Séminaire d'arithmétique à Lyon

Stark's Conjectures and Hilbert's 12th Problem

by Dr Samit Dasgupta


In this talk we will discuss two central problems in algebraic number theory and their interconnections: explicit class field theory and the special values of L-functions.  The goal of explicit class field theory is to describe the abelian extensions of a ground number field via analytic means intrinsic to the ground field; this question lies at the core of Hilbert's 12th Problem.  Meanwhile, there is an abundance of conjectures on the special values of L-functions at certain integer points.  Of these, Stark's Conjecture has special relevance toward explicit class field theory.  I will describe two recent joint results with Mahesh Kakde on these topics.  The first is a proof of the Brumer-Stark conjecture away from p=2. This conjecture states the existence of certain canonical elements in CM abelian extensions of totally real fields.  The second is a proof of an exact formula for Brumer-Stark units that has been developed over the last 15 years.  We show that these units together with other easily written explicit elements generate the maximal abelian extension of a totally real field, thereby giving a p-adic solution to the question of explicit class field theory for these fields.