L-Functions and Geometric Methods in Langlands Duality

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Although the ancient Greeks saw arithmetic -- what we now call number theory -- and geometry as two distinct sciences, with very different methods, contemporary mathematicians have understood for many years that they are really two aspects of a single science. This science, whose ramifications extend throughout mathematics, and whose applications are manifest in every corner of modern physics, computer science, and even the most recent developments in statistics, is as old as civilization, but it also takes on a completely new character at least once per generation. The current project is a study of Langlands duality, a synthesis of number theory and the geometry of symmetry that has been one of the most familiar names for this science for nearly two generations. For most of this period, the emblem of this synthesis has been the theory of L-functions, a computable system for encoding properties in number theory and geometry that matches the two aspects of this common science. More recently, new and sophisticated geometric techniques promise to reframe the synthesis from above, as part of a larger synthesis that ultimately extends to the most speculative ideas in mathematical physics. The PI plans to revisit Langlands duality from this new geometric standpoint, and to understand the place of L-functions in this broader picture.The project is a contribution to the arithmetic theory of automorphic forms, in the setting of the Langlands program, motivated in part by the geometric Langlands program. The specific goals of the present project are the study of the local Langlands correspondence, using trace formula methods, L-functions, and Galois theory to reduce the local Langlands correspondence to a conjecture on "incorrigible" representations; the study of derived aspects of the Langlands program, inspired by both in specific cases arising from coherent cohomology and in the development of a unified geometric framework for both Langlands reciprocity for cohomology of locally symmetric spaces and Venkatesh's conjectures on the actions of derived deformation rings on cohomology; the construction of p-adic square root L-functions in families, using the higher Hida theory introduced by Pilloni and Boxer; the completion of the proof of an automorphic version of Deligne's conjecture on special values of tensor product L-functions; and the development of a character theory for mod p representations of p-adic groups. In connection with this project, the PI is actively collaborating with colleagues in France, Austria, Germany, England, and Spain, as well as in the United States and Canada. The methods involved in the present project combine techniques from arithmetic geometry and automorphic forms, especially the trace formula, categorical representation theory, and the new methods developed by Venkatesh, Darmon, and Gaitsgory.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.