# Number Theory Seminar

This seminar is held on Wednesdays at 3pm in 210 Deady Hall unless otherwise stated.

### Winter Quarter, 2017

- January 18,
**Rahul Krishna** (Northwestern University)

A New Approach to Waldspurger’s Formula
**Abstract**: I present a new trace formula approach to Waldspurger’s formula for toric periods of automorphic forms on $PGL_2$. The method is motivated by interpreting Waldspurger’s result as a period relation on $SO_2 \times SO_3$, which leads to a strange comparison of relative trace formulas. I will explain the local results needed to carry out this comparison, and discuss some optimistic dreams for extending these results to high rank orthogonal groups.

- February 1,
**Sean Howe** (University of Chicago)

Overconvergent modular forms and the p-adic Jacquet-Langlands correspondence
**Abstract**: We describe an explicit transfer of Hecke eigensystems from overconvergent modular forms to quaternionic modular forms, answering an old question of Serre and connecting with recent work of Knight and Scholze. The transfer depends on a construction of overconvergent modular forms using the infinite level modular curve, and we sketch this construction and explain how certain features of the p-adic theory of modular forms (e.g., p-adic weights) arise naturally from the equivariant geometry of the projective line.

### Spring Quarter, 2017

- April 4,
**Nuno Freitas** (University of British Columbia)

**Special Time:**10am
- April 12,
**Cathy Hsu** (UO)
- May 9,
**Zhiwei Yun** (Yale)

### Fall Quarter, 2016

- October 18,
**Naomi Tanabe** (Dartmouth College)

**Special Time/Location**: Tuesday, 2pm in 206 Deady Hall

Identifying a Hilbert modular form by the central L-values of

its twists
**Abstract**: We discuss the twisted moments of Rankin-Selberg $L$-functions at $s=1/2$, associated to two Hilbert modular forms. The study shows that the central $L$-values of Rankin-Selberg convolutions

uniquely determine an underlying modular form. We establish such results both in level and weight aspects. This is joint work with Alia Hamieh.

- October 26,
**Jeff Vaaler** (University of Texas, Austin)

The p-adic Selberg Integral
- November 8,
**Elena Mantovan** (Caltech)

**Special Time/Location**: Tuesday, 2pm in 206 Deady Hall

Towards a geometric realization of p-adic automorphic forms on unitary Shimura varieties at non-split unramified primes
**Abstract**: A crucial construction in the study of p-adic automorphic forms on Shimura varieties is their realization as global functions on the ordinary Igusa tower.

Following Hida’s approach, we consider the cases when the ordinary locus is empty, and explore the relation between p-adic automorphic forms and global functions on the \mu-ordinary Igusa tower.

This is joint work in progress with Ellen Eischen.

- November 9,
**Amos Turchet** (University of Washington)

Uniformity results in Diophantine Geometry
**Abstract**: In 1997 Caporaso, Harris and Mazur proved that Lang’s Conjecture implies that the number of K-rational points in a smooth projective curve of general type defined over a number field K is uniformly bounded by a constant depends only on K and on the genus of the curve. This breakthrough it is still regarded either as one of the most fascinating implication of Lang Conjecture or as a hint that the Conjecture is too good to be true. Various generalization in higher dimensions have been obtained in the subsequent years. All these result rely on a purely geometric statement, now known as Fibered Power Theorem, that is based on the geometry of the moduli space of the object involved. In this talk I will give a survey of these fascinating results, explaining why their natural generalization to integral points fails, and how to recover analogous geometric and arithmetic statements for quasi-projective curves and surfaces of log general type. This is joint work with Kenny Ascher.

- November 16,
**Vivek Pal** (University of Oregon)

Euler Systems
**Abstract**: Euler systems, in particular, the Heeger point Euler system have been integral to our approach to the Birch and Swinnerton-Dyer (BSD) conjecture for elliptic curves. I will describe Kolyvagin’s Euler system and how it can be used to approach the BSD conjecture. Then, I will describe work in progress on constructing such an Euler system associated to higher dimensional cycles in the function field setting.

**Previous year**: 2015