Lewis-Zagier correspondence for higher order forms

Series expansions for Maass forms on the full modular group from the Farey transfer operators

We deepen the study of the relations previously established by Mayer, Lewis and Zagier, and the authors, among the eigenfunctions of the transfer operators of the Gauss and the Farey maps, the solutions of the Lewis-Zagier three-term functional equation and the Maass forms on the modular surface $PSL(2,\Z)\backslash \HH$. In particular we introduce an "inverse" of the integral transform studied by Lewis and Zagier, and use it to obtain new series expansions for the Maass cusp forms and the non-holomorphic Eisenstein series restricted to the imaginary axis. As corollaries we obtain further info…

高阶Herglotz-Zagier函数上的Hecke型作用

Hecke-type action on higher order Herglotz-Zagier function

本文证明Lewis与Zagier构造的某类函数中的首个例子是Hecke本征形式，并据此确定Hecke算子对高阶Herglotz-Zagier函数导数的作用，导出相关多项函数方程。

The Lewis Correspondence for submodular groups

The Lewis Correspondence attaching a unique "period function" to each Maass wave form for SL(2,Z) is extended to subgroups of finite index. This is achieved by using vector-valued forms.

关于扭曲周期函数与 Dirichlet L-函数在临界线上加权均方均值的注记

On twisted period functions and Moments of a weighted mean square of Dirichlet L-functions on the critical line

本文将周期函数等工具推广至任意本原特征的 Dirichlet L-函数，给出了加权均方测度矩的闭式表达式，完整刻画了临界线上的模性质，并导出了适用于所有本原特征的无限个非平凡正性条件。

A Gross--Kohnen--Zagier Type Theorem for Higher-Codimensional Heegner Cycles

We prove that Heegner cycles of codimension m+1 inside Kuga-Sato type varieties of dimension 2m+1 are coefficients of modular forms of weight 3/2+m in the appropriate quotient group. The main technical tool for generating the necessary relations is a Borcherds style theta lift with polynomials. We also show how this lift defines a new singular Shimura-type correspondence from weakly holomorphic modular forms of weight 1/2-m to meromorphic modular forms of weight 2m+2.

The Geometric Theta Correspondence for Hilbert Modular Surfaces

In a series of papers we have been studying the geometric theta correspondence for non-compact arithmetic quotients of symmetric spaces associated to orthogonal groups. It is our overall goal to develop a general theory of geometric theta liftings in the context of the real differential geometry/topology of non-compact locally symmetric spaces of orthogonal and unitary groups which generalizes the theory of Kudla-Millson in the compact case. In this paper we study in detail the geometric theta lift for Hilbert modular surfaces. In particular, we will give a new proof and an extension (to all f…