# Number Theory Seminar

This seminar is held on Tuesdays at 10am in 119 Fenton Hall unless otherwise stated.

### Spring Quarter, 2017

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

Fermat type equations and symplectic isomorphisms of the p-torsion of elliptic curves
**Abstract**: Wiles’ proof of Fermat’s Last Theorem gave birth to the ‘modular method’ to study Diophantine equations. Since then many other equations were solved using generalizations of this method. However, the success of the generalizations relies on a final “contradiction step” which is invisible in the original proof.

In this talk, we will discuss why developing methods to distinguish Galois representations is relevant to this contradiction step. In particular, we will explain how the “symplectic argument” can be used to succeed in this last step. We will illustrate the method with example of application to special cases of the Generalized Fermat equation

x^r + y^q = z^p.

- April 12,
**Cathy Hsu** (UO)

Higher Eisenstein Congruences
**Abstract**: Let $p\geq 3$ be prime. For squarefree level $N>6,$ we use a commutative algebra result of Berger, Klosin, and Kramer to bound the depth of Eisenstein congruences modulo $p$ (from below) by the $p$-adic valuation of the numerator of $\frac{\varphi(N)}{24}$. We then show that if $N$ has at least three prime factors and some prime $p\geq 5$ divides $\varphi(N),$ the Eisenstein ideal is not locally principal. Time-permitting, we will illustrate these results with explicit computations as well as discuss generalizations to other families of modular forms.

- April 18,
**Roger Heath-Brown** (Oxford University)

Vinogradov’s Mean Value Problem and the Riemann Zeta-function
**Abstract**: We will dsicuss Vinogradov’s Mean Value problem, and the exciting recent progress by Wooley and by Bourgain, Demeter and Guth. A key application of these results is to provide new bounds on the size of the Riemann Zeta-function, and on exponential sums in general.

- April 25,
**Mckenzie West** (Reed College)

The Geometry of a Family of K3 Surfaces
- April 28,
**Adam Hughes** (Austin)

Conditional Expectation, Id\`{e}le Groups, and the Fundamental Theorem of Galois Theory
**Abstract**: We consider the topological space of primes of $\overline{\Bbb Q}$ and show how to consider algebraic numbers as integrable functions on this space. By way of application we will compute the expectation for certain numbers and relative field extensions and discuss how they relate to the field, id\`{e}le norms, and the inclusion reversing bijection given by the Fundamental Theorem of Galois Theory. Time permitting we will also mention applications of this representation to multiplicative approximation.

- May 1,
**Ellen Eischen** (UO)

**Special Time/Location:** Monday at 9:30am in 206 Deady

p-adic L-functions: What, How, and Why
**Abstract**: L-functions play a central role in number theory. In this talk, I will give an overview of p-adic L-functions (including what a p-adic L-function is) and an approach to constructing them, which relies on relating them to families of modular forms. I will start with the earliest examples of p-adic L-functions (due to Serre, Leopoldt, and Kubota) and conclude with a recently completed construction of myself, Harris, Li, and Skinner.

- May 9,
**Zhiwei Yun** (Yale)

**Special Time/Location:** Tuesday at 3pm in 104 Deady

Intersection numbers and higher derivatives of L-functions for function fields
**Abstract**: In joint work with Wei Zhang, we prove a higher derivative analogue of the Waldspurger formula and the Gross-Zagier formula in the function field setting. Our formula relates the self-intersection number of certain cycles on the moduli of Shtukas for GL(2) to higher derivatives of automorphic L-functions for GL(2).

- May 25,
**Paul Pollack** (University of Georgia)

Torsion of CM elliptic curves: Analytic aspects
**Abstract**: For each positive integer d, let T(d) denote the supremum of all orders of groups E(F)[tors] appearing for an elliptic curve E defined over a degree d number field F. A celebrated theorem of Merel asserts that T(d) is finite for all d. However, the known quantitative results in this direction are far from the conjectured truth. Let T_{CM}(d) be defined the same way as T(d), but with the restriction to CM elliptic curves. I will discuss some recent statistical results concerning T_{CM}(d) and related functions. Perhaps surprisingly, the “anatomy of integers” (as pioneered by Paul Erdos) plays a key role in the proofs. Joint work with Abbey Bourdon and Pete L. Clark.

- June 6,
**Kevin Mcgown** (California State University, Chico)

Genus numbers of cubic fields
**Abstract**: The class number is among the most important invariants associated to a number field, but it is very difficult to study. Conjecturally, its behavior (at the “good” primes) is governed by the heuristics of Cohen-Lenstra-Martinet. By contrast, the genus number is the part of the class number that is easier to understand. It is very natural to ask about the density of genus number one fields among all number fields of a fixed degree and signature. We do not impose the restriction that our fields are Galois, and consequently the genus theory is a little more subtle. The simplest situation where this statistical question has not been previously addressed is that of cubic fields. We prove that approximately 96.23% of cubic fields, ordered by discriminant, have genus number one. Finally, we show that a positive proportion of totally real cubic fields with genus number one fail to be norm-Euclidean. This is joint work with Amanda Tucker.

### 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.

### 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