This term we will have a study group on the work of Dimitrov--Gao--Habegger and Kühne on uniformity in the Mordell conjecture. The first half of the schedule is meant to be an introduction to the area for non-specialists. In the second half, we will try to introduce some of the ideas from functional transcendence, dynamics and the moduli of abelian varieties which go into the proof.

The first talk will be an introduction to the history and statement of the results, with a vague hint at the methods of proof. A plan of the rest of the study group can be found here: https://sites.google.com/site/ class='hl'>netandogra/seminars/uniform-mordell

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.

---------------------------

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.