Number Theory

The endoscopic classification of representations of quasi-split symplectic and orthogonal groups is a celebrated result of Arthur (extended to quasi-split unitary groups by Mok) which has had wide applications to representation theory and number theory. It is a collection of many interrelated…

A result of Jan Nekovář says that the Galois action on p-adic intersection cohomology of Hilbert modular varieties with coefficients in automorphic local systems is semisimple.  We will explain a new proof of this result for the non-CM part of the cohomology.  Interestingly, Nekovář’s…

For a prime p, let {C_n} be a Z_p-tower of smooth projective curves over a finite field of characteristic p. The p-part of the class group of the function field of C_n is known to depend on n in a regular way.  In characteristic p, the p-divisible group of the Jacobian of C_n is a geometric…

Consider a collection of forms of odd degree with rational coefficients. Birch proved in 1957 that if the number of variables is sufficiently large, then the forms must have a nontrivial rational zero. The bounds resulting from Birch's proof, however, are so large that he has described…

I will explain a construction of p-adic variations of twistor structure and how it relates to other recent developments in relative p-adic Hodge theory including: geometric Sen theory, the p-adic Simpson correspondence, and analytic prismatization. After introducing the basic theory and…

The "fundamental curve" of arithmetic over nonarchimedean local fields is the Fargues–Fontaine curve. You'd think the "fundamental curve" of arithmetic over global function fields is... just the associated curve over F_q. In this talk, I will instead advocate for a different "curve", in the…

Despite a lot of progress, the question of whether all the finite groups PSL(n,q) of all n by n matrices with coefficients in F_q and of determinant 1, modulo center, occur as Galois groups of a Galois extension of Q remains open. We study the cohomology of curves with an automorphism with F_p-…

Abstract:   In this talk we review results on several types of harmonic weak Maass forms that are related to newforms of even integer weight.  

Starting with an integral weight newform, we briefly present different constructions of integral weight harmonic weak Maass forms via (…

The sum-product phenomenon concerns an incongruence between additive and multiplicative structure over finite sets of integers. Results of this type often involve showing that a finite set of integers either gives rise to many pairwise sums or many pairwise products. Such results have been…

For smooth manifolds, the Gysin map of a closed immersion is defined as the cohomology applied on the Pontryagin–Thom collapse map, which collapses the ambient manifold to the one-point compactification of the tubular neighborhood of the closed submanifold. In this talk, I will present a version…