The Analytic Theory of Division Fields and Spectral L-functions

University Of California-Los Angeles, Los Angeles CA

Investigators

Abstract

Abstract for award DMS-0355564 of Duke Duke and his students will work on two broad sets of problems: to develop the analytic theory of division fields of elliptic curves and abelian varieties and to understand the number-theoretic significance of special values of spectral L-functions and their derivatives. The first part may be described as ``nonabelian" analytic number theory and is motivated by applications to the mod p reductions of elliptic curves and abelian varieties. So far progress has been possible assuming the Generalized Riemann Hypothesis (GRH), and its removal it one of the many challenges. The second part is motivated by some analogies Duke has observed between the generation of number fields by units occurring as special values of derivatives of L-functions (e.g. the ``Stark conjectures") and the special values of derivatives of L functions associated to Riemann surfaces. A general goal of the proposal is to extend the reach of analytic number theory, both within number theory at large and also within mathematics more generally. It has become evident over the last few years that the techniques of analytic number theory have a much broader applicability than was earlier appreciated. Such techniques also have potential importance in practical applications, especially in cryptography and related areas.