# Number Theory Seminar

This seminar is held on Mondays at 3pm via a Zoom Meeting during Spring term. The meeting information can be found at the Number Theory Seminar page.

### Spring Quarter, 2021

- April 26,
**Gil Moss** (University of Utah)

Moduli of Langlands parameters
**Abstract**: The local Langlands correspondence connects representation of p-adic groups to Langlands parameters, which are certain representations of Galois groups of local fields. In recent work with Dat, Helm, and Kurinczuk, we have shown that Langlands parameters, when viewed through the right lens, occur naturally within a moduli space over Z[1/p], and we can say some things about the geometry of this moduli space. This geometry should be reflected in the representation theory of p-adic groups, on the other side of the local Langlands correspondence. The “local Langlands in families” conjecture describes the moduli space of Langlands parameters in terms of the center of the category of representations of the p-adic group– it was established for GL(n) in 2018. The goal of the talk is to give an overview of this picture, including current work in-progress, with some discussion of the relation with recent work of Zhu and Fargues-Scholze.

- May 3,
**Brandon Williams** (RWTH Aachen)

**Special Time:** 1:30-2:30 PM

Borcherds products and a ring of Hermitian modular forms
**Abstract**: We will compute the ring of modular forms for the group U(2, 2) over the integers in

. The main tool is Borcherds products and their application to Hermitian modular forms due to Dern.

- May 10,
**Sam Mundy** (Columbia University)

The Skinner–Urban method and the symmetric cube Bloch–Kato conjecture
**Abstract**: The Skinner–Urban method is a general method which can be used to construct nontrivial elements in the Bloch–Kato Selmer groups attached to certain Galois representations. After giving a historical overview of the method as well as techniques which preceded it, I will briefly explain how it can be used to construct nontrivial elements in the Selmer group for the symmetric cube of the Galois representation attached to a level 1 modular form, under certain standard conjectures. This will take us through the theory of automorphic forms and Galois representations for the exceptional group G_2.

- May 17,
**David Lowry-Duda** (ICERM)

Visualizing modular forms
**Abstract**: We investigate ways to visualize modular forms. A good visualization of a modular form should reveal some of the highly symmetric structure of the modular form. But different methods of visualization shine different spotlights on the modular form. In this talk, we examine different methods of making and studying these visualizations. Further, we’ll examine both classical and non-classical modular forms in a variety of different visualizations. There will be lots and lots of pictures!

- May 24,
**Lassina Dembélé** (University of Luxembourg)

**Special Time:** 2:00-3:00 PM

### Winter Quarter, 2021

- February 17,
**Ari Shnidman** (Hebrew University of Jerusalem)

Rational points on twist families of curves
**Abstract**: A curve C of genus g > 1 has finite automorphism group G. If C is defined over a number field F, we consider a twist family of C, which is a family of curves over F each of which is isomorphic C over the algebraic closure of F. For example, the family C_d : x^6 + y^6 = d is a family of twists of the degree 6 fermat curve C_1. In this talk, I’ll present some results which show that for various twist families, a large proportion of twists have very few F-rational points. For example, we can show that C_d(Q) is empty for at least 66% of integers d. Our proofs generally have two steps: bound the average rank of the Jacobian using a 3-descent, and then apply Chabauty-like methods to bound the number of rational points when the rank is small.

- March 3,
**Razan Taha** (Purdue University)

p-adic Measures for Reciprocals of L-functions of Totally Real Fields
**Abstract**: In 2014, Gelbart, Miller, Panchishkin, and Shahidi introduced an analog to part of the Langlands-Shahidi method by constructing certain p-adic L-functions using the non-constant Fourier coefficients of Eisenstein series. In this talk, we extend their work to totally real number fields. We construct p-adic measures which interpolate the special values of p-adic L-functions of a totally real field K at negative integers. These measures are defined by analyzing the non-constant terms of partial Eisenstein series of the Hilbert modular group.

### Fall Quarter, 2020

- October 21,
**Joe Webster** (UO)

The p-adic Mehta Integral
**Abstract**: The Mehta integral is the canonical partition function for 1-dimensional log-Coulomb gas in a harmonic potential well. Mehta and Dyson showed that it also determines the joint probability densities for the eigenvalues of Gaussian random matrix ensembles, and Bombieri later found its explicit form. We introduce the p-adic analogue of the Mehta integral as the canonical partition function for a p-adic log-Coulomb gas, discuss its underlying combinatorial structure, and find its explicit formula and domain.

- November 4,
**Yujie Xu** (Harvard)
- November 11,
**Mathilde Gerbelli-Gauthier** (University of Chicago)

Cohomology of Arithmetic Groups and Endoscopy
**Abstract**: How fast do Betti numbers grow in a congruence tower of compact arithmetic manifolds? The dimension of the middle degree of cohomology is proportional to the volume of the manifold, but away from the middle the growth is known to be sub-linear in the volume. I will explain how automorphic representations and the phenomenon of endoscopy provide a framework to understand and quantify this slow growth. Specifically, I will discuss how to obtain some explicit bounds in the case of unitary groups using Arthur’s stable trace formula.

- November 18,
**Melissa Emory** (University of Toronto)

A multiplicity one theorem for general spin groups
**Abstract**: A classical problem in representation theory is how a representation of a group decomposes when restricted to a subgroup. In the 1990s, Gross-Prasad formulated an intriguing conjecture regarding the restriction of representations, also known as branching laws, of special orthogonal groups. Gan, Gross and Prasad extended this conjecture, now known as the local Gan-Gross-Prasad (GGP) conjecture, to the remaining classical groups. There are many ingredients needed to prove a local GGP conjecture. In this talk, we will focus on the first ingredient: a multiplicity at most one theorem. Aizenbud, Gourevitch, Rallis and Schiffmann proved a multiplicity (at most) one theorem for restrictions of irreducible representations of certain p-adic classical groups and Waldspurger proved the same theorem for the special orthogonal groups. We will discuss work that establishes a multiplicity (at most) one theorem for restrictions of irreducible representations for a non-classical group, the general spin group. This is joint work with Shuichiro Takeda.

- November 25,
**Jon Aycock** (UO)

Overconvergent Differential Operators Acting on Modular Forms
**Abstract**: In the 1970’s, Katz used Damerell’s formula to construct p-adic L-functions for CM fields by interpolating differential operators. However, these operators were only defined over the ordinary locus, leading to restrictions on p. Recently, a geometric theory of overconvergent modular forms has given a way around these restrictions. I will describe how to do this in the case of modular forms, and then give a brief template for the Hilbert case.

- December 2,
**Ananth Shankar** (University of Wisconsin)

A finiteness criterion for 2-dimensional representations of surface groups
**Abstract**: Let C be a a complex algebraic curve of genus $\geq 1$, and let $\pi$ be its fundamental group. Let $\rho: \pi\rightarrow \GL_2(\C)$ be a semisimple 2-dimensional representation, such that $\rho(\alpha)$ has finite order for every simple closed loop $\alpha.$ We will prove that $\rho$ has finite image. If time permits, we will mention applications of this result to the Grothendieck-Katz p-curvature conjecture. This is joint work with Anand Patel and Junho Peter Whang.

**Previous years**: 2020 2019 2018 2017 2016