Colloquium - Britton Lecture Series - Samit Dasgupta - Stark’s Conjectures and Hilbert’s 12th Problem

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Speaker: Samit Dasgupta, (Duke University)

Title: Stark’s Conjecturesand Hilbert’s 12th Problem

Abstract: In this talk we will discuss two centralproblems in algebraic number theory and their interconnections: explicit classfield theory and the special values of L-functions. The goal of explicitclass field theory is to describe the abelian extensions of a ground numberfield via analytic means intrinsic to the ground field; this question lies atthe core of Hilbert's 12th Problem. Meanwhile, there is an abundance ofconjectures on the values of L-functions at certain special points. Ofthese, Stark's Conjecture has relevance toward explicit class fieldtheory. I will describe two recent joint results with Mahesh Kakde onthese topics. The first is a proof of the Brumer-Stark conjecture awayfrom p=2. This conjecture states the existence of certain canonical elements inabelian extensions of totally real fields. The second is a proof of anexact formula for Brumer-Stark units that has been developed over the last 15years. We show that these units together with other easily writtenexplicit elements generate the maximal abelian extension of a totally realfield, thereby giving a p-adic solution to the question of explicit class fieldtheory for these fields. This lecture will be accessible to a broad audience.
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