On Some Analytic Problems Connected to the Relative Trace Formula

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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.