{2}=J_{2}\oplus K_{2}$. Then $\dim J_{1}=\dim J_{2}$ and the pairing between $K_{1}$ and $K_{2}$ is perfect. (We may have $K_{i}=\{0\}$. This is the case, e.g., when $g^{\prime}=1$.) We set ${\Lambda^{\prime}_{\mathbb{Q}}}=K_{1}\oplus K_{2}$ and ${\Lambda^{\prime\prime}_{\mathbb{Q}}}=({\Lambda^{\prime}_{\mathbb{Q}}})^{\perp}$, which are nondegenerate subspaces of ${\Lambda_{\mathbb{Q}}}$ with ${\Lambda_{\mathbb{Q}}}={\Lambda^{\prime}_{\mathbb{Q}}}\perp{\Lambda^{\prime\prime}_{\mathbb{Q}}}$. By definition $J_{1}$ and $J_{2}$ are isotropic subspaces of ${\Lambda^{\prime\prime}_{\mathbb{Q}}}$ with $J_{1}\cap J_{2}=\{0\}$ and $(J_{1},J_{2})\equiv 0$. We can take another isotropic subspace $J_{0}$ of ${\Lambda^{\prime\prime}_{\mathbb{Q}}}$ of the same dimension as $J_{1},J_{2}$ such that the pairings $(J_{0},J_{1})$ and $(J_{0},J_{2})$ are perfect. We set $I_{3}=J_{0}\oplus K_{2}$ and $I_{4}=J_{0}\oplus K_{1}$. Then $I_{3},I_{4}$ are isotropic subspaces of ${\Lambda_{\mathbb{Q}}}$ of the same dimension as $I_{1},I_{2}$. By construction the pairings $(I_{1},I_{3})$ and $(I_{2},I_{4})$ are perfect, and we have $I_{3}\cap I_{4}\neq\{0\}$. Then we can apply the result of §3.2 to $I_{1}$ vs $I_{3}$ and to $I_{2}$ vs $I_{4}$, and when $K_{i}\neq\{0\}$ the result of §3.1 to $I_{3}$ vs $I_{4}$. (When $K_{i}=\{0\}$, so that $I_{3}=I_{4}$, the latter process is skipped.) If $Z_{3},Z_{4}$ are the cusps of ${X_{\Gamma}}$ associated to $I_{3},I_{4}$ respectively, this shows that

[TABLE]

in $CH_{k}({X_{\Gamma}})_{{\mathbb{Q}}}$. This completes the proof of Theorem 1.2.

Remark 3.1*.*

Summing up the argument in the case $g^{\prime}>1$, we see that if $Z_{1}$ and $Z_{2}$ are not top dimensional, we can obtain their rational equivalence through a chain of higher dimensional cusps of length $\leq 5$.

4. The unitary case

In this section we prove Theorem 1.3. We first recall modular varieties of unitary type (cf. [8], [11]). Let $K={\mathbb{Q}}(\sqrt{-D})$ be an imaginary quadratic field with $R=\mathcal{O}_{K}$ its ring of integers (or more generally an order in $K$). By a Hermitian lattice over $R$ we mean a finitely generated torsion-free $R$-module $\Lambda$ equipped with a nondegenerate Hermitian form $(\>,\>):\Lambda\times\Lambda\to R$. We denote ${\Lambda_{K}}=\Lambda\otimes_{R}K$ and ${\Lambda_{\mathbb{C}}}=\Lambda\otimes_{R}{\mathbb{C}}$, which are Hermitian spaces over $K,{\mathbb{C}}$ respectively and in which $\Lambda$ is naturally embedded. We may assume without loss of generality that the signature $(p,q)$ of $\Lambda$ satisfies $p\leq q$.

Let ${{\rm U}(\Lambda)}$ be the unitary group of $\Lambda$, namely the group of $R$-linear isomorphisms $\Lambda\to\Lambda$ preserving the Hermitian form. This is the same as $K$-linear isomorphisms ${\Lambda_{K}}\to{\Lambda_{K}}$ preserving the lattice $\Lambda$ and the Hermitian form. We write ${\rm SU}(\Lambda)$ for the subgroup of ${{\rm U}(\Lambda)}$ of determinant $1$. For a natural number $N$ we write ${{\rm U}(\Lambda,N)}$ for the kernel of the reduction map ${{\rm U}(\Lambda)}\to{\rm GL}(\Lambda/N\Lambda)$. A subgroup ${\Gamma}$ of ${{\rm U}(\Lambda)}$ is called a congruence subgroup if it contains ${{\rm U}(\Lambda,N)}$ for some level $N$.

Let $G_{\Lambda}=G(p,{\Lambda_{\mathbb{C}}})$ be the Grassmannian parametrizing $p$-dimensional ${\mathbb{C}}$-linear subspaces of ${\Lambda_{\mathbb{C}}}$. The Hermitian symmetric domain ${\mathcal{D}_{\Lambda}}$ attached to $\Lambda$ is defined as the open locus

[TABLE]

of subspaces $V$ to which restriction of the Hermitian form is positive definite. When $p=0$, this is one point; when $p=1$, this is a ball in ${{\mathbb{P}}}{\Lambda_{\mathbb{C}}}\simeq{{\mathbb{P}}}^{q}$.

Rational boundary components of ${\mathcal{D}_{\Lambda}}$ correspond to isotropic $K$-subspaces $I$ of ${\Lambda_{K}}$. For each such $I$ we associate the locus $\mathcal{D}_{I}\subset G_{\Lambda}$ of those $V$ which contains $I$ and for which $(\>,\>)|_{V}$ is positive semidefinite with kernel $I_{{\mathbb{C}}}$. If we consider ${\Lambda_{K}^{\prime}}=I^{\perp}/I$, this is a nondegenerate $K$-Hermitian space of signature $(p-r,q-r)$ where $r=\dim_{K}I$, and $\mathcal{D}_{I}$ is naturally isomorphic to the Hermitian symmetric domain $\mathcal{D}_{\Lambda^{\prime}}$ attached to ${\Lambda_{K}^{\prime}}$ by sending $[V]\in\mathcal{D}_{I}$ to $[V/I_{{\mathbb{C}}}]$. The union

[TABLE]

is equipped with the Satake topology ([1], [5]). By Baily-Borel [1], the quotient space ${X_{\Gamma}}={\Gamma}\backslash{\mathcal{D}_{\Lambda}^{\ast}}$ has the structure of a normal projective variety and contains ${\Gamma}\backslash{\mathcal{D}_{\Lambda}}$ as a Zariski open set.

The proof of Theorem 1.3 proceeds by induction on $q$. The case $q=1$ is the Manin-Drinfeld theorem: we explain this in §4.1. The inductive argument is done in §4.2. Since this is similar to the symplectic case, we will be brief in §4.2.

4.1. On the case $q=1$

Let $q=1$. Then $r=p=q=1$, so ${\Lambda_{K}}$ is the (unique) $K$-Hermitian space of signature $(1,1)$ containing an isotropic vector, and ${\mathcal{D}_{\Lambda}}$ is the unit disc in ${{\mathbb{P}}}{\Lambda_{\mathbb{C}}}\simeq{{\mathbb{P}}}^{1}$. The group ${{\rm SU}(\Lambda_{K})}$ is naturally isomorphic to ${{\rm SL}_{2}(\mathbb{Q})}$, and $\Gamma\cap{{\rm SU}(\Lambda)}$ is mapped to a conjugate of a congruence subgroup of ${{\rm SL}_{2}(\mathbb{Z})}$ under this isomorphism. This is a classical fact, but since we could not find a suitable reference for the second assertion, we supplement below a self-contained account for the convenience of the reader. Theorem 1.3 in the case $q=1$ then follows from the Manin-Drinfeld theorem, because we have a natural finite morphism from $X_{\Gamma\cap{{\rm SU}(\Lambda)}}$ to ${X_{\Gamma}}$.

We embed $K={\mathbb{Q}}(\sqrt{-D})$ into the matrix algebra ${M_{2}(\mathbb{Q})}$ by sending $\sqrt{-D}$ to $J_{D}=\begin{pmatrix}0&-D\\ 1&0\end{pmatrix}$. Left multiplication by $J_{D}$ makes ${M_{2}(\mathbb{Q})}$ a $2$-dimensional $K$-linear space. We have a $K$-Hermitian form on ${M_{2}(\mathbb{Q})}$ defined by

[TABLE]

where for $B=\begin{pmatrix}a&b\\ c&d\end{pmatrix}$ we write $B^{\ast}=\begin{pmatrix}d&-b\\ -c&a\end{pmatrix}$. We denote ${\Lambda_{K}}={M_{2}(\mathbb{Q})}$ when we want to stress this $K$-Hermitian structure. Then ${\Lambda_{K}}$ has signature $(1,1)$ and contains an isotropic vector, e.g., $\begin{pmatrix}1&0\\ 0&0\end{pmatrix}$. Right multiplication by ${{\rm SL}_{2}(\mathbb{Q})}$ on ${M_{2}(\mathbb{Q})}$ is $K$-linear and preserves this Hermitian form. This defines a homomorphism

[TABLE]

which in fact is an isomorphism (see e.g., [16] §2).

Let $\Lambda\subset{\Lambda_{K}}$ be a full $R$-lattice. We shall show that for every level $N$ the image of ${{\rm SU}(\Lambda,N)}={{\rm SU}(\Lambda_{K})}\cap{{\rm U}(\Lambda,N)}$ by (4.1) is conjugate to a congruence subgroup of ${{\rm SL}_{2}(\mathbb{Z})}$. Let

[TABLE]

This is an order in ${M_{2}(\mathbb{Q})}$ (see [12] §2.2). Then ${{\rm SU}(\Lambda)}=\mathcal{O}^{1}$, where for any subset $\mathcal{S}$ of ${M_{2}(\mathbb{Q})}$ we write $\mathcal{S}^{1}=\mathcal{S}\cap{{\rm SL}_{2}(\mathbb{Q})}$. Take a maximal order $\mathcal{O}_{max}$ of ${M_{2}(\mathbb{Q})}$ containing $\mathcal{O}$. Since $\mathcal{O}$ is of finite index in $\mathcal{O}_{max}$, there exists a natural number $N_{0}$ such that $N_{0}\mathcal{O}_{max}\subset\mathcal{O}$. Therefore

[TABLE]

Since $(I+N\mathcal{O})^{1}\subset{{\rm SU}(\Lambda,N)}$, this implies that

[TABLE]

Since every maximal order of ${M_{2}(\mathbb{Q})}$ is conjugate to $M_{2}({\mathbb{Z}})$, there exists $g\in{\rm GL}_{2}({\mathbb{Q}})$ such that

[TABLE]

This proves our claim.

4.2. Inductive step

Let $q\geq 2$. Suppose that Theorem 1.3 is proved for all Hermitian lattices of signature $(p^{\prime},q^{\prime})$ with $p^{\prime}\leq q^{\prime}<q$. We then prove the theorem for Hermitian lattices of signature $(p,q)$ with $p\leq q$. Since the argument is similar to the symplectic case, we will just indicate the outline. Let $I_{1}\neq I_{2}$ be two isotropic $K$-subspaces of ${\Lambda_{K}}$ of the same dimension, say $r$, and $Z_{1},Z_{2}\subset{X_{\Gamma}}$ the associated cusps. We make the following classification:

1. (1)

$I_{1}\cap I_{2}\neq\{0\}$; 2. (2)

the pairing between $I_{1}$ and $I_{2}$ is perfect; 3. (3)

$I_{1}\cap I_{2}=\{0\}$ but the pairing between $I_{1}$ and $I_{2}$ is not perfect.

(1) This is similar to §3.1. In this case $Z_{1}$ and $Z_{2}$ are joined by the cusp associated to $I_{1}\cap I_{2}$, to which we can apply the hypothesis of induction.

(2) The case $r=1$ is similar to §3.2.2. If we set $\Lambda_{K}^{\prime}=I_{1}\oplus I_{2}$ and $\Lambda_{K}^{\prime\prime}=(\Lambda_{K}^{\prime})^{\perp}$, these are nondegenerate of signature $(1,1)$ and $(p-1,q-1)$ respectively. Then $Z_{1}$ and $Z_{2}$ are joined by the embedding $\mathcal{D}_{\Lambda^{\prime}}\times\mathcal{D}_{\Lambda^{\prime\prime}}\hookrightarrow{\mathcal{D}_{\Lambda}}$. We can apply the Manin-Drinfeld theorem to $\mathcal{D}_{\Lambda^{\prime}}$.

The case $r>1$ is similar to §3.2.1. We can interpolate $Z_{1}$ and $Z_{2}$ by a third cusp by taking a proper subspace $J_{1}\neq\{0\}$ of $I_{1}$ and setting $I_{3}=J_{1}\oplus(J_{1}^{\perp}\cap I_{2})$. Then we can apply the result of the case (1) to $I_{1}$ vs $I_{3}$ and to $I_{3}$ vs $I_{2}$.

(3) This is similar to §3.3. We take splittings $I_{1}=J_{1}\oplus K_{1}$ and $I_{2}=J_{2}\oplus K_{2}$ such that $(J_{1},I_{2})\equiv 0$, $(J_{2},I_{1})\equiv 0$ and $(K_{1},K_{2})$ perfect. We choose an isotropic subspace $J_{0}$ from $(K_{1}\oplus K_{2})^{\perp}$ with $(J_{1},J_{0})$ and $(J_{2},J_{0})$ perfect, and put $I_{3}=J_{0}\oplus K_{2}$ and $I_{4}=J_{0}\oplus K_{1}$. Then we apply the case (2) to $I_{1}$ vs $I_{3}$ and to $I_{4}$ vs $I_{2}$, and the case (1) to $I_{3}$ vs $I_{4}$ when $K_{i}\neq\{0\}$. This proves Theorem 1.3.

Remark 4.1*.*

As in the symplectic case, we see that when $Z_{1},Z_{2}$ are not top dimensional, their rational equivalence can be obtained through a chain of higher dimensional cusps of length $\leq 5$.

5. Modular units and higher Chow cycles

Let ${\Gamma}$, ${\mathcal{D}_{\Lambda}}$ and ${X_{\Gamma}}$ be as in the previous sections. As a consequence of Theorems 1.1 – 1.3, we can associate to each pair of maximal cusps of ${X_{\Gamma}}$ a nonzero higher Chow cycle of the modular variety ${Y_{\Gamma}}={\Gamma}\backslash{\mathcal{D}_{\Lambda}}$. This gives a higher dimensional analogue of modular units ([10]) from the viewpoint of algebraic cycles.

Let $Z_{1}\neq Z_{2}$ be two cusps of ${X_{\Gamma}}$ of the same dimension, say $k$. By our result, we have $[Z_{1}]=\alpha[Z_{2}]$ in $CH_{k}({X_{\Gamma}})_{{\mathbb{Q}}}$ for some $\alpha\neq 0\in{\mathbb{Q}}$. On the other hand, we can also view $Z_{1},Z_{2}$ as $k$-cycles on the boundary $\partial{X_{\Gamma}}={X_{\Gamma}}-{Y_{\Gamma}}$, which is an equidimensional reduced closed subscheme of ${X_{\Gamma}}$.

Lemma 5.1**.**

*When the cusps $Z_{1},Z_{2}$ are not top dimensional, the equality $[Z_{1}]=\alpha[Z_{2}]$ holds already in $CH_{k}(\partial{X_{\Gamma}})_{{\mathbb{Q}}}$. *

Proof.

When $Z_{1},Z_{2}$ are not top dimensional, the proof of Theorems 1.1 – 1.3 and Remarks 2.3, 3.1, 4.1 show that we can connect $Z_{1}$ and $Z_{2}$ by a chain of higher dimensional cusps. To be more precise, we have (congruence) modular varieties $X_{1},\cdots,X_{N}$, their cusps $Z_{i}^{+},Z_{i}^{-}\subset X_{i}$ of dimension $k$, and a finite morphism $f_{i}:X_{i}\to{X_{\Gamma}}$ onto a cusp of ${X_{\Gamma}}$, such that $f_{i}(Z_{i}^{-})=f_{i+1}(Z_{i+1}^{+})$ for each $i$ and $f_{1}(Z_{1}^{+})=Z_{1}$, $f_{N}(Z_{N}^{-})=Z_{2}$. By induction on dimension, we have $[Z_{i}^{+}]=\alpha_{i}[Z_{i}^{-}]$ in $CH_{k}(X_{i})_{{\mathbb{Q}}}$ for some $\alpha_{i}\in{\mathbb{Q}}$. Since $f_{i}$ factors through $X_{i}\to\partial X_{{\Gamma}}\subset X_{{\Gamma}}$, then $[f_{i}(Z_{i}^{+})]=\alpha_{i}^{\prime}[f_{i}(Z_{i}^{-})]$ in $CH_{k}(\partial{X_{\Gamma}})_{{\mathbb{Q}}}$ for some $\alpha_{i}^{\prime}\in{\mathbb{Q}}$. It follows that $[Z_{1}]=(\prod_{i}\alpha_{i}^{\prime})[Z_{2}]$ in $CH_{k}(\partial{X_{\Gamma}})_{{\mathbb{Q}}}$. ∎

Consider the localization exact sequence of higher Chow groups ([3], [4]) for the Baily-Borel compactification

[TABLE]

The first few terms of this sequence are written as

[TABLE]

where $\delta$ is the connecting map. By Lemma 5.1, the ${\mathbb{Q}}$-linear subspace of $CH_{k}(\partial{X_{\Gamma}})_{{\mathbb{Q}}}$ generated by the $k$-dimensional cusps has dimension $1$ if $k$ is not the maximal dimension of cusps. On the other hand, when $k=\dim\partial{X_{\Gamma}}$, the $k$-dimensional(=maximal) cusps are irreducible components of $\partial{X_{\Gamma}}$, so $CH_{k}(\partial{X_{\Gamma}})_{{\mathbb{Q}}}$ is freely generated over ${\mathbb{Q}}$ by those cusps. Let $t$ be the number of maximal cusps of ${X_{\Gamma}}$. Since the image of $i_{\ast}\colon CH_{k}(\partial{X_{\Gamma}})_{{\mathbb{Q}}}\to CH_{k}({X_{\Gamma}})_{{\mathbb{Q}}}$ has dimension $1$ by Theorems 1.1 – 1.3, we find that

[TABLE]

Let us construct some explicit elements of $CH_{k}({Y_{\Gamma}},1)_{{\mathbb{Q}}}$ whose image by $\delta$ generate ${\rm Im}(\delta)={\rm Ker}(i_{\ast})$.

Let $Z_{1}\neq Z_{2}$ be two maximal cusps of ${X_{\Gamma}}$, say of dimension $k=\dim\partial{X_{\Gamma}}$. As above, we have $i_{\ast}(Z_{1}-\alpha Z_{2})=0$ in $CH_{k}({X_{\Gamma}})_{{\mathbb{Q}}}$ for some $\alpha\in{\mathbb{Q}}$. We will construct an element of $CH_{k}({Y_{\Gamma}},1)_{{\mathbb{Q}}}$ whose image by $\delta$ is $Z_{1}-\alpha Z_{2}$ in $CH_{k}(\partial{X_{\Gamma}})_{{\mathbb{Q}}}$. (Such an element must be nonzero because $Z_{1}-\alpha Z_{2}$ is nonzero in $CH_{k}(\partial{X_{\Gamma}})_{{\mathbb{Q}}}$.) Recall from the proof of Theorems 1.1 – 1.3 that, in a basic case, we have a compactified modular curve $X^{\prime}=X_{\Gamma^{\prime}}$, its two cusps $p_{1},p_{2}\in X^{\prime}$, a $k$-dimensional compactified modular variety $X^{\prime\prime}=X_{\Gamma^{\prime\prime}}$, and a finite morphism $f\colon X^{\prime}\times X^{\prime\prime}\to{X_{\Gamma}}$ such that $f(p_{i}\times X^{\prime\prime})=Z_{i}$. (In the orthogonal case $X^{\prime\prime}$ is one point when $k=0$ and a modular curve when $k=1$; in the symplectic case $X^{\prime\prime}$ is a Siegel modular variety of genus $g-1$; in the unitary case $X^{\prime\prime}$ is associated to a unitary group of signature $(p-1,q-1)$.) The general case is a chain of such basic cases. For simplicity we assume that $(Z_{1},Z_{2})$ is such a basic pair.

By the Manin-Drinfeld theorem for $X^{\prime}$, there exists a modular function $F$ on $X^{\prime}$ such that ${\rm div}(F)=\beta(p_{1}-p_{2})$ for some natural number $\beta$. Let $Y^{\prime}\subset X^{\prime}$ and $Y^{\prime\prime}\subset X^{\prime\prime}$ be the modular varieties before compactification. We can view $F$ as an element of $\mathcal{O}^{\ast}(Y^{\prime})=CH_{0}(Y^{\prime},1)$. Then $\delta(F)=\beta(p_{1}-p_{2})$ for the connecting map $\delta\colon CH_{0}(Y^{\prime},1)\to CH_{0}(\partial X^{\prime})$. Let $\pi:Y^{\prime}\times Y^{\prime\prime}\to Y^{\prime}$ be the projection and, by abuse of notation, $f:Y^{\prime}\times Y^{\prime\prime}\to{Y_{\Gamma}}$ be the restriction of $f:X^{\prime}\times X^{\prime\prime}\to{X_{\Gamma}}$. We can pullback the higher Chow cycle $F$ by the flat morphism $\pi$ and then take its pushforward by the finite morphism $f$. The result, $f_{\ast}\pi^{\ast}F$, is an element of $CH_{k}({Y_{\Gamma}},1)$.

Proposition 5.2**.**

We have ${\mathbb{Q}}\delta(f_{\ast}\pi^{\ast}F)={\mathbb{Q}}(Z_{1}-\alpha Z_{2})$ in $CH_{k}(\partial{X_{\Gamma}})_{{\mathbb{Q}}}$.

Proof.

We take a desingularization $\tilde{X}^{\prime\prime}\to X^{\prime\prime}$ of $X^{\prime\prime}$, and let $\tilde{Y}^{\prime\prime}\subset\tilde{X}^{\prime\prime}$ be the inverse image of $Y^{\prime\prime}$. We have the commutative diagram

[TABLE]

Here various $\delta$ are the connecting maps of each localization sequence, $\tilde{\pi}\colon X^{\prime}\times\tilde{X}^{\prime\prime}\to X^{\prime}$ the projection, $\partial(X^{\prime}\times\tilde{X}^{\prime\prime})=X^{\prime}\times\tilde{X}^{\prime\prime}-Y^{\prime}\times\tilde{Y}^{\prime\prime}$, $\tilde{j}\colon Y^{\prime}\times\tilde{Y}^{\prime\prime}\hookrightarrow Y^{\prime}\times\tilde{X}^{\prime\prime}$ the open immersion, $\tilde{i}:\partial X^{\prime}\times\tilde{X}^{\prime\prime}\hookrightarrow\partial(X^{\prime}\times\tilde{X}^{\prime\prime})$ the closed embedding, and $\tilde{f}\colon X^{\prime}\times\tilde{X}^{\prime\prime}\to{X_{\Gamma}}$ the proper morphism induced from $f$. If we send ${\mathbb{Q}}F\subset CH_{0}(Y^{\prime},1)_{{\mathbb{Q}}}$ through this diagram to $CH_{k}(\partial{X_{\Gamma}})_{{\mathbb{Q}}}$, the image is ${\mathbb{Q}}(Z_{1}-\alpha Z_{2})$. The assertion follows by noticing that $\tilde{f}_{\ast}\tilde{j}^{\ast}\tilde{\pi}^{\ast}=f_{\ast}\pi^{\ast}$. ∎

In this way, as a “lift” from the modular unit $F$, we obtain an explicit nonzero element of $CH_{k}({Y_{\Gamma}},1)_{{\mathbb{Q}}}$ whose image by $\delta$ is $Z_{1}-\alpha Z_{2}$. If we run $(Z_{1},Z_{2})$ over all basic pairs of maximal cusps, we obtain a set of nonzero elements of $CH_{k}({Y_{\Gamma}},1)_{{\mathbb{Q}}}$ whose image by $\delta$ generate ${\rm Im}(\delta)={\rm Ker}(i_{\ast})$. In general, by this construction we could obtain more than $t-1$ higher Chow cycles on ${Y_{\Gamma}}$. This is because

1. (1)

the choice of $X^{\prime}\times X^{\prime\prime}\to{X_{\Gamma}}$ is not necessarily unique for the given pair $(Z_{1},Z_{2})$, and 2. (2)

the number of basic pairs could be larger than $t-1$.

The point (1) amounts to the situation that two pairs $(I_{1},I_{2})$, $(I_{1}^{\prime},I_{2}^{\prime})$ of isotropic subspaces are not ${\Gamma}$-equivalent as pairs although $I_{1}$ is ${\Gamma}$-equivalent to $I_{1}^{\prime}$ and $I_{2}$ is ${\Gamma}$-equivalent to $I_{2}^{\prime}$ respectively. A typical situation of (2) is that for three cusps $Z_{1},Z_{2},Z_{3}$, all pairs $(Z_{1},Z_{2})$, $(Z_{2},Z_{3})$, $(Z_{3},Z_{1})$ are basic.

If the span $V\subset CH_{k}({Y_{\Gamma}},1)_{{\mathbb{Q}}}$ of all higher Chow cycles constructed in this way has dimension $\geq t$, the kernel of $\delta\colon V\to CH_{k}(\partial{X_{\Gamma}})_{{\mathbb{Q}}}$ would then give rise to a nontrivial subspace of $CH_{k}({X_{\Gamma}},1)_{{\mathbb{Q}}}$.