Arithmetic of L-values

Trustees Of Boston University, Boston

Investigators

Abstract

Arithmetic of L-values PI: Glenn Stevens Proposal 0071065 Project Abstract The principal investigator proposes to continue his work on special values of L-functions and their relation to arithmetic geometry. The proposed research would improve our understanding of both automorphic forms and p-adic cohomology by providing new tools for studying the former and by providing concrete examples of the latter. Specifically, the research would extend the applicability of Explicit Reciprocity Laws to include p-adic analytic families of galois representations. This, in turn, would provide a new tool for studying the arithmetic properties of two-variable p-adic L-functions. It would also clarify how analytic families of galois representations degenerate at semistable non-crystalline points and would describe monodromy at such points in terms of a deformation in the weight direction, thus enlarging the standard picture of monodromy in potentially useful ways. This research would also complement Kato's recent work on values of L-functions and K-theory of modular curves by providing tools for deforming Kato's theory in p-adic analytic families. In related work, the PI hopes to develop a p-adic Eichler-Shimura correspondence that would relate his theory of overconvergent modular symbols to Katz's theory of overconvergent modular forms and to construct analytic families of non-ordinary half-integral weight modular forms by generalizing a p-adic theta lifting developed in earlier work of the PI. Finally, the PI intends to generalize these ideas to automorphic forms on other reductive algebraic groups. This research offers promising tools for the construction of p-adic analytic families of non-ordinary automorphic representations together with natural deformation spaces of Galois representations, and multivariable p-adic L-functions. This is connected with a number of investigations, including p-adic monodromy, Jochnowitz's conjectures on the square root of the theta operator, and the p-adic deformation theory of Galois representations. The investigations of this proposal belong to the general mathematical area of Arithmetic Geometry. This ultramodern research area combines two of the oldest branches of mathematics: number theory and geometry. New insights arising out of this combination are producing increasingly powerful tools to solve longstanding problems like Fermat's Last Theorem, which have resisted the strongest efforts of over three centuries of mathematicians. In addition, though Arithmetic Geometry is sometimes regarded as the purest of pure mathematics, it has also been developing insightful new techniques leading to dramatic progress in such applied areas as error-correcting codes and cryptography.