Some Results on the Schwartz Space of $Γ\backslash G$

Schwartz函数的渐近性质

Asymptotics of Schwartz functions

本文研究分裂、单连通、几乎单代数群G的极大抛物子群P的仿射闭包X_P上的Schwartz空间，证明两种定义在经典群和G2情形下等价，并建立Poisson求和公式。

The space of homogeneous probability measures on $\overline{Γ\backslash X}^S_{\rm max}$ is compact

In this paper we prove that the space of homogeneous probability measures on the maximal Satake compactification of an arithmetic locally symmetric space is compact. As an application, we explain some consequences for the distribution of weakly special subvarieties of Shimura varieties.

Q-binomials and non-continuity of the p-adic Fourier transform

Let F be a finite extension of Q_p. We show that every Schwartz function on F, with values in an algebraic closure of Q_p, is the uniform limit of a sequence of Schwartz functions, whose Fourier transforms tend uniformly to 0. The proof uses the notion of q-binomial coefficients and some classical identities between them.

On the growth of the number of totally geodesic surfaces in some hyperbolic $3$-manifolds

Let $d$ be a positive square-free integer $\equiv 3 \pmod{4}$ such that there is no invariant of the ideal class group $\mathbb{Q}\lbrack \sqrt{-d}\rbrack$ which is divisible by $4$. We prove an asymptotic formula for the number of immersed totally geodesic surfaces in $Γ_{-d}\backslash \mathbb{H}^3$ having area less than $X$.

Product formulas on a unitary group in three variables

We obtain some basic partial differential operators connected with nonholomorphic automorphic forms on $Γ\backslash U(2, 1)/K$. We give the corresponding Eisenstein series of weight $k$ and automorphic Green functions of weight $k$. Moreover, we get the product formulas connected with them. As a consequence of the product formulas, we give the functional equations of Eisenstein series associated to a convex cocompact subgroup of U(2,1).

Bottom of the Length Spectrum of Arithmetic Orbifolds

We prove that cocompact arithmetic lattices in a simple Lie group are uniformly discrete if and only if the Salem numbers are uniformly bounded away from $1$. We also prove an analogous result for semisimple Lie groups. Finally, we shed some light on the structure of the bottom of the length spectrum of an arithmetic orbifold $Γ\backslash X$ by showing the existence of a positive constant $δ(X)>0$ such that squares of lengths of closed geodesics shorter than $δ$ must be pairwise linearly dependent over $\mathbb Q$.