Arithmetic expressions of Selberg's zeta functions for congruence subgroups

Higher Selberg zeta functions for congruence subgroups

As a generalization of the results [KW3],we study the functional equation of the higher Selberg zeta function for congruence subgroups. To obtain the gamma factor of this function, we introduce a higher Dirichlet $L$-function. Then we determine the gamma factor explicitly in terms of the Barnes triple gamma function and the higher Dirichlet $L$-function.

Selberg's zeta functions for congruence subgroups of modular groups in SL(2,R) and SL(2,C)

It is known that the Selberg zeta function for the modular group has an expression in terms of the class numbers and the fundamental units of the indefinite binary quadratic forms. In the present paper, we generalize such a expression to any congruence subgroup of the modular groups in SL(2,R) and SL(2,C).

秩数为 Rankin-Selberg 赤字积分和分裂非温控 Gan-Gross-Prasad 猜想

Residues of Rankin-Selberg Zeta integrals and the split non-tempered Gan-Gross-Prasad conjectures

本文通过 Zeta 积分残差正则化方法，解决了非温控 Gan-Gross-Prasad 猜想及其细化版本在局部域上的问题。

GL₂×GL₂秩纳积分的积分性：对于分裂表示

Integrality of $\mathrm{GL}_2\times\mathrm{GL}_2$ Rankin-Selberg integrals for ramified representations

本文研究GL₂×GL₂秩纳积分在分裂表示下的积分性，引入了(π₁×π₂)积分数据的概念，并改进了雅克布-兰格兰德的GCD结果，与新形式的傅里叶系数的积分性相容。

Test vectors for Rankin-Selberg $L$-functions

We study the local zeta integrals attached to a pair of generic representations $(π,τ)$ of $GL_n\times GL_m$, $n>m$, over a $p$-adic field. Through a process of unipotent averaging we produce a pair of corresponding Whittaker functions whose zeta integral is non-zero, and we express this integral in terms of the Langlands parameters of $π$ and $τ$. In many cases, these Whittaker functions also serve as a test vector for the associated Rankin-Selberg (local) $L$-function.

Analysis on Symmetric and Locally Symmetric Spaces (Multiplicities, Cohomology and Zeta functions)

The goal of the course was a review of results mainly due to M. Olbrich and the first author. We consider a discrete cocompact subgroup $Γ$ of a semisimple Lie group $G$. We relate the group cohomology of $Γ$ with coefficients in the maximal globalization of a representation of $G$ with the multiplicities of unitary representations of $G$ in $L^2(G/Γ)$. Explicit calculations are given in the case $G=SL(2,R)$. In the rank-one case we state a version of the Selberg trace formula, discuss the singularities of the Selberg zeta-functions, and state their relation with the cohomology of $Γ$ in princ…