Quantitative Oppenheim conjecture for $S$-arithmetic quadratic forms of rank $3$ and $4$

TL;DR

This paper extends the quantitative Oppenheim conjecture results to $S$-arithmetic quadratic forms of rank 3 and 4, showing that almost all such forms satisfy the conjecture despite exceptions in low dimensions.

Contribution

It generalizes the quantitative Oppenheim conjecture to the $S$-arithmetic setting for quadratic forms of rank 3 and 4, including almost all forms.

Findings

Almost all $S$-arithmetic quadratic forms of rank 3 and 4 satisfy the conjecture.

Extension of previous results from real to $S$-arithmetic quadratic forms.

The result holds despite known exceptions in low-dimensional cases.

Abstract

The celebrated result of Eskin, Margulis and Mozes (1998) and Dani and Margulis (1993) on quantitative Oppenheim conjecture says that for irrational quadratic forms $q$ of rank at least 5, the number of integral vectors $\mathbf v$ such that $q(\mathbf v)$ is in a given bounded interval is asymptotically equal to the volume of the set of real vectors $\mathbf v$ such that $q(\mathbf v)$ is in the same interval. In dimension $3$ or $4$, there are exceptional quadratic forms which fail to satisfy the quantitative Oppenheim conjecture. Even in those cases, one can say that almost all quadratic forms hold that two asymptotic limits are the same ([Eskin-Margulis-Mozes'98, Theorem 2.4]). In this paper, we extend this result to the $S$-arithmetic version.

Equations

$$\begin{array}{rl}{\mathbb{Z}}_S& =\{m{p}_1^{n_1}\cdots p_s^{n_s}:m,n_1,\dots ,n_s\in \mathbb{Z}\}\end{array}$$

$$\sigma(p)=\left\{\begin{array}[]{ll}-1&\text{if}\;p<\infty,\\ 1&\text{if}\;p=\infty.\\ \end{array}\right.$$

$${\Omega }_p=\left\{\mathbf{v}\in {\mathbb{Q}}_p^n:\Vert \mathbf{v}{\Vert }_p<{\rho }_p(\mathbf{v}/\Vert \mathbf{v}{\Vert }_p^{\sigma })\right\},$$

$${\rho }_p(u{v}_p)=\rho (v_p).$$

$$\underset{\mathsf{T}\to \infty }{\lim }\mathbf{N}(\mathsf{T})=C({\mathsf{q}}_S,\Omega )\mu ({\mathsf{I}}_S)|\mathsf{T}{|}^{n-2},$$

$$\mathbf{N}({\mathsf{q}}_S,\Omega ,{\mathsf{I}}_S)(\mathsf{T})\sim {\lambda }_{{\mathsf{q}}_S,\Omega }\mu ({\mathsf{I}}_S)|\mathsf{T}{|}^{n-2},$$

$$\begin{split}&\left\{\begin{array}[]{l}q_{\infty}(x_{1},x_{2},x_{3})=2x_{1}x_{3}+x_{2}^{2},\\ q_{p}(x_{1},x_{2},x_{3})=2x_{1}x_{3}+a_{1}x_{2}^{2},\;p\in S_{f},\\ \end{array}\right.\hskip 62.51395pt\text{if}\;\operatorname{rank}(\mathsf{q}^{0}_{S})=3,\\ &\left\{\begin{array}[]{l}q_{\infty}(x_{1},x_{2},x_{3},x_{4})=2x_{1}x_{4}+a_{1}x_{2}^{2}+a_{2}x_{3}^{2},\\ q_{p}(x_{1},x_{2},x_{3},x_{4})=2x_{1}x_{4}+a_{1}x_{2}^{2}+a_{2}x_{3}^{2},\;p\in S_{f},\end{array}\right.\quad\text{if}\;\operatorname{rank}(\mathsf{q}^{0}_{S})=4,\\ \end{split}$$

$$\mathsf{K}=K_{\infty }\times \prod _{p\in S_f}{K}_p=\left(\mathrm{SO}(q_{\infty }^0)\cap \mathrm{SO}(n)\right)\times \prod _{p\in S_f}\left(\mathrm{SO}(q_p^0)\cap {\mathrm{SL}}_n({\mathbb{Z}}_p)\right)$$

$$\mathsf{A}=\left\{{\mathsf{a}}_{\mathsf{t}}=\prod _{p\in S}{a}_{t_{p_i}}:\mathsf{t}=(t_{\infty },t_1,\dots ,t_s)\in {\mathbb{R}}_{\ge 0}\times p_1^{\mathbb{Z}}\times \cdots \times p_s^{\mathbb{Z}}\right\},$$

$$T_{\infty }=e^{t_{\infty }}\text{and}{T}_p=p^{t_p},\forall p\in S_f,$$

$$d_{\infty }(g_1,g_2)=d_{\infty }(K_{\infty }{g}_1,K_{\infty }{g}_2):=d_{{\mathbb{H}}^3}(\mathrm{p}(g_1),\mathrm{p}(g_2)).$$

$$d_{\infty }(g_1,g_2):=d_{{\mathbb{H}}^2}(\mathrm{p}(g_1),\mathrm{p}(g_2)).$$

$$(x,y,z,w)\mapsto\left(\begin{array}[]{cc}x&y\\ z&w\end{array}\right).$$

$$(g_{1},g_{2}).\left(\begin{array}[]{cc}x&y\\ z&w\end{array}\right)=g_{1}\left(\begin{array}[]{cc}x&y\\ z&w\end{array}\right)g^{t}_{2}.$$

$$K_{\infty }\setminus \mathrm{SO}(2,2){\simeq }_{loc}(\mathrm{SO}(2)\times \mathrm{SO}(2))\setminus ({\mathrm{SL}}_2(\mathbb{R})\times {\mathrm{SL}}_2(\mathbb{R}))\simeq {\mathbb{H}}_1^2\times {\mathbb{H}}_2^2,$$

$$d_{\infty }=d_{\mathrm{SO}(2,2)}:=\max \{d_{{\mathbb{H}}_1^2},d_{{\mathbb{H}}_2^2}\}.$$

$$\left|\left\{k\in K_{\infty }:d_{\infty }(a_{t}k{a}_t^{-1},1)\le r\right\}\right|<C_{\infty }{e}^{-2t+r},$$

$$\begin{array}{rl}\left|\left\{k\in K_{\infty }:d_{\infty }(a_{t}k{a}_t^{-1},1)\le r\right\}\right|& =\left|\left\{k\in K_{\infty }:d_{\infty }(a_{t}k,a_t)\le r\right\}\right|\\ & =\left|\left\{y\in S_{2t}:d_{{\mathbb{H}}^3}(y,e^{2t}i)\le r\right\}\right|.\end{array}$$

$$\cosh (d(x,y))=1+{\sinh }^2(2t)\cdot (1-\cos \theta )=1+2{\sinh }^2(2t){\sin }^2(\frac{\theta }{2}),$$

$$\left|\left\{k\in K_{\infty }:d_{\infty }(a_{t}k{a}_t^{-1},1)\le r\right\}\right|\le C_{\infty }{e}^{2(-2t+r/2)}$$

$$\begin{array}{rl}& \left|\left\{k\in K_{\infty }:d_{\infty }(a_{t}k{a}_t^{-1},1)\le r\right\}\right|\\ & =\left|\{k_1\in \mathrm{SO}(2):d_{\mathbb{H}}(b_t{k}_1,b_t)\le r\}\times \{k_2\in \mathrm{SO}(2):d_{\mathbb{H}}(b_t{k}_2,b_t)\le r\}\right|\\ & <C{\left(e^{-t+r/2}\right)}^2=C{e}^{-2t+r}.\end{array}$$

$${\mathrm{GL}}_n({\mathbb{Q}}_p)={\mathrm{GL}}_n({\mathbb{Z}}_p)\cdot \hat{A}\cdot {\mathrm{GL}}_n({\mathbb{Z}}_p)={\mathrm{GL}}_n({\mathbb{Z}}_p)\cdot {\hat{A}}^{+}\cdot {\mathrm{GL}}_n({\mathbb{Z}}_p),$$

$$\begin{array}{rl}\hat{A}& =\{\mathrm{diag}(p^{m_1},p^{m_2},\dots ,p^{m_n}):m_1,m_2,\dots ,m_n\in \mathbb{Z}\},\\ {\hat{A}}^{+}& =\{\mathrm{diag}(p^{m_1},p^{m_2},\dots ,p^{m_n}):0\le m_1\le m_2\le \cdots \le m_n\}\subseteq \hat{A}.\end{array}$$

$$g.({\mathbb{Z}}_p{\mathbf{v}}_1\oplus \cdots \oplus {\mathbb{Z}}_p{\mathbf{v}}_n)={\mathbb{Z}}_p({\mathbf{v}}_{1}g)\oplus \cdots \oplus {\mathbb{Z}}_p({\mathbf{v}}_{n}g),$$

$$p{L}_0⊊L_1⊊\cdots ⊊L_k⊊L_0.$$

$$\begin{array}{rl}& [pL]=[⟨p{\mathbf{v}}_1,p{\mathbf{v}}_2,\dots ,p{\mathbf{v}}_n⟩]⊊[⟨{\mathbf{v}}_1,p{\mathbf{v}}_2,\dots ,p{\mathbf{v}}_n⟩]⊊\\ & [⟨{\mathbf{v}}_1,{\mathbf{v}}_2,p{\mathbf{v}}_3,\dots ,p{\mathbf{v}}_n⟩]⊊[⟨{\mathbf{v}}_1,\dots ,{\mathbf{v}}_{n-1},p{\mathbf{v}}_n⟩⊊[⟨{\mathbf{v}}_1,\dots ,{\mathbf{v}}_n⟩]=[L].\end{array}$$

$$\mathcal{V}(\mathcal{A})=\{[ag]\in {\mathrm{GL}}_n({\mathbb{Z}}_p)\setminus {\mathrm{GL}}_n({\mathbb{Q}}_p):a\in \hat{A}\}$$

$$\{[\mathrm{diag}(p^{m_1},\dots ,p^{m_n})]:m_i\in {\mathbb{Z}}_{\ge 0}\}.$$

$$\mathrm{p}:\left[\left(\begin{array}[]{cccc}p^{m_{1}}&v_{12}&\cdots&v_{1n}\\ &p^{m_{2}}&\cdots&v_{2n}\\ &&\ddots&\vdots\\ &&&p^{m_{n}}\end{array}\right)\right]\mapsto[\mathrm{diag}(p^{m_{1}},p^{m_{2}},\ldots,p^{m_{n}})],$$

$${\mathrm{p}}^{-1}(\{a(m):m\in \mathbb{Z}\})$$

Full text

In dimension $3$ or $4$, there are exceptional quadratic forms which fail to satisfy the quantitative Oppenheim conjecture. Even in those cases, one can say that almost all quadratic forms hold that two asymptotic limits are the same ([7, Theorem 2.4]). In this paper, we extend this result to the $S$-arithmetic version.

1. Introduction

History

The Oppenheim conjecture, proved by Margulis [15], says that the image set $q(\mathbb{Z}^{n})$ of integral vectors of an isotropic irrational quadratic form $q$ of rank at least 3 is dense in the real line (See [17] for the original statement of Oppenheim conjecutre).

Let $(a,b)$ be a bounded interval and let $\Omega\subset{\mathbb{R}}^{n}$ be a convex set containing the origin. Dani and Margulis [6] and Eskin, Margulis and Mozes [7] established a quantitative version of Oppenheim conjecture: for a quadratic form $q$, let $\mathbf{V}_{(a,b),\Omega}(T)$ be the volume of the region $T\Omega\cap q^{-1}(a,b)$ and $\mathbf{N}_{(a,b),\Omega}(T)$ be the number of integral vectors in $T\Omega\cap q^{-1}(a,b)$. They found that if an irrational isotropic quadratic form $q$ is of rank greater than or equal to 4 and is not a split form, then $\mathbf{V}_{(a,b),\Omega}(T)$ approximates $\mathbf{N}_{(a,b),\Omega}(T)$ as $T$ goes to infinity. Moreover, in this case, the image set $q({\mathbb{Z}}^{n})$ is equidistributed in the real line.

Their works heavily rely on the dynamical properties of orbits of a certain semisimple Lie group, generated by its unipotent one-parameter subgroups, on the associated homogeneous space.

The $S$-arithmetic space is one of the optimal candidates to consider a generalization of their work, since it has a lattice ${\mathbb{Z}}_{S}^{n}$ which is similar to the integral lattice ${\mathbb{Z}}^{n}$ in ${\mathbb{R}}^{n}$. Borel and Prasad [4] extend Margulis’ theorem to the $S$-arithmetic version and Han, Lim and Mallahi-Karai [12] proved the quantitative version of $S$-arithmetic Oppenheim conjecture. See also [27] and [13] for flows on $S$-arithmetic symmetric spaces. Ratner [19] generalized her measure rigidity of unipotent subgroups in the real case to the cartesian product of real and $p$-adic spaces.

Main statements

Consider a finite set $S_{f}=\{p_{1},\ldots,p_{s}\}$ of odd primes and let $S=\{\infty\}\cup S_{f}$. For each $p\in S$, denote by ${\mathbb{Q}}_{p}$ the completion field of ${\mathbb{Q}}$ with respect to the $p$-adic norm. If $p=\infty$, the norm $\|\cdot\|_{\infty}$ is the usual Euclidean norm and ${\mathbb{Q}}_{\infty}={\mathbb{R}}$. Define ${\mathbb{Q}}_{S}=\prod_{p\in S}{\mathbb{Q}}_{p}$. The diagonal embedding of

[TABLE]

is a uniform lattice subgroup in an additive group ${\mathbb{Q}}_{S}$. We will define an $S$-lattice $\Delta$ in ${\mathbb{Q}}_{S}^{n}$ as a free ${\mathbb{Z}}_{S}$-module of rank $n$. An $S$-quadratic form $\mathsf{q}_{S}$ is a collection of quadratic forms $q_{p}$ defined over ${\mathbb{Q}}_{p}$, $p\in S$. We call $\mathsf{q}_{S}$ nondegenerate or isotropic if every $q_{p}$ in $\mathsf{q}_{S}$ is nondegenerate or isotropic, respectively. We say that $\mathsf{q}_{S}$ is rational if there is a single rational quadratic form $q$ such that $\mathsf{q}_{S}=(\lambda_{p}q)_{p\in S}$, where $\lambda_{p}\in{\mathbb{Q}}_{p}-\{0\}$ and $\mathsf{q}_{S}$ is irrational if it is not rational.

Notation. For a nonzero vector $\mathbf{v}=(v_{p})_{p\in S}\in{\mathbb{Q}}_{S}^{n}$, the $p$-adic norm $\|\mathbf{v}\|_{p}$ refers the $p$-adic norm $\|v_{p}\|_{p}$ of $p$-adic component $v_{p}\in{\mathbb{Q}}_{p}^{n}$. We will use the following notation

[TABLE]

We use the notation $\mathbf{v}/\|\mathbf{v}\|_{p}^{\sigma}:=(v_{p}/|v_{p}|_{p}^{\sigma})_{p\in S}$ for a vector consisting of unit vectors in each place $p\in S$.

For $p\in S$, we fix a star-shaped convex set $\Omega_{p}$ on ${\mathbb{Q}}_{p}^{n}$ centered at the origin defined as

[TABLE]

where $\rho_{p}$ is a positive function on the set of unit vectors in ${\mathbb{Q}}_{p}^{n}$ and let $\Omega=\prod_{p\in S}\Omega_{p}$. If $p<\infty$, we will assume that $\rho_{p}$ is $({\mathbb{Z}}_{p}-p{\mathbb{Z}}_{p})$-invariant: for any $u\in{\mathbb{Z}}_{p}-p{\mathbb{Z}}_{p}$ and for any unit vector $v_{p}\in{\mathbb{Z}}_{p}^{n}-p{\mathbb{Z}}_{p}^{n}$,

[TABLE]

Let $I_{p}\subset{\mathbb{Q}}_{p}$ be a bounded convex set of the form $(a,b)$ if $p=\infty$ and $a+p^{b}{\mathbb{Z}}_{p}$ if $p<\infty$. We call $I_{p}$ a $p$-adic interval. Define an $S$-interval $\mathsf{I}_{S}=\prod_{p\in S}I_{p}$. Denote by $\mathsf{T}=(T_{p})_{p\in S}\in{\mathbb{R}}_{\geq 0}\times\prod_{p\in S_{f}}(p^{{\mathbb{Z}}}\cup\{0\})$ the collection of radius parameters $T_{p}$, $p\in S$. Consider $\mathsf{T}\Omega=\prod_{p\in S}T_{p}\Omega_{p}$ the dilation of $\Omega$ by $\mathsf{T}$.

We are interested in the number $\mathbf{N}(\mathsf{T})=\mathbf{N}(\mathsf{q}_{S},\Omega,\mathsf{I}_{S})(\mathsf{T})$ of vectors $\mathbf{v}$ in ${\mathbb{Z}}_{S}^{n}\cap\mathsf{T}\Omega$ such that $\mathsf{q}_{S}(\mathbf{v})\in\mathsf{I}_{S}$. Let $\mu$ be the product of Haar measures $\mu_{p}$ on ${\mathbb{Q}}_{p}$, $p\in S$. We assume that $\mu_{\infty}$ is the usual Lebesgue measure on ${\mathbb{R}}$ and $\mu_{p}({\mathbb{Z}}_{p})=1$ when $p<\infty$. Let $\mathbf{V}(\mathsf{T})=\mathbf{V}(\mathsf{q}_{S},\Omega,\mathsf{I}_{S})(\mathsf{T})$ be the volume of the set $\{\mathbf{v}\in\mathsf{T}\Omega:\mathsf{q}_{S}(\mathbf{v})\in\mathsf{I}_{S}\}$ with respect to $\mu^{n}$ on ${\mathbb{Q}}_{S}^{n}$.

Recall that a quadratic form of rank $4$ is split if it is equivalent to $x_{1}x_{4}-x_{2}x_{3}$. In [12], when an isotropic irrational quadratic form $\mathsf{q}_{S}$ is of rank $n\geq 4$ and does not contain a split form, then as $\mathsf{T}\rightarrow\infty$, $\mathbf{N}(\mathsf{T})$ is approximated to $\mathbf{V}(\mathsf{T})$. Here, we say that $\mathsf{T}\rightarrow\infty$ if $T_{p}\rightarrow\infty$ for all $p\in S$. Moreover, it is possible to estimate $\mathbf{V}(\mathsf{T})$ in terms of $\mathsf{I}_{S}$ and $\mathsf{T}$. As a result, there is a constant $C(\mathsf{q}_{S},\Omega)>0$ such that

[TABLE]

where $|{\mathsf{T}}|=\prod_{p\in S}T_{p}$.

When $\mathsf{q}_{S}$ is of rank $3$ or $\mathsf{q}_{S}$ is of rank $4$ and contains a split form, there exist quadratic forms such that $\mathbf{N}(\mathsf{T})$ fails to approximate $\mathbf{V}(\mathsf{T})$. For instance, there is an irrational quadratic form $\mathsf{q}_{S}$ of rank $3$ and a sequence $(\mathsf{T}_{j})$ for a given $\varepsilon>0$ so that $\mathbf{N}(\mathsf{T}_{j})\geq|\mathsf{T}_{j}|(\log|\mathsf{T}_{j}|)^{1-\varepsilon}$ (see [7, Theorem 2.2] and [12, Lemma 9.2]).

Even in the low dimensional cases, one can expect that for generic isotropic quadratic forms, $\mathbf{N}(\mathsf{T})$ approximates $\mathbf{V}(\mathsf{T})$, which is our main theorem.

Here, the term generic is with respect to the following measure: fix some quadratic form $\mathsf{q}^{0}_{S}$. One can identify ${\mathrm{SO}}(\mathsf{q}^{0}_{S})\setminus\mathrm{SL}_{n}({\mathbb{Q}}_{S})$ with the space of quadratic forms of the same discriminant with $\mathsf{q}^{0}_{S}$. Under this identification, one can assign a natural $\mathrm{SL}_{n}({\mathbb{Q}}_{S})$-invariant measure on the space of quadratic forms.

Theorem 1.1**.**

For almost all isotropic quadratic forms $\mathsf{q}_{S}$ of rank 3 or 4, as $\mathsf{T}\rightarrow\infty$,

[TABLE]

where $n=\operatorname{rank}(\mathsf{q}_{S})$ and $\lambda_{\mathsf{q}_{S},\Omega}$ is a constant depending on a quadratic form $\mathsf{q}_{S}$ and a convex set $\Omega$.

For real quadratic forms of signature $(2,2)$, Eskin, Margulis and Mozes describe quadratic forms such that $\mathbf{N}_{a,b,\Omega}(T)$ does not approximate to $\mathbf{V}_{a,b,\Omega}(T)$ ([8]).

On the generic quadratic forms and their Oppenheim conjecture-type problems, there is a work of Bourgain [5] for real diagonal quadratic forms using analytic number theoretical method. Ghosh and Kelmer [9] showed another version of quantitative version of Oppenheim problem for real generic ternary quadratic forms and Ghosh, Gorodnik and Nevo [10] extended this result for more general setting, such as for generic characteristic polynomial maps. Recently, Athreya and Margulis [2] provided the bound of error terms $\mathbf{N}_{(a,b),\Omega}(T)-\mathbf{V}_{(a,b),\Omega}(T)$ for almost every real quadratic forms of rank at least 3.

For a pair of a quadratic form $q$ and a linear form $L$ of ${\mathbb{R}}^{n}$, Gorodnik [11] showed the density of $\{(q(\mathbf{v}),L(\mathbf{v})):\mathbf{v}\in\mathcal{P}({\mathbb{Z}}^{n})\}\subseteq{\mathbb{R}}^{2}$ under the certain assumption and Lazar [14] generalized his result for the $S$-arithmetic setting. Sargent showed the density of $M(\{\mathbf{v}\in{\mathbb{Z}}^{n}:q(\mathbf{v})=a\})\subseteq{\mathbb{R}}$, where $q$ is a rational quadratic form and $M$ is a linear map ([21]). In [22], he showed the quantitative version of his result.

In Section 2, we briefly review the symmetric space of the real and $p$-adic Lie groups. In Section 3, we introduce an $S$-arithmetic alpha function, defined on $\mathrm{SL}_{n}({\mathbb{Q}}_{S})/\mathrm{SL}_{n}({\mathbb{Z}}_{S})$ and equidistribution properties. We prove the main theorem in Section 4.

Acknowledgments

I would like to thank Seonhee Lim and Keivan Mallahi-Karai for suggesting this problem and providing valuable advices. This paper is supported by the Samsung Science and Technology Foundation under project No. SSTF-BA1601-03.

2. Symmetric spaces

Let us denote by $\mathsf{G}$ an $S$-arithmetic group of the form $\mathsf{G}=\prod_{p\in S}G_{p}$, where $G_{p}$ is a semisimple Lie group defined over ${\mathbb{Q}}_{p}$, $p\in S$. An element of $\mathsf{G}$ is $\mathsf{g}=(g_{p})_{p\in S}=(g_{\infty},g_{1},\ldots,g_{s})$, where $g_{i}\in G_{p_{i}}$. In most of cases, $\mathsf{G}$ will be $\mathrm{SL}_{n}({\mathbb{Q}}_{S}):=\prod_{p\in S}\mathrm{SL}_{n}({\mathbb{Q}}_{p})$, $n\geq 3$. Consider the lattice subgroup $\Gamma=\mathrm{SL}_{n}({\mathbb{Z}}_{S})$ of $\mathsf{G}$ and the symmetric space $\mathsf{G}/\Gamma$, which can be embedded in the space of unimodular $S$-lattices in ${\mathbb{Q}}_{S}^{n}$.

The notation $\mathsf{q}^{0}_{S}$ refers to a standard quadratic form, which is the collection of quadratic forms such that

[TABLE]

where $a_{1},a_{2}\in\{\pm 1\}$ if $p=\infty$ and $a_{1},a_{2}\in\{\pm 1,\pm u_{0},\pm p,\pm pu_{0}\}$ if $p<\infty$. Here $u_{0}$ is some fixed square-free integer in ${\mathbb{Q}}_{p}$ (See [24, Section 4.2]).

In ${\mathrm{SO}}(\mathsf{q}^{0}_{S})$, we fix a maximal compact subgroup $\mathsf{K}$ of ${\mathrm{SO}}(\mathsf{q}^{0}_{S})$

[TABLE]

and a diagonal subgroup

[TABLE]

where $a_{t_{\infty}}=\mathrm{diag}(e^{-t_{\infty}},1,\ldots,1,e^{t_{\infty}})$ and $a_{t_{p}}=\mathrm{diag}(p^{t_{p}},1,\ldots,p^{-t_{p}})$, $p\in S_{f}$. These groups $\mathsf{K}$ and $\mathsf{A}$ will be heavily used throughout the paper.

We assume that $\mathsf{T}$ and $\mathsf{t}$ have the relation

[TABLE]

unless otherwise specified. Also we briefly denote by $T_{i}$ or $t_{i}$ instead of $T_{p_{i}}$ or $t_{p_{i}}$ respectively.

In this section, we take $G_{\infty}=\mathrm{SL}_{n}({\mathbb{R}})$ and $G_{p}=\mathrm{GL}_{n}({\mathbb{Q}}_{p})$, $p<\infty$, where $n=3$ or $4$. Corresponding maximal compact subgroups are $\hat{K}_{\infty}={\mathrm{SO}}(n)$ and $\hat{K}_{p}=\mathrm{GL}_{n}({\mathbb{Z}}_{p}),$ respectively.

2.1. The 3-dimensional hyperbolic space ${\mathbb{H}}^{3}$

Let $q^{0}_{\infty}$ be a standard isotropic quadratic form of rank $n$, $n=3,4$. The special orthogonal subgroup $H_{\infty}={\mathrm{SO}}(q^{0}_{\infty})$ is one of the well-known linear groups ${\mathrm{SO}}(2,1)$, ${\mathrm{SO}}(3,1)$ and ${\mathrm{SO}}(2,2)$. Let us examine the symmetric space of $H_{\infty}$ quotiented by $K_{\infty}$ and define a metric invariant by right multiplication.

Case i) $H_{\infty}={\mathrm{SO}}(3,1)$: Set ${\mathbb{H}}^{3}$ to be the 3-dimensional hyperbolic space ${\mathbb{H}}^{3}=\{z+ti:z=x+yj\in{\mathbb{C}}\;\text{and}\;t\in{\mathbb{R}}_{>0}\;\}$. The group ${\mathrm{SO}}(3,1)$ is locally isomorphic to $\mathrm{PSL}_{2}({\mathbb{C}})$, which is the group $\mathrm{Isom}^{+}({\mathbb{H}}^{3})$ of orientation-preserving isometries of ${\mathbb{H}}^{3}$. Since the stabilizer of the point $i$ in $\mathrm{PSL}_{2}({\mathbb{C}})$ is the maximal compact subgroup $\mathrm{{PSU}}(2)$, we may identify the symmetric space $K_{\infty}\setminus{\mathrm{SO}}(3,1)$, which is isomorphic to $\mathrm{{PSU}}(2)\setminus\mathrm{PSL}_{2}({\mathbb{C}})$, with the hyperbolic space ${\mathbb{H}}^{3}$. Let $\mathrm{p}:{\mathrm{SO}}(3,1)\rightarrow K_{\infty}\setminus{\mathrm{SO}}(3,1)\simeq{\mathbb{H}}^{3}$ be the projection given by $\mathrm{p}(g)=g.i.$ Define the metric $d_{\infty}$ of $K_{\infty}\setminus{\mathrm{SO}}(3,1)$ by

[TABLE]

Case ii) $H_{\infty}={\mathrm{SO}}(2,1)$: Consider the isomorphism $K_{\infty}\setminus{\mathrm{SO}}(2,1)\simeq{\mathrm{PSO}}(2)\setminus\mathrm{PSL}_{2}({\mathbb{R}})\simeq{\mathbb{H}}^{2}=\{x+ti:x\in{\mathbb{R}}\;\text{and}\;t\in{\mathbb{R}}_{>0}\>\}$. We will use the notation $\mathrm{p}$ for the projection ${\mathrm{SO}}(2,1)\rightarrow{\mathbb{H}}^{2}$ as well. We define the metric $d_{\infty}$ of $K_{\infty}\setminus{\mathrm{SO}}(2,1)$ by

[TABLE]

Case iii)$H_{\infty}={\mathrm{SO}}(2,2)$: In the case of $H_{\infty}={\mathrm{SO}}(2,2)$, consider the isomorphism between real vector spaces ${\mathbb{R}}^{4}$ and $\mathcal{M}_{2}({\mathbb{R}})$ given by

[TABLE]

The split form $xw-yz$ of ${\mathbb{R}}^{4}$ corresponds to the determinant of \left(\begin{array}[]{cc}x&y\\ z&w\end{array}\right) in $\mathcal{M}_{2}({\mathbb{R}})$. Moreover, there is the local isomorphism from $\mathrm{SL}_{2}({\mathbb{R}})\times\mathrm{SL}_{2}({\mathbb{R}})$ to ${\mathrm{SO}}(2,2)$, which is induced by the action

[TABLE]

Hence we deduce that

[TABLE]

where ${\mathbb{H}}^{2}_{1}\simeq{\mathbb{H}}^{2}\simeq{\mathbb{H}}^{2}_{2}$. Note also that the action of $a_{t}$ splits into the action of $(b_{t},b_{t})$, where $b_{t}=\mathrm{diag}(e^{t/2},e^{-t/2})$. Put

[TABLE]

Lemma 2.1**.**

Let $H_{\infty}$ be one of ${\mathrm{SO}}(2,1)$, ${\mathrm{SO}}(3,1)$ or ${\mathrm{SO}}(2,2)$. Let $K_{\infty}$ be a maximal compact subgroup of $H_{\infty}$. Then there exists a constant $C_{\infty}>0$ such that for any $t>0$ and for any $r\in(0,2t)$,

[TABLE]

where $|\cdot|$ is the normalized Haar measure on $K_{\infty}$.

Proof.

Assume that $H_{\infty}={\mathrm{SO}}(3,1)\cong\mathrm{SL}_{2}({\mathbb{C}})$ and $K_{\infty}\cong\mathrm{{SU}}(2)$. For each $t>0$, $K_{\infty}$ acts transitively on the hyperbolic sphere $S_{2t}\subset{\mathbb{H}}^{3}$ of radius $2t$ centered at $i$.

Since $d_{\infty}$ is $H_{\infty}$-invariant,

[TABLE]

The $K_{\infty}$-invariant measure of $S_{2t}$ is identified with the normalized Lebesgue measure of the unit sphere $\SS^{2}$ which is isomorphic to $\partial{\mathbb{H}}^{3}$.

Let $x=e^{2t}i$ and let $\theta$ be the angle between two geodesics from $i$ to $x$ and from $i$ to $y$ (Figure 1). By the hyperbolic law of cosines

[TABLE]

and since $d(x,y)\leq r$, we obtain that $\theta$ is bounded by $e^{-2t+r/2}$. Hence

[TABLE]

for some constant $C_{\infty}>0$.

The case of $H_{\infty}={\mathrm{SO}}(2,1)\cong\mathrm{SL}_{2}({\mathbb{R}})$ follows immediately from the compatibility between two embeddings ${\mathrm{SO}}(2,1)\hookrightarrow{\mathrm{SO}}(3,1)$ and ${\mathbb{H}}^{2}\hookrightarrow{\mathbb{H}}^{3}$ with the projection $\mathrm{p}$ (see also [7, Lemma 3.12]).

If $H_{\infty}={\mathrm{SO}}(2,2)$, using the local isometry (2.2), $K_{\infty}\cong{\mathrm{SO}}(2)\times{\mathrm{SO}}(2)$ acts transitively on the product $S_{t}\times S_{t}$ of two hyperbolic spheres in ${\mathbb{H}}^{2}\times{\mathbb{H}}^{2}$. Thus we have

[TABLE]

∎

2.2. A tree in the building $\mathcal{B}_{n}$

For a prime $p$ and $n\geq 2$, let us briefly recall the Euclidean building $\mathcal{B}_{n}$, whose vertex set $\mathcal{V}(\mathcal{B}_{n})$ is isomorphic to $\mathrm{GL}_{n}({\mathbb{Z}}_{p})\setminus\mathrm{GL}_{n}({\mathbb{Q}}_{p})$. Recall the Cartan decomposition ([18, Theorem 3.14])

[TABLE]

where

[TABLE]

Consider the space of free ${\mathbb{Z}}_{p}$-modules $L$ of rank $n$ in ${\mathbb{Q}}_{p}^{n}$ on which $\mathrm{GL}_{n}({\mathbb{Q}}_{p})$ acts by

[TABLE]

where $g\in\mathrm{GL}_{n}({\mathbb{Q}}_{p})$ and $\mathbf{v}_{1},\ldots,\mathbf{v}_{n}\in{\mathbb{Q}}_{p}^{n}$ are ${\mathbb{Z}}_{p}$-basis of $L$. Two rank-$n$ free ${\mathbb{Z}}_{p}$-modules $L_{1}$ and $L_{2}$ are said to be equivalent if $L_{1}=p^{m}L_{2}$ for some $m\in{\mathbb{Z}}$.

The building $\mathcal{B}_{n}$ is an $(n-1)$-dimensional CW complex whose vertices are equivalence classes of free ${\mathbb{Z}}_{p}$-modules $[L]$ of rank $n$. Vertices $[L_{0}]$, $\ldots$, $[L_{k}]$ in $\mathcal{B}_{n}$ form a $k$-simplex if there are representatives $L_{0}$, $\ldots$, $L_{k}$ in ${\mathbb{Q}}_{p}^{n}$ such that

[TABLE]

In particular, at each vertex $[L]$ in $\mathcal{B}_{n}$, the number of adjacent vertices with $[L]$ equals the number of proper nontrivial subspaces of the space $({\mathbb{Z}}_{p}/p{\mathbb{Z}}_{p})^{n}$. Note that at most $(n-1)$ vertices form a simplex in $\mathcal{B}_{n}$: there is an ${\mathbb{Z}}_{p}$-generating set $\{\mathbf{v}_{1},\mathbf{v}_{2},\ldots,\mathbf{v}_{n}\}$ of $L$ such that

[TABLE]

Vertices $[L]$ in $\mathcal{B}_{n}$ is also denoted by $n$ by $n$ matrices whose rows are ${\mathbb{Z}}_{p}$ generators of $L$. By right multiplication of $\mathrm{GL}_{n}({\mathbb{Z}}_{p})$, they are represented by upper triangle matrices whose diagonal entries are in $p^{{\mathbb{Z}}_{\geq 0}}$.

In $\mathcal{B}_{n}$, an apartment $\mathcal{A}$ is defined as follows. The vertex set $\mathcal{V}(\mathcal{A})$ is

[TABLE]

for some $g\in\mathrm{GL}_{n}({\mathbb{Q}}_{p})$. A $k$-simplicial complex is in $\mathcal{A}$ if its vertices are all contained in $\mathcal{A}$. Denote by $\mathcal{A}_{0}$ the apartment whose vertex set is

[TABLE]

Note that $\mathcal{A}$ is isometric to ${\mathbb{R}}^{n-1}$ and $\mathcal{V}(\mathcal{A}_{0})$ is a lattice in ${\mathbb{R}}^{n-1}$ (See Figure 2), which is the reason that $\mathcal{B}_{n}$ is called a Euclidean building. There is the natural covering map $\mathrm{p}:\mathcal{B}_{n}\rightarrow\mathcal{A}_{0}$ given by

[TABLE]

where the matrix in the above map is a reduced type, i.e., $m_{i}\geq 0$ and $\gcd\{p^{m_{j}},v_{ij}\}=1$, $1\leq i<j\leq n$.

We give a distance $d$ on $\mathcal{V}(\mathcal{B}_{n})$ by $d([L_{1}],[L_{2}])=\ell$, where $\ell$ is the minimal number of edges connecting $[L_{1}]$ and $[L_{2}]$.

Let $a:{\mathbb{Z}}\rightarrow\mathcal{V}(\mathcal{A}_{0})$ be any geodesic in the vertex set of $\mathcal{A}_{0}$. Then the inverse image

[TABLE]

is a tree in $\mathcal{B}_{n}$. If the above set $\{a(m)\}$ is generated by one element, then $\mathrm{p}^{-1}(\{a(m)\})$ is a regular tree. More basic properties about buildings can be found in [1], [20] for Euclidean buildings, and see also [25] for Bruhat-Tits trees.

Lemma 2.2**.**

Let $H_{p}$ be a ${\mathbb{Q}}_{p}$-rank one connected and simply connected semisimple algebraic subgroup of $\mathrm{GL}_{n}({\mathbb{Q}}_{p})$. Denote the Cartan decomposition of $H_{p}$ by $H_{p}=K_{p}\cdot\hat{A}\cdot K_{p}$, where $K_{p}=H_{p}\cap\mathrm{GL}_{n}({\mathbb{Z}}_{p})$. Then the symmetric space $K_{p}\setminus H_{p}$ is a tree.

Proof.

Note that there is the natural embedding from $K_{p}\setminus H_{p}$ to $\mathrm{GL}_{n}({\mathbb{Z}}_{p})\setminus\mathrm{GL}_{n}({\mathbb{Q}}_{p})$ given by $K_{p}g\mapsto\mathrm{GL}_{n}({\mathbb{Z}}_{p})g$, $g\in H_{p}$. For $g\in H_{p}$, we denote $\mathrm{GL}_{n}({\mathbb{Z}}_{p})g$ by $K_{p}g$ and $\{\mathrm{GL}_{n}({\mathbb{Z}}_{p})g:g\in H_{p}\}$ by $K_{p}\setminus H_{p}$.

Since $H_{p}$ is of ${\mathbb{Q}}_{p}$-rank one, up to an appropriate conjugation, $K_{p}\hat{A}$ is generated by some element $a=\mathrm{diag}(p^{m_{1}},\ldots,p^{m_{n}})$ with $m_{1}\leq\cdots\leq m_{n}$ and $m_{1}\neq 0$. Hence we may assume that $K_{p}\hat{A}=\langle a\rangle$. Choose a geodesic $\{a(m):m\in{\mathbb{Z}}\}$ in $\mathcal{V}(\mathcal{A}_{0})$ containing $K_{p}\hat{A}$. Then $K_{p}\setminus H_{p}$ is a subset of a tree $\mathrm{p}^{-1}(\{a(m):m\in{\mathbb{Z}}\})$. ∎

Let us show that for a $4$-dimensional nondegenerate isotropic quadratic form $q_{p}$ on ${\mathbb{Q}}^{4}_{p}$, if it is not a split form, then the special orthogonal group ${\mathrm{SO}}(q_{p})$ is of ${\mathbb{Q}}_{p}$-rank one, similar to the case of real.

Lemma 2.3**.**

Let $q_{p}$ be an isotropic non-split quadratic form of rank $4$. If we denote ${\mathrm{SO}}(q_{p})=K\cdot\hat{A}\cdot K$, where $K={\mathrm{SO}}(q_{p})\cap\mathrm{SL}_{4}({\mathbb{Z}}_{p})$, then $\hat{A}$ is isomorphic to the diagonal subgroup $\left\{\mathrm{diag}(p^{-t},1,1,p^{t}):t\in{\mathbb{Z}}_{\geq 0}\right\}$.

Proof.

Without loss of generality, let $q_{p}(x,y,z,w)=2xw+\alpha_{1}y^{2}+\alpha_{2}z^{2}$, where $\alpha_{1},\alpha_{2}\in\{1,u_{0},p,u_{0}p\}$ for some fixed $1\leq u_{0}\leq p-1$, square-free over ${\mathbb{Q}}_{p}$. Since $q_{p}$ is not a split form, $\alpha_{2}/\alpha_{1}\neq-1$. Hence, up to a change of variables, we may further assume that

1. (1)

$\alpha_{2}/\alpha_{1}\in\{p,pu_{0},pu^{-1}_{0}\}$ if $-1$ is square-free; 2. (2)

$\alpha_{2}/\alpha_{1}\in\{p,pu_{0},pu^{-1}_{0},u_{0}\}$ otherwise.

Since any semisimple element of a connected algebraic group is contained in a maximal abelian subgroup and two maximal abelian subgroups are conjugate to each other ([26, Theorem 6.3.5]), we may assume that

[TABLE]

If the ${\mathbb{Q}}_{p}$-rank of $\hat{A}$ is larger than one, there is an element $a=(a_{ij})_{1\leq i,j\leq 4}\in\hat{A}-\mathrm{SL}_{4}({\mathbb{Z}}_{p})$ such that $a\neq\mathrm{diag}(p^{-t},1,1,p^{t})$ for any $t$.

From the commutativity, $\mathrm{diag}(p^{-t},1,1,p^{t})\hskip 1.8063pta=a\hskip 1.8063pt\mathrm{diag}(p^{-t},1,1,p^{t})$ for any $t\in{\mathbb{Z}}$, it follows that

[TABLE]

Since $a\in{\mathrm{SO}}(q_{p})$, it satisfies that for any $(x,y,z,w)\in{\mathbb{Q}}_{p}^{4}$,

[TABLE]

Since $a_{11}$, $a_{44}$ have no restriction except $a_{11}a_{44}=1$, by multiplying appropriate $(p^{-t},1,1,p^{t})$ and $(u,1,1,u^{-1})$ to $a$, where $t\in{\mathbb{Z}}$ and $u\in{\mathbb{Z}}_{p}-p{\mathbb{Z}}_{p}$, we may assume that $a_{11}=a_{44}=1$. Hence we need to find $(a_{22},a_{23},a_{32},a_{33})\notin{\mathbb{Z}}_{p}^{4}$ such that (a) $a^{2}_{22}+(\alpha_{2}/\alpha_{1})a^{2}_{32}=1$, (b) $a_{22}a_{23}+(\alpha_{2}/\alpha_{1})a_{32}a_{33}=0$, (c) $(\alpha_{1}/\alpha_{2})a^{2}_{23}+a^{2}_{33}=1$.

If $\alpha_{2}/\alpha_{1}=p$, $pu_{0}$, $pu^{-1}_{0}$, let us denote $\alpha_{2}/\alpha_{1}=pu$, where $u\in\{1,u_{0},u^{-1}_{0}\}$.

Suppose that $(a_{22},a_{32})\notin{\mathbb{Z}}_{p}^{2}$ and let $a^{2}_{22}+pua^{2}_{32}=\sum_{i\geq m}c_{i}p^{i}$, where $m\in{\mathbb{Z}}$ is the first term such that $c_{m}\neq 0$. Denote by $\nu:{\mathbb{Q}}_{p}\rightarrow{\mathbb{Z}}$ the $p$-adic valuation. Then

1. (1)

If $|a_{22}|_{p}\geq|a_{32}|_{p}$, then $m$ is even and $\nu(a_{22})<0$. Then $c_{m}=\left(a_{22}p^{m/2}\right)^{2}\mod p^{m}$ and $m=2\nu(a_{22})<0$ which contradicts to (a). 2. (2)

If $|a_{22}|_{p}<|a_{32}|_{p}$, then $m$ is odd and $\nu(a_{32})<0$. Then $c_{m}=u\left(a_{3}2p^{(m-1)/2}\right)^{2}\mod p^{m}$ and $m=2\nu(a_{32})+1<0$ which is also impossible according to (a).

Hence $(a_{22},a_{32})\in{\mathbb{Z}}_{p}^{2}$. Similarly, $(a_{23},a_{33})\in{\mathbb{Z}}_{p}^{2}$.

Now, assume $\alpha_{2}/\alpha_{1}=u_{0}$. In this case, $-1$ is a square. If there exists $(a_{22},a_{23},a_{32},a_{33})$ satisfying (a) through (c), then $u_{0}=-(a_{22}/a_{32})^{2}$ which contradict to the fact that $u_{0}$ is square-free. ∎

If $q_{p}$ is an isotropic quadratic form of rank 3, ${\mathrm{SO}}(q_{p})$ is isomorphic to $\mathrm{SL}_{2}({\mathbb{Q}}_{p})$. Then the symmetric space $K_{p}\setminus{\mathrm{SO}}(q_{p})$ is isomorphic to a subtree of the $(p+1)$ regular tree $\mathcal{T}_{p}$. By Lemma 2.2 and Lemma 2.3, if $q_{p}$ is an isotropic non-split quadratic form of rank 4, then $K_{p}\setminus{\mathrm{SO}}(q_{p})$ is a subtree of the building $\mathcal{B}_{n}$. Lastly, if $q_{p}$ is a split form of rank 4, then ${\mathrm{SO}}(q_{p})$ is locally isomorphic to $\mathrm{SL}_{2}({\mathbb{Q}}_{p})\times\mathrm{SL}_{2}({\mathbb{Q}}_{p})$ as in (2.2). Then $K_{p}\setminus{\mathrm{SO}}(q_{p})$ is properly embedded to the product $\mathcal{T}_{p}\times\mathcal{T}_{p}$ of two $(p+1)$-regular trees.

Corollary 2.4**.**

Let $q^{0}_{p}$ be a standard $p$-adic isotropic quadratic form of rank $n=3$ or $4$. Let $K_{p}={\mathrm{SO}}(q_{p})\cap\mathrm{SL}_{n}({\mathbb{Z}}_{p})$. For any $r\in(0,2t)$,

[TABLE]

Proof.

Suppose that $q_{p}$ is an isotropic quadratic form of rank 3 or of rank 4 and non-split. By Lemma 2.2 and Lemma 2.3, $K_{p}\setminus{\mathrm{SO}}(q_{p})$ is isomorphic to a tree. Since $d_{p}$ is right $K_{p}$-invariant, the left hand side of (2.4) is

[TABLE]

For any $k\in K_{p}$, consider the geodesic segment $[K_{p},K_{p}a_{t}k]$ in $\mathcal{T}_{p}$. If $d_{p}(a_{t}k,a_{t})\leq r$, since $K_{p}\setminus{\mathrm{SO}}(q_{p})$ is a tree, $[K_{p},K_{p}a_{t}k]$ and $[K_{p},K_{p}a_{t}]$ have the common segment of length at least $2t-\lceil r/2\rceil$. Therefore

[TABLE]

where $\lceil r\rceil$ is the largest integer less than or equal to $r$.

Since ${\mathrm{SO}}(2xz-y^{2})({\mathbb{Q}}_{p})$ is embedded in ${\mathrm{SO}}(q_{p})$, for any $t\in{\mathbb{N}}$, there are at least $(p+1)p(p^{2})^{(t-1)-\lceil r/4\rceil}$ vertices in the sphere of radius $2t-\lceil r/2\rceil$ in $K_{p}\setminus{\mathrm{SO}}(q_{p})$.

In the split case, the action of $a_{2t}=\mathrm{diag}(p^{2t},1,1,p^{-2t})$ on $K_{p}\setminus{\mathrm{SO}}(q_{p})$ is converted to the action of $(b_{t},b_{t})$ on $\mathrm{SL}_{2}({\mathbb{Z}}_{p})\setminus\mathrm{SL}_{2}({\mathbb{Q}}_{p})\times\mathrm{SL}_{2}({\mathbb{Z}}_{p})\setminus\mathrm{SL}_{2}({\mathbb{Q}}_{p})\cong\mathcal{T}_{p}\times\mathcal{T}_{p}$, where $b_{t}=\mathrm{diag}(p^{t},p^{-t})$. Then similar to the case of ${\mathrm{SO}}(2,2)({\mathbb{R}})$, we obtain the inequality (2.4). ∎

3. Equidistribution property for generic points

For $x\in\mathsf{G}/\Gamma$, let $\Delta_{x}$ be an $S$-lattice in ${\mathbb{Q}}_{S}^{n}$ associated to $x$, i.e., if $x=\mathsf{g}\Gamma$, $\Delta_{x}=\mathsf{g}{\mathbb{Z}}^{n}_{S}$. We call $L\subseteq{\mathbb{Q}}^{n}_{S}$ a $\Delta_{x}$-rational subspace if $L$ is generated by elements in $\Delta_{x}$. If $L$ is a $\Delta_{x}$-rational subspace, $\Delta_{x}\cap L$ is an $S$-lattice in $L$. Let $d(L)=d_{\Delta_{x}}(L)$ be a covolume of $\Delta_{x}\cap L$ in $L$. If $\{\mathbf{v}_{1},\ldots,\mathbf{v}_{j}\}$ is an ${\mathbb{Z}}_{S}$-basis of a $j$-dimensional $\Delta_{x}$-rational subspace $L$, then $d(L)$ is given by

[TABLE]

Here for each $p\in S$, the $p$-adic norm $\|\cdot\|_{p}$ of ${\mathbb{Q}}_{p}$ is canonically extended to $\bigwedge^{j}({\mathbb{Q}}_{p}^{n})$, which we also denote by $\|\cdot\|_{p}$.

Define a function $\alpha_{S}:\mathsf{G}/\Gamma\rightarrow{\mathbb{R}}_{>0}$ by

[TABLE]

The following lemma is originated from [23], stated for lattices in ${\mathbb{R}}^{n}$. The statement and the proof of the $S$-arithmetic version can be found in [12].

Lemma 3.1**.**

For a bounded function $f:{\mathbb{Q}}_{S}^{n}\rightarrow{\mathbb{R}}$ of compact support, the Siegel transform $\tilde{f}$ of $f$ is defined by

[TABLE]

Then there is a constant $C_{f}>0$ such that $\tilde{f}(x)<C_{f}\alpha_{S}(\Delta_{x})$ for any $x\in\mathsf{G}/\Gamma$.

By Lemma 3.1 and Proposition 3.2, one can easily deduce that a Siegel transform $\tilde{f}$ is in $\mathcal{L}^{s}$.

Proposition 3.2**.**

[12, Lemma 3.9]** The function $\alpha_{S}$ on $\mathrm{SL}_{n}({\mathbb{Q}}_{S})/\mathrm{SL}_{n}({\mathbb{Z}}_{S})$ belongs to every $\mathcal{L}^{s}$, $1\leq s<n$.

In this section, we prove the equidistribution theorem for functions bounded by $\alpha_{S}$. We first recall the Howe-Moore theorem for $\mathrm{SL}_{n}({\mathbb{Q}}_{S})$. The statement for more general settings can be found in [3] or [16].

Theorem 3.3**.**

Consider $\mathsf{G}=\mathrm{SL}_{n}({\mathbb{Q}}_{S})$ and $\Gamma=\mathrm{SL}_{n}({\mathbb{Z}}_{S})$, $n\geq 3$. Take a maximal subgroup $\hat{\mathsf{K}}$ of $\mathsf{G}$ as $\hat{\mathsf{K}}=\underset{p\in S}{\prod}{\hat{K}_{p}}={\mathrm{SO}}(n)\times\underset{p\in S_{f}}{\prod}\mathrm{SL}_{n}({\mathbb{Z}}_{p})$. Then, there exist constants $\lambda_{p}>0$, $p\in S$ satisfying the following: if $f_{1},f_{2}\in\mathcal{L}^{2}_{0}(\mathsf{G}/\Gamma)$ satisfy either

1. (1)

both $f_{1}$ and $f_{2}$ are smooth of compact support or 2. (2)

both $f_{1}$ and $f_{2}$ are left $\hat{\mathsf{K}}$-invariant,

then there exists a constant $C_{f_{1},f_{2}}>0$ so that

[TABLE]

where $\mathsf{g}=(g_{p})_{p\in S}\in\mathsf{G}$.

Recall that $\mathsf{a}_{t}$ is a diagonal element of $\mathsf{A}$ defined in (2.1).

Lemma 3.4**.**

Let $\mathsf{q}_{S}$ be a nondegenerate isotropic quadratic form of rank $n=3$ or $4$. Let $\mathsf{K}$ be a maximal compact subgroup of ${\mathrm{SO}}(\mathsf{q}_{S})$ as in Section 3. Then there exist $\mathsf{t}_{0}\succ 0$ and a constant $C_{S}>0$ such that for $\mathsf{t}\succ\mathsf{t}_{0}$ and for any $r>0$,

[TABLE]

Proof.

Let $s=|S_{f}|$ and consider $(k_{1},\ldots,k_{s})\in{\mathbb{Z}}_{\geq 0}$ such that $\sum_{i}k_{i}\leq r$. From Lemma 2.1 and Lemma 2.2,

[TABLE]

The number of such nonnegative vectors $(k_{1},\ldots,k_{s})$ is bounded above by $r^{s}$. Since $e<p$ for any odd prime $p$, the inequality (3.1) holds for $C_{S}:=C_{\infty}\prod_{i}p_{i}/(p_{i}+1)$. ∎

We say that a sequence $(\mathsf{t}_{j})$ is divergent if $(\mathsf{t}_{j})$ escapes any bounded subset of ${\mathbb{R}}_{>0}\times\prod_{p\in S_{f}}p^{{\mathbb{Z}}}$ as $j\rightarrow\infty$. Recall that in Theorem 1.1, we are interested in the asymptotic limit of the given quantity when every component $t_{p}$ of $\mathsf{t}$ goes to infinity. However, for the proof of the main theorem, we need to consider more general divergence in the next proposition and corollary.

Proposition 3.5**.**

Let $\mathsf{q}_{S}$ and $\mathsf{K}$ be as in Lemma 3.4 and let $\phi$ be a continuous function on $\mathsf{G}/\Gamma=\mathrm{SL}_{n}({\mathbb{Q}}_{S})/\mathrm{SL}_{n}({\mathbb{Z}}_{S})$. Assume that there are constants $s\in(0,n/2)$ and $C_{\phi,s}>0$ for which

[TABLE]

Then for any nonnegative bounded function $\nu$ on $\mathsf{K}$ and a divergent sequence $(\mathsf{t}_{j})$,

[TABLE]

for almost all $x_{0}\in\mathrm{SL}_{n}({\mathbb{Q}}_{S})/\mathrm{SL}_{n}({\mathbb{Z}}_{S})$.

Proof.

Let $\varepsilon>0$ be arbitrary. For any function $f$ on $\mathsf{G}/\Gamma$, define

[TABLE]

For $\mathsf{t}\succ 0$, let us denote by $\mathsf{s}\hskip 0.72229pt(\mathsf{t})=\sum_{p\in S}t_{p}$. We claim that one can find $\mathsf{t}_{0}\succ 0$ and constants $C,\;\lambda>0$ such that for any $\mathsf{t}\succ\mathsf{t}_{0}$,

[TABLE]

Then (3.3) implies that

[TABLE]

Let $A(r)=\left\{x\in\mathsf{G}/\Gamma:\alpha_{S}(x)>r\right\}$. Choose a smooth non-negative function $g_{r}:\mathsf{G}/\Gamma\rightarrow[0,1]$ such that

1. (1)

$g_{r}$ is $\mathsf{K}$-invariant, i.e., $g_{r}(\mathsf{k}x)=g_{r}(x)$ for all $\mathsf{k}\in\mathsf{K}$; 2. (2)

$g_{r}(x)=1$ if $x$ is in $A(r+1)$ and $g_{r}(x)=0$ if $x$ is outside of $A(r)$.

Take $\phi_{1}=\phi g_{r}$ and $\phi_{2}=\phi(1-g_{r})$. Then $\phi=\phi_{1}+\phi_{2}$ and $\phi_{2}$ is compactly supported. Choose $\theta>0$ such that $1\leq s+\theta<n/2$. Then for any $x\in\mathsf{G}/\Gamma$,

[TABLE]

Put $\psi=C_{\phi}r^{-\theta}\alpha^{s+\theta}_{S}(x)$. By the above inequalities and Proposition 3.2,

[TABLE]

Let $f\in\mathcal{L}^{2}_{0}(\mathsf{G}/\Gamma)$ be either a smooth function of compact support or a left $\mathsf{K}$-invariant function. Later, we will put $f=\phi_{2}-\int_{\mathsf{K}}\phi_{2}$ or $\psi-\int_{\mathsf{K}}\psi$. By Theorem 3.3, there are $\lambda_{f}$, $C_{f}>0$ such that for any $\mathsf{g}\in\mathsf{G}$,

[TABLE]

where $\mathsf{g}=(g_{p})_{p\in S}$. For instance, we can take $\lambda_{f}=\min\{\lambda_{p},p\in S\}$, where $\lambda_{p}$’s are in Proposition 3.3. Note that

[TABLE]

where $\hat{\nu}(\mathsf{k}):=\nu(\mathsf{k}^{-1})$, $\forall\mathsf{k}\in\mathsf{K}$. By Lemma 3.4,

[TABLE]

Hence by Cauchy-Schwartz inequality, we have that

[TABLE]

It is deduced from (3.4) that

[TABLE]

Hence by combining (3.5) and (3.6), there are $\mathsf{t}^{1}_{0}\succ 0$ and a constant $\lambda^{1}=\lambda^{1}(f,\mathsf{t}^{1}_{0})>0$ such that for $\mathsf{t}\succ\mathsf{t}^{1}_{0}$,

[TABLE]

Now let us show following inequalities:

[TABLE]

[TABLE]

for some positive constants $C_{1}$ and $C_{2}$. It is obvious that (3.7) and (3.8) imply (3.2).

First we note that $\alpha_{S}(\mathsf{g}x)/\alpha_{S}(x)\leq\max\{\prod_{p\in S}\|\wedge^{i}(\mathsf{g})\|_{p}:i=1,\ldots,n\}$. Let $\mathsf{t}_{\tau}=(\tau,0,\ldots,0)$. Then there is $M>0$ such that for all $x\in\mathsf{G}/\Gamma$, for all $\tau\in[0,1]$, $\alpha^{s+\theta}_{S}(\mathsf{a}_{\mathsf{t}_{\tau}}x)\leq M\alpha_{S}^{s+\theta}(x)$. Hence we have

[TABLE]

Choose $r>0$ sufficiently large so that $\|\phi_{1}\|_{1}\leq\|\psi\|_{1}\leq\varepsilon/(8\max(\nu*\hat{\nu})M)$. Then

[TABLE]

By taking $f=\psi-\int_{\mathsf{G}/\Gamma}\psi$ in (3.6) and using the Chebyshev’s inequality, there are $\mathsf{t}^{\psi}_{0}\succ 0$ and $\lambda_{\psi}>0$ such that for any $\mathsf{t}\succ\mathsf{t}^{\psi}_{0}$,

[TABLE]

Hence using the geometric series argument, there is a constant $C_{1}>0$ such that the inequality (3.7) holds.

[TABLE]

For (3.8), note that since $\phi_{2}$ is compactly supported, $\phi_{2}$ is uniformly continuous. Hence there is $\delta>0$ such that

[TABLE]

for all $\mathsf{t}\succ 0$, $x\in\mathsf{G}/\Gamma$ and $\tau\in[0,\delta]$. Again by (3.6) and Chebyshev’s inequality, there are $\mathsf{t}^{\phi_{2}}_{0}\succ 0$ and $\lambda_{\phi_{2}}>0$ such that for all $\mathsf{t}\succ\mathsf{t}^{\phi_{2}}_{0}$,

[TABLE]

Then (3.8) follows from the similar geometric argument used above. ∎

From now on, a function $f_{p}$ on ${\mathbb{Q}}_{p}^{n}$, $p\in S$, is always assumed to be compactly supported. If $p<\infty$, we additionally assume that $f_{p}$ is $({\mathbb{Z}}_{p}-p{\mathbb{Z}}_{p})$-invariant:

[TABLE]

Let $\nu$ be the product of non-negative continuous functions $\nu_{p}$ on the unit sphere in ${\mathbb{Q}}_{p}^{n}$, $p\in S$. For $p\in S_{f}$, we also assume that

[TABLE]

Define a function $J_{f}$ for $f=\prod_{p\in S}f_{p}$ by $J_{f}=\prod_{p\in S}J_{f_{p}}$, where

[TABLE]

where $x_{n}$ is determined by the equation $\zeta_{p}=q^{0}(p^{-r},x_{2},\ldots,x_{n-1},x_{n})$ (If $p=\infty$, replace $p^{-r}$ by $r$). By Lemma 3.6 in [7] and Lemma 4.1 in [12], for sufficiently small $\varepsilon>0$, there are $c(K_{p})>0$ and $t_{p}^{0}>0$ for each $p\in S$, such that if $t_{p}>t^{0}_{p}$,

[TABLE]

[TABLE]

Recall that $\sigma=1$ for the infinite place and $\sigma=-1$ for the finite place. Define

[TABLE]

By Lemma 3.6 in [7] and Lemma 4.1 in [12], there is a constant $c(\mathsf{K})>0$ such that for sufficiently small $\varepsilon>0$ and sufficiently large $\mathsf{t}\succ 0$,

[TABLE]

Corollary 3.6**.**

Let $(\mathsf{T}_{j}=(T_{j,p})_{p\in S})_{j\in{\mathbb{N}}}$ be a divergent sequence. Define

[TABLE]

Let $\Omega,\mathsf{I}_{S}=(I_{p})_{p\in S}$ be as in Theorem 1.1. Then for almost all nondegenerate isotropic quadratic form $\mathsf{q}_{S}=(q_{p})_{p\in S}$, there is a constant $C=C(S^{\prime})>0$ such that

[TABLE]

where $n$ is the rank of $\mathsf{q}_{S}$.

Proof.

Since we want to show the upper bound, for simplicity, we may assume that $\Omega$ is the product of unit balls in ${\mathbb{Q}}_{p}^{n}$, $p\in S$.

Let $\varepsilon>0$ be given. For each $p\in S$, let $\mathcal{C}_{p}$ be a compact set of the space of nondegenerate isotropic quadratic forms over ${\mathbb{Q}}_{p}$ of a given signature. Let $\mathcal{C}=\prod_{p\in S}\mathcal{C}_{p}$.

Let $\mathsf{g}_{\mathsf{q}_{S}}$ be an element of $\mathrm{SL}_{n}({\mathbb{Q}}_{S})$ such that $\mathsf{q}_{S}(\mathbf{v})=\mathsf{q}^{0}_{S}(\mathsf{g}_{\mathsf{q}_{S}}\mathbf{v})$ for all $\mathbf{v}\in{\mathbb{Q}}_{S}^{n}$. Then for each $p\in S$, there is $\beta_{p}=\beta_{p}(\mathcal{C}_{p})>0$ such that if $\mathsf{q}_{S}\in\mathcal{C}$, $\beta_{p}^{-1}\leq\|\mathsf{g}_{\mathsf{q}_{S}}\mathbf{v}\|_{p}/\|\mathbf{v}\|^{\sigma}_{p}\leq\beta_{p}$ for all $\mathbf{v}\in{\mathbb{Q}}_{S}^{n}$.

Choose bounded continuous functions $f_{p}$, $p\in S$, of compact support such that

[TABLE]

If $\mathbf{v}$ satisfies that $T_{p}/2\leq\|\mathbf{v}\|_{p}\leq T_{p}$ and $q_{p}(\mathbf{v})\in I_{p}$, then

[TABLE]

for $\mathsf{g}=\mathsf{g}_{\mathsf{q}_{S}}$ with $\mathsf{q}_{S}\in\mathcal{C}$. By (3.11) and (3.12), there is $\mathsf{T}^{0}=(T^{0}_{p})_{p\in S}\succ 0$ such that for each $p\in S$ and for all $T_{p}>T^{0}_{p}$,

[TABLE]

For $p\in S^{\prime}$, we may further assume that $T_{j,p}\leq T^{0}_{p}$ for all $j$.

Finally for each $p\in S$, choose a nonnegative bounded function $g_{p}$ on ${\mathbb{Q}}_{p}^{n}$ of compact support such that if $\|\mathbf{v}\|_{p}\leq T^{0}_{p}$ and $t_{p}\leq\log_{p}(T_{p}^{0})$,

[TABLE]

Take $h_{S^{\prime}}=\prod_{p\in S^{\prime}}g_{p}\times\prod_{p\in S-S^{\prime}}f_{p}$ and let $\widetilde{h_{S^{\prime}}}$ be the Siegel transform of $h_{S^{\prime}}$ defined in Lemma 3.1. Then for $\mathsf{T}=(T_{p})$ such that $T_{p}>T_{p}^{0}$, $\forall p\in S-S^{\prime}$, we obtain that

[TABLE]

By Proposition 3.5, for almost all quadratic form $\mathsf{q}_{S}$, as $\mathsf{t}$ diverges,

[TABLE]

Hence there is a constant $C^{\prime}>0$ such that for all $\mathsf{T}_{j}$,

[TABLE]

Therefore we have

[TABLE]

∎

4. The proof of the main theorem

The proof of Theorem 1.1 is similar to that of Theorem 1.5 in [12] except we use Proposition 3.5 instead of Theorem 7.1 in [12]. Let $\mathcal{C}$ be any compact subset of the space of isotropic quadratic forms with equal signature and let $\varepsilon>0$ be given. Let $\phi$ and $\nu$ be compactly supported functions defined as before.

Theorem 7.1 in [12] says that there is $\mathsf{t}_{0}=\mathsf{t}_{0}(\mathcal{C})\succ 0$ such that except on a finite union of orbits of ${\mathrm{SO}}(\mathsf{q}^{0}_{S})$, $x\in\mathcal{C}$ satisfies the following: for all $\mathsf{t}\succ\mathsf{t}_{0}$,

[TABLE]

Proposition 4.1**.**

Let $\mathsf{q}_{S}$ be an isotropic quadratic form of rank $n=3$ or $4$. Let $f=\prod f_{p}$ and $\nu=\prod\nu_{p}$ be as before. Assume further that $f$ satisfies the following condition: there is a nonnegative continuous function $f^{+}$ of compact support on ${\mathbb{Q}}_{S}^{n}$ such that $\operatorname{supp}(f)\subset\operatorname{supp}(f^{+})^{\circ}$, where $A^{\circ}$ is an interior of $A$ and

[TABLE]

Then there exists $\mathsf{t}_{0}\succ 0$ such that if $\mathsf{t}\succ\mathsf{t}_{0}$,

[TABLE]

Proof.

We first claim that there is a constant $c>0$ such that

[TABLE]

Since the interior of $\operatorname{supp}(f^{+})$ contains $\operatorname{supp}(f)$, there is $\rho>0$ such that

[TABLE]

Note that since $J_{f^{+}}$ is compactly supported, the set $\Pi$ is finite. Take $\varepsilon>0$ such that $\varepsilon<\rho/2$. Then by (3.13), for sufficiently large $\mathsf{t}$, we have that

[TABLE]

By (4.1), $\rho/2\;|\Pi|\leq c(\mathsf{K})M|\mathsf{T}|^{n-2}$. This implies (4.3).

Now (4.2) follows from applying (3.13) by putting $\varepsilon/(c(\mathsf{K})\rho M)$ instead of $\varepsilon$ and taking summation over all $\mathbf{v}\in{\mathbb{Z}}_{S}^{n}$. ∎

Recall that a convex set $\Omega$ is defined using a non-negative continuous function $\rho=\prod_{p\in S}\rho_{p}$, where $\rho_{p}$ is a positive function on the unit sphere of ${\mathbb{Q}}_{p}^{n}$, $p\in S$.

Define the shell $\hat{\Omega}$ of $\Omega$ by

[TABLE]

Note that when $p<\infty$, the inequality in the above definition is in fact the equality: $\|\mathbf{v}\|_{p}=\rho_{p}(\mathbf{v}/\|\mathbf{v}\|^{\sigma}_{p})$.

The following proposition was originally stated for $\Omega$ but the proof can be easily modified for $\hat{\Omega}$.

Proposition 4.2**.**

[12, Proposition 1.2]** There is a constant $\lambda=\lambda(\mathsf{q}_{S},\Omega)>0$ such that as $\mathsf{T}\rightarrow\infty$,

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Lemma 4.3**.**

[12, Lemma 5.2]** Let $f$ and $\nu$ be as in Proposition 4.1. Take

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and set $h(\mathbf{v},\zeta)=\prod_{p\in S}h_{p}(\mathbf{v}_{p},\zeta_{p})$. Then we have

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Proof of Theorem 1.1.

Since $f$ in Proposition 4.1 is compactly supported, there is $f^{+}$ such that $\operatorname{supp}(f_{p})\subset\operatorname{supp}(f^{+}_{p})^{\circ}$. By Proposition 3.5 with $\phi=f^{+}$ and $\nu\equiv 1$,

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for almost all $x=\mathsf{g}\Gamma\in\mathsf{G}/\Gamma$. By Lemma 3.1 and Proposition 3.2, the integral of $\widetilde{f^{+}}$ over $\mathsf{G}/\Gamma$ is finite. Hence one can apply Proposition 4.1 for almost all $x\in\mathsf{G}/\Gamma$.

We also remark that the set of functions of the form $J_{f}\nu$, where $f=\prod f_{p}$, $\nu=\prod\nu_{p}$ with (3.9) and (3.10) respectively, is a generating set of

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Hence Proposition 4.1 holds for functions in $\mathcal{L}$ as well (see details in [12]). Define

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The characteristic function $\mathbbm{1}_{\mathsf{T}\hat{\Omega}\times\mathsf{I}_{S}}(\mathsf{g}\mathbf{v},\zeta)$ is contained in $\mathcal{L}$. Let $\varepsilon>0$ be given. Take continuous functions $h^{a},h^{b}\in\mathcal{L}$, depending on $\mathsf{T}$, $\varepsilon$ and $\mathsf{q}_{S}$ such that

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From (4.2), (3.2) and (4.4), for $h=h^{a}$, $h^{b}$ and any $\varepsilon>0$, there is $\mathsf{T}_{0}\succ 0$ such that if $\mathsf{T}\succ\mathsf{T}_{0}$,

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for almost every $\mathsf{g}\Gamma\in\mathsf{G}/\Gamma$. By the definition of $L(h)$ and rescaling $\mathsf{T}_{0}\succ 0$ if necessary, we also obtain that

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Combining (4.6) and (4.7) with (4.5), if we regard $h$ as the characteristic function of $\mathsf{T}\hat{\Omega}\times\mathsf{I}_{S}$,

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Since

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the lemma follows from Proposition 4.2 and the classical argument of geometric series, if we obtain the following: as $\mathsf{T}\rightarrow\infty$, the summation over all $\mathsf{S}=(S_{\infty},S_{1},\ldots,S_{s})=(2^{-n_{0}}T_{\infty},p^{-n_{1}}_{1}T_{1},\ldots,p^{-n_{s}}_{s}T_{s})$ with $\mathsf{S}\nsucc\mathsf{T}_{0}=(T^{0}_{\infty},T^{0}_{1},\ldots,T^{0}_{s})$ is

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For this, by rescaling $\mathsf{T}^{0}$ if necessary, let us assume that Corollary 3.6 holds for any $S^{\prime}\subsetneq S$. Denote the set $\{\mathsf{S}=(2^{-n_{0}}T_{\infty},p^{-n_{1}}_{1}T_{1},\ldots,p^{-n_{s}}_{s}T_{s})\prec\mathsf{T}:\mathsf{S}\nsucc\mathsf{T}_{0}\}$ by $\cup\{\Psi_{S^{\prime}}:S^{\prime}\subseteq S\}$, where

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Then for $S^{\prime}\subsetneq S$, by Corollary 3.6, there is a constant $C(S^{\prime})>0$ such that

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where $\mathsf{T}_{S^{\prime}}=(T^{\prime}_{p})_{p\in S}$. Here $T^{\prime}_{p}=T^{0}_{p}$ for $p\in S^{\prime}$ and $T^{\prime}_{p}=T_{p}$ for $p\in S-S^{\prime}$. Hence the left hand side of (4.8) is $o(\prod_{p\in S-S^{\prime}}T_{p}^{n-2})$. If $S^{\prime}=S$, since

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it is obviously $o(|\mathsf{T}|^{n-2})$. ∎