Problems in The Theory of Automorphic Forms and L-functions

Purdue University, West Lafayette IN

Investigators

Abstract

Taking the lead from the recent progress in Langlands program, the investigator proposes a number of projects, both towards making new progress on functoriality as well as benefiting from what is available. The first includes a study of Langlands "Beyond Endoscopy" by means of both regular and relative trace formulas as well as the possibility of using other Poincar\'e series besides Eisenstein series on infinite dimensional groups with the hope of capturing the new adjoint actions that happen in the dual setting, since it now appears that Eisenstein series on these groups do not lead to any new L-functions. The second set of projects includes establishing the strong transfer from general spin groups as well as the transfer from quasisplit special orthogonal groups to GL(n); special value results for L-functions by means of Harder-Mahnkopf periods through functoriality as well as an attempt in using the Langlands-Shahidi method to obtain such results via certain ideas of Harder; a general theory of Bessel functions dictated by the investigator's work on local coefficients with an eye on proving stability for root numbers of symmetric and exterior square L-functions of GL(n), among others, as well as equality of root numbers obtained from different methods. Finally the investigator will study the singular residues of certain local intertwining operators hoping to interpret them as certain weighted orbital integrals, as well as other problems in representation theory of local groups and Lfunctions. Most of these projects are joint with other mathematicians. Langlands Program is a vast collection of problems and conjectures which connects objects of arithmetic or geometric nature to those of analytic character. Such reciprocities are usually called "Functoriality". One example of this appeared in a fundamental way in the celebrated proof of Fermat's Last Theorem by Wiles. Throughout his career the investigator has developed a theory usually called the Langlands-Shahidi method, which through collaboration with a number of mathematicians, has recently led to a number of new and surprising cases of functoriality with consequences such as new bounds on eigenvalues of Laplacian on certain hyperbolic Riemann surfaces. The present proposal suggests a number of problems to try to extend functoriality to a larger class of cases as well as using them to establish new results in number theory and group representations. Among them is understanding the transcendental nature of values of L-functions, generalizations of Riemann zeta functions, at certain integers and half-integers, in line with integral values of the latter, as well as analyzing other analytic objects of arithmetic significance. The proposal involves training of graduate students and postdocs and collaboration with younger investigators.