UCSD Number Theory Seminar (Math 209)

April 4

Organizational meeting

April 11

Claus Sorensen (UC San Diego)
Projective smooth representations mod $p$

This talk will be colloquial and geared towards people from other fields. I will talk about smooth mod $p$ representations of $p$-adic Lie groups. In stark contrast to the complex case, these categories typically do not have any (nonzero) projective objects. For reductive groups this is a byproduct of a stronger result on the derived functors of smooth induction. The talk is based on joint work with Peter Schneider.

April 18

Aranya Lahiri (UC San Diego)
Distribution algebras of p-adic groups

My goal will be to motivate why looking at distribution algebras associated to p-adic lie groups is natural in the context of number theory. More specifically I will try to briefly outline their importance in the p-adic Langlands program. And then I will give a simple example of an overconvergent distribution algebra of certain kinds of p-adic groups with an eye towards illuminating techniques used in my work Dagger groups and p-adic distribution algebras (joint w/ Matthias Strauch and Claus Sorensen).

April 25

Jon Aycock (UC San Diego)
Congruences Between Automorphic Forms

We will introduce an analytic notion of automorphic forms. These automorphic forms encode arithmetic data by way of their Fourier theory, and we will explore two different families of automorphic forms which have interesting congruences between their Fourier coefficients.

May 2

Wei Yin (UC San Diego)
Higher Coates-Sinnott Conjectures for CM-Fields

The classical Coates-Sinnott Conjecture and its refinements predict the deep relationship between the special values of L-functions and the structure of the étale cohomology groups attached to number fields. In this talk, we aim to delve deeper along this direction to propose what we call the “Higher Coates-Sinnott Conjectures" which reveal more information about these two types of important arithmetic objects. We introduce the conjectures we formulate and our work towards them. This is joint work with C. Popescu.

May 9

Nandagopal Ramachandran (UC San Diego)
Euler factors in Drinfeld modules

In this talk, I'll first give a quick introduction to the theory of Drinfeld modules and talk about an equivariant $L$-function associated to Drinfeld modules as defined by Ferrara-Higgins-Green-Popescu in their work on the ETNC. As is usual, these $L$-functions are defined as an infinite product of Euler factors, and the main focus of this talk is a result relating these Euler factors to a certain quotient of Fitting ideals of some algebraically relevant modules. This is joint work with Cristian Popescu.

May 16

Bryan Hu (UC San Diego)
Critical values of the adjoint L-function of U(2,1) in the quaternionic case

We will discuss questions surrounding automorphic L-functions, particularly Deligne’s conjecture about critical values of motivic L-functions. In particular, we study the adjoint L-function of U(2,1). Hundley showed that a certain integral, involving an Eisenstein series on the exceptional group G_2, computes this L-function at unramified places. We discuss the computation of this integral at the archimedean place for quaternionic modular forms, and how this relates to Deligne's conjecture.

May 23

Christian Klevdal (UC San Diego)
Local systems on Shimura varieties

A large area of modern number theory (the Langlands program) studies a deep correspondence between the representation theory of Galois groups, algebraic varieties and certain analytic objects (automorphic forms). Many spectacular theorems have come from this area, for example the key insight in Wiles' proof of Fermat's last theorem was a connection between elliptic curves, modular forms and Galois representations. The goal of this talk is to explain how geometric constructions, particularly related to Shimura varieties, arise naturally in the Langlands program. I will then talk about joint work with Stefan Patrikis, stating that Galois representations arising from certain Shimura varieties satisfy the properties predicted by the correspondence introduced above.

May 30

Ellen Eischen (Oregon)
Algebraic and p-adic aspects of L-functions, with a view toward Spin L-functions for GSp_6

I will discuss recent developments and ongoing work for algebraic and p-adic aspects of L-functions. Interest in p-adic properties of values of L-functions originated with Kummer’s study of congruences between values of the Riemann zeta function at negative odd integers, as part of his attempt to understand class numbers of cyclotomic extensions. After presenting an approach to studying analogous congruences for more general classes of L-functions, I will conclude by introducing ongoing joint work of G. Rosso, S. Shah, and myself (concerning Spin L-functions for GSp_6). I will explain how this work fits into the context of earlier developments, while also indicating where new technical challenges arise. All who are curious about this topic are welcome at this talk, even without prior experience with p-adic L-functions or Spin L-functions.

June 6

Chris Xu (UC San Diego)
Rational points on modular curves via the moduli interpretation

In theory, Chabauty-Coleman provides an explicit method to obtain rational points on any curve, so long as its genus exceeds its Mordell-Weil rank. In practice, when applied to modular curves, we often encounter difficulties in finding a suitable plane model, which only worsens as the genus increases. In this talk we describe how to skip this step and instead work directly with the coarse moduli space. This is joint work with Steve Huang and Jun Bo Lau.