The principal objective of the proposal is to study arithmetic aspects of automorphic forms with the aim of understanding the special values of automorphic L-functions. The proposal has several projects which fall under the general theme of applying recent progress in the Langlands program to study Deligne's conjecture on the special values of L-functions. Some of these projects are about understanding the special values of various Rankin-Selberg L-functions, and then via Langlands functoriality, to understand special values of symmetric power L-functions attached to a modular form. An important aspect of the proposal is that the investigator will be developing a cohomological point of view of certain analytic theories of L-functions, like the Langlands-Shahidi method, with the hope of proving arithmetic theorems on their special values. Another major aim of the proposal is to study the arithmetic of automorphic forms on the some other groups: like inner forms of the general linear group, and the group of symplectic similitudes. This will be carried out with a view toward understanding cuspidal cohomology and rational structures on certain models of automorphic representations, which will allow us to study the special values of L-functions attached to these groups, thus substantially widening the class of L-functions for which special value results can be proved. The proposal also aims to develop a framework which will explain known cuspidality criteria, and predict new criteria, for various instances of Langlands functoriality.
Numbers are a fundamental aspect of reality and have been investigated for thousands of years. One of the central themes in modern number theory involves the concept of an L-function. These L-functions encode within them all knowledge about numbers and their underlying patterns. The investigator's work, on the special values of L-functions, will allow us to decode such information and hence deepen our understanding of numbers. The subject of L-functions involves, and in turn has applications to, several other branches of mathematics like representation theory, harmonic analysis, algebraic and differential geometry, not to mention important and practical applications to cryptography, coding theory, physics, and electrical engineering. The investigator is actively involved in providing research experience for undergraduates. He is also committed to the mentoring and training of graduate students and postdocs by involving them in several projects of the proposal. The investigator intends to write a monograph on special values of L-functions with the aim of bringing advanced undergraduate and beginning graduate students to the frontiers of research.
