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Monday, October 05, 2020
12:30 PM - 2:00 PM
Online Event

Joint Berkeley-Caltech-Stanford Number Theory Seminar

Stark's Conjectures and Hilbert's 12th Problem
Samit Dasgupta, Department of Mathematics, Duke University,

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 state a conjectural exact formula for these Brumer-Stark units that has been developed over the last 15 years.  I will conclude with a description of work in progress that aims to prove this conjecture and thereby give a solution to Hilbert's 12th problem.

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