Title: Stark's Conjectures and Hilbert's 12th Problem

Date: 02/02/2021

Time: 4:00 PM - 5:00 PM

Place: Online (virtual meeting)

Contact: Aaron D Levin ()

In this talk we will discuss two central problems in algebraic number theory and their interconnections: explicit class field theory (also known as Hilbert's 12th Problem), 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. 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 my recent proof, joint with Mahesh Kakde, 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. Next I will describe our proof of an exact formula for these Brumer-Stark units that had been developed by many authors over the last 15 years. We show that the Brumer-Stark units along with other elementary quantities generate the maximal abelian extension of totally real number fields, thereby giving a solution to Hilbert's 12th problem for these fields.