# Number Theory Seminar

This seminar is held on Wednesdays at 3pm via a Zoom Meeting during Spring term.

### Spring Quarter, 2020

- May 20,
**Stella Gastineau** (Boston College)

Diving into the Shallow End
**Abstract**: In 2013, Reeder-Yu gave a construction of supercuspidal representations by starting with stable characters coming from the shallowest depth of the Moy-Prasad filtration. In this talk, we will be diving deeper—but not too deep. In doing so, we will construct examples of supercuspidal representations coming from a larger class of “shallow” characters. Using methods similar to Reeder-Yu, we can begin to make predictions about the Langlands parameters for these representations.

- May 27,
**Chi-Yun Hsu** (UCLA)

Construction of Euler Systems for GSp4×GL2
**Abstract**: An Euler system is a collection of norm-compatible first Galois cohomology classes with the Galois groups varying over cyclotomic fields. By constructing an Euler system, one can bound the Selmer group of Galois representations. We construct Euler systems for the Galois representations coming from automorphic representations of GSp4×GL2. The strategy follows the work of Loeffler-Zerbes-Skinner in the case of GSp4, using automorphic input to show norm compatibility. This is a work in progress with Zhaorong Jin and Ryotaro Sakamoto.

- June 3,
**Christelle Vincent** (University of Vermont)

On the equidistribution of joint shapes of rings and their resolvents
**Abstract**: In her thesis, Piper H showed that “shapes of number fields” are “equidistributed” under certain mild and expected conditions. The proof uses Bhargava’s parametrization of cubic, quartic, and quintic rings, which itself works by attaching to each such ring one or more “resolvent ring” and parametrizing rings with a choice of resolvent.

A natural question then arises: Are the shapes of a ring and its resolvent independent of one another; in other words is the ordered pair of shapes equidistributed too? What if we replace rings and a choice of resolvent ring with fields and their resolvent fields?

In this talk we will introduce the notion of the shape of a ring or number field, briefly define what we mean by equidistribution in this context, and describe the resolvent ring of a quartic ring. We will then give a glimpse of the difficulties in extending a result about rings and their resolvents to fields and their resolvents. This is joint work with Piper H.

### Winter Quarter, 2020

- January 8,
**John Bergdall** (Bryn Mawr College)

Reductions of some 2-dimensional Galois representations
**Abstract**: Many problems in the arithmetic of modular forms are studying using techniques from the theory of Galois representations. For instance, congruences between modular forms can be “detected” by studying p-adic families Galois representations on which “reduction modulo p” is constant. Thus, there is great interest in extracting the reduction of Galois representations associated with modular forms from the arithmetic of the modular form. In our particular case, we will explain how to use the weight k and the p-th Fourier coefficient to determine the reduction modulo p of a modular Galois representation, in certain cases. Our result improves a result of Berger, Li, and Zhu from 2004. The techniques are in fact completely local, making use of work of Fontaine and Kisin in p-adic Hodge theory, and we will explain as broadly as possible how the local perspective helps solve problem.

- January 29,
**Liubomir Chiriac** (Portland State University)

Summing Fourier coefficients over polynomials values
**Abstract**: Functions of number-theoretic interest are often studied on average. This talk is concerned with mean values of Fourier coefficients of modular forms over polynomials. While much progress has been done in this direction for polynomials of degree at most two, rather little is known beyond that. Here we will present an approach to obtain upper bounds for sums involving polynomials of arbitrary degree. We will also discuss specific examples to illustrate our results.

- February 19,
**Francesc Castella** (UCSB)

Heegner points in Hida families and non-vanishing of central derivatives
**Abstract**: Let F be a cuspidal Hida family, and let F_k run over the arithmetic specializations of F of even weight. A deep conjecture of Greenberg predicts that, expect for finitely many k, the cyclotomic p-adic L-function attached to F_k should vanish to order exactly 0 or 1 at the center, depending on the generic sign of F_k. When this sign is +1, by the work of Skinner-Urban the non-vanishing of central values predicted by Greenberg’s conjecture follows from the torsion-ness of a certain Selmer group attached to F. In this talk, I’ll describe an analogous result when the generic sign is -1. Based on a joint work with Xin Wan.

- March 4,
**Daniel E. Martin** (University of Colorado)

The geometry of imaginary quadratic fields
**Abstract**: If O is a ring of integers, the group SL_n(O) is generated by its elementary matrices except when n=2 and O is non-Euclidean, imaginary quadratic. Besides the five Euclidean cases, SL_2(O) presentations are computed algorithmically for imaginary quadratic fields. We will take a geometric perspective on these groups that produces an explicit generating set and a new algorithm for finding the corresponding relations. Applications to continued fractions and the ideal class group will be highlighted.

- March 11,
**Carl Wang-Erickson** (University of Pittsburgh)

Bi-ordinary modular forms
**Abstract**: It is well-understood that p-ordinary Hecke eigenforms give rise to global 2-dimensional Galois representations which become reducible with an unramified quotient after restriction to a decomposition group at p. Coleman and Greenberg independently asked for a characterization of those p-ordinary forms whose associated Galois representation is also semi-simple after restriction to this decomposition group. It has been suspected that all such p-ordinary forms have complex multiplication, a restrictive global property. We will discuss joint work with Francesc Castella, in which we give a criterion for this suspicion to be true, and give a construction of “bi-ordinary” modular forms as a tool to explore the case that the criterion fails.

**Previous years**: 2019 2018 2017 2016