P-adic Analysis in Algebraic Geometry over finite fields

University Of Rochester, Rochester NY

Investigators

Abstract

A large portion of this proposal considers L-functions attached to geometric Galois representations arising from families of exponential sums. Previous studies have focused on techniques from l-adic representation theory since they are especially amenable to degree computations via Euler characteristic and bounds for the weights of the zeros and poles. The PI proposes a p-adic study of such L-functions using techniques inspired by Dwork, Adolphson, Sperber, Robba, Wan, et al.. This would be a general study including many common archetypes, such as the generalized Airy family, the hyper-Kloosterman family, and the Dwork family. Other aspects of the proposal focus upon a continuation of the p-adic study of the zeta function of divisors, and a continuation of the study into the arithmetic similarities and differences of mirror manifolds. Galois representations encode many deep and significant problems within number theory. In fact, it was through their study that the celebrated Fermat's Last theorem was finally settled in the affirmative by Wiles et al. in 1995. More recently, Serre's conjecture, which implies Fermat, and the Sato-Tate conjecture, also came down to statements about Galois representations. Galois representations are frequently studied by their associated L-function. This is simply a function which encodes much, if not essentially all, of the Galois representation. Even the most elementary questions about these L-functions are still largely unknown. For instance, the Riemann hypothesis is a conjecture describing the precise location of the zeros of the L-function attached to a specific Galois representation. In this proposal, the PI plans to study Galois representations which arise from geometric situations using, and extending, tools from p-adic analysis.