# Seminars & Events for Princeton University/IAS Number Theory Seminar

##### 2^∞-Selmer groups, 2^∞-class groups, and Goldfeld's conjecture

Take E/Q to be an elliptic curve with full rational 2-torsion (satisfying some extra technical assumptions). In this talk, we will show that 100% of the quadratic twists of E have rank less than two, thus proving that the BSD conjecture implies Goldfeld's conjecture in these families. To do this, we will extend Kane's distributional results on the 2-Selmer groups in these families to 2^k-Selmer groups for any k>1. In addition, using the close analogy between 2^k-Selmer groups and 2^{k+1}-class groups, we will prove that the 2^{k+1}-class groups of the quadratic imaginary fields are distributed as predicted by the Cohen-Lenstra heuristics for all k>1.

##### Cohomology of p-adic Stein spaces

I will discuss a comparison theorem that allows us to recover p-adic (pro-)etale cohomology of p-adic Stein spaces with semistable reduction over local rings of mixed characteristic from complexes of differential forms. To illustrate possible applications, I will show how it allows us to compute cohomology of Drinfeld half-space in any dimension and of its coverings in dimension one. This is a joint work with Pierre Colmez and Gabriel Dospinescu.

##### Kloosterman sums and Siegel zeros

Kloosterman sums arise naturally in the study of the distribution of various arithmetic objects in analytic number theory. The 'vertical' Sato-Tate law of Katz describes their distribution over a fixed field F_p, but the equivalent 'horizontal' distribution as the base field varies over primes remains open. We describe work showing cancellation in the sum over primes if there are exceptional Siegel-Landau zeros. This is joint work with Sary Drappeau, relying on a fun blend of ideas from algebraic geometry, the spectral theory of automorphic forms and sieve theory.

##### Unlikely intersections for algebraic curves in positive characteristic

Please follow this link for the abstract: http://www.math.ias.edu/seminars/abstract?event=131079

##### TBA - Joachim Schwermer

##### The arithmetic intersection conjecture

The Gan-Gross-Prasad conjecture relates the non-vanishing of a special value of the derivative of an L-function to the non-triviality of a certain functional on the Chow group of a Shimura variety. Beyond the one-dimensional case, there is little hope for proving this conjecture. I will explain a variant of this conjecture (suggested by Wei Zhang) which seems more accessible and report on progress on it. This is joint work with B. Smithling and W. Zhang.