Arithmetical Algebraic Geometry

University Of Arizona, Tucson AZ

Investigators

Abstract

Dr. Ulmer proposes three projects in arithmetical algebraic geometry, related to Galois representations, modular forms, and elliptic curves, both over number fields and over function fields. The first project proposed is to study the reduction modulo a prime p of certain representations of the Galois group of the p-adic numbers, using the recent work of Colmez and Fontaine on p-adic Hodge theory. In the second project, Dr. Ulmer has constructed a subgroup of the local points at suitable places on an elliptic curve over a function field; this subgroup contains the global points. He proposes to use these local points to study the conjecture of Birch and Swinnerton-Dyer for elliptic curves over function fields. The third project Dr. Ulmer proposes is to study a new class of problems in the cohomology of varieties which are inspired by classical non-vanishing results for L-series. The new questions come by reinterpreting vanishing results using Grothendieck's analysis of L-functions and lead to purely geometric questions. In some instances these questions can be treated using monodromy results of Katz and Sarnak. This proposal falls into the general area of arithmetical algebraic geometry - a subject that blends two of the oldest areas of mathematics: number theory and geometry. This combination has proved extraordinarily fruitful, having recently solved problems that withstood the efforts of generations. Among its many consequences are new error correcting codes which are used in computer storage devices like compact disks and hard drives and secure information transmission schemes which are used for financial transactions on the internet.