On Some Analytic Problems Connected to the Relative Trace Formula

这个项目属于数论领域。更具体地说，它是在自守形式和表示论领域。自守形式中最雄心勃勃的事业是朗兰兹计划，特别是函子提升问题。不严格地说，它假设自守形式从较小的群上形成较大的群。这将数论中一些最突出的开放问题结合在一起，到目前为止，朗兰兹纲领的进展是围绕着三个相互关联的主题完成的：自守形式的显式构造（例如使用θ-核或Eisenstein级数的剩余的傅立叶系数），L-函数理论与逆定理相结合，以及各种形式的迹公式。本项目的重点是迹公式方法。所有迹公式都起源于用两种不同的方式表示核函数--几何和光谱。应用函式时，两个不同群体的轨迹公式进行比较。该项目涉及的分析方面，对称空间中一个迹公式谱展开（也称为相对迹公式），由Jacquet开创。这是在诸如二次基变化的情况下建立泛函的重要一步，并表征在某些周期子群上具有非零周期积分的尖点形式。该项目有3个部分，它是与UCLA的Jonathan Rogawski合作的（前两部分），史蒂夫·拉利斯和埃尔韦·雅凯（第三部分）。该项目是数论领域的基础研究。更具体的领域被称为自守形式和表示论。在这个领域最雄心勃勃的企业是朗兰兹计划。简而言之，其目标是在“生活”在完全不同的世界上的各种对象之间建立关系，这些对象似乎没有什么共同点。这种关系非常深刻。一个引人注目的和相对较新的例子是模形式和椭圆曲线之间的对应关系，这是怀尔斯证明费马大定理的关键。粗略地说，椭圆曲线是一个“甜甜圈“，它被描述为两个3次方程的轨迹，4个变量。另一方面，在最简单的情况下，阿萨序列，通过将其所有因子之和赋给每个整数而得到模形式及其相关对象除了在数学的许多分支中具有固有的重要性外，还在物理学、密码学和通信系统中得到了应用。

The project is in the realm of Number Theory. More specifically, it is inthe area of automorphic forms and representation theory.The most ambitious enterprise in automorphic forms is theLanglands program,and, especially, the issue of functorial lifting.Loosely speaking, it postulates the formation of automorphic forms ona bigger group from those on a smaller group.This bounds together some of the most outstanding open problems in NumberTheory, such as the Artin's Conjectureand the Ramanujan Hypothesis.So far, the advance in the Langlands program was accomplished alongthree major themes which are inter-related: explicit constructions ofautomorphic forms(e.g. using theta-kernels or Fourier coefficients of residues ofEisenstein series), the theory of L-functions combined withconverse theorems, and the trace formula in its various guises.The focus in this project is on the trace formula approach.All trace formulas have their origin in expressing the kernel functionin two different ways -- geometrically and spectrally.The application to functoriality comes aboutwhen trace formulas of two different groups are compared.The project deals with analytical aspects of the spectral expansionof a trace formula for {\em symmetric spaces} (also called arelative trace formula), inaugurated by Jacquet.This is a major step in establishing functoriality in such instancesas quadratic base change, and characterizing cusp forms havinga non-zero period integral over certain period subgroups.The project has 3 parts and it is a collaboration withJonathan Rogawski from UCLA (first 2 parts), andSteve Rallis and Herve Jacquet (third part).The project is a basic research in the realm of Number Theory.The more specific area is calledautomorphic forms and representation theory.The most ambitious enterprise in this field is theLanglands program.In a nutshell, the goal is to establish relationships betweenvarious objects which "live" on completely different worlds,and seemingly have little in common.Such relations are extremely deep.A spectacular and relatively recent example is a correspondence betweenmodular forms and elliptic curves, which wasa keystone in Wiles' proof of Fermat's Last Theorem.Roughly speaking, an elliptic curve is a "doughnut"which is described as the locus of two equationsof degree 3 in 4 variables.A modular form on the other hand can be thought of,in the simplest cases, asa sequence which is obtained by assigning to each integerthe sum of all its divisors (or powers of them).Beside their inherent significance in many branches of mathematics,modular forms and their related objects have found applicationsin physics, cryptography and communication systems.