Speaker: Netan Dogra

Title: Rational points and p-adic integrals on families of curves.

Abstract: If X is a curve of genus >1 over a number field, then the set of rational points of X is finite. It is a big open problem to understand how this finite set varies with X. I will explain what this has to do with p-adic integration, and how a suitable notion of 'p-adic integration in families' enables us to say some new things.

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Speaker: Matthew Honnor

Title: Formulas for Brumer--Stark Units

Abstract: In the 1980's, Tate stated the Brumer--Stark conjecture which, for a totally real field $F$ with prime ideal $\mathfrak{p}$, conjectures the existence of a $\mathfrak{p}$-unit called the Brumer--Stark unit. This unit has $\mathfrak{P}$ order equal to the value of a partial zeta function at 0, for a prime $\mathfrak{P}$ above $\mathfrak{p}$. There have been three formulas conjectured for the Brumer--Stark unit by Dasgupta and Dasgupta--Spie\ss. In this talk, I will present forthcoming joint work with Dasgupta which shows that these three formulas are equivalent.