Talk 10248 data/Spring

Number Theory Seminar
Wednesday, January 17, 2018, 12:00pm, Physics 119
Samit Dasgupta (UC Santa Cruz, Mathematics)
Stark's Conjectures and Hilbert's 12th Problem

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 of the Gross-Stark conjecture, a p-adic version of Stark's Conjecture that relates the leading term of the Deligne-Ribet p-adic L-function to a determinant of p-adic logarithms of p-units in abelian extensions. Next I will state my refinement of the Gross-Stark conjecture that gives an exact formula for Gross-Stark units. I will conclude with a description of work in progress that aims to prove this conjecture and thereby give a p-adic solution to Hilbert's 12th problem. [video]

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