L-functions, Fourier Transforms, and Gamma Factors

Purdue University, West Lafayette IN

Investigators

Abstract

An important goal in any mathematical theory is to understand one set of objects by means of another one. When the two sets are in a one-to-one correspondence, then one may call the correspondence a reciprocity law. One deep example of a reciprocity law is that of Artin and Langlands, which is a vast generalization of Quadratic Reciprocity Law discovered by Gauss that is important for solving equations over the integers modulo a prime, and which is the genesis of the more general Langlands Program. While a general reciprocity law within the Langlands program is still far from properly formulated, even for objects over rational numbers, one can consider the "Langlands Functoriality Principle," which is a consequence of Artin-Langlands reciprocity and is a conjecture that is currently at the core of Langlands program. This project deals with both reciprocity and functoriality and will establish new techniques and tools to study them. In more detail, the PI will compare the Fourier transform defined by Braverman-Kazhdan and the Hankel transform defined by Ngo, for the standard L-functions for classical groups. In the long run this will lead to a full theory of L-functions for cusp forms on any reductive group and any irreducible representation of its L-group. This will not only lead to fairly general cases of functoriality through converse theorems, but will also provide suitable Poisson summation formulas that are needed in the Beyond Endoscopy approach to functoriality. In terms of reciprocity, a number of cases where the equality of Artin factors with the factors defined by the Langlands-Shahidi method through the local Langlands correspondence (local reciprocity) for GL(n) will be established, following the techniques used in the cases of exterior and symmetric square factors for GL(n) proved earlier. Projects involving p-adic L-functions, local coefficients matrices for covering groups, and Rankin products of L-functions for GSpin groups will also be pursued. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.