# Number Theory Seminar

This seminar is held on Wednesdays at 3pm in 221 Friendly Hall unless otherwise stated.

### 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)
- February 26,
**Ashay Burungale** (Caltech)
- 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)

### Spring Quarter, 2020

- April 1,
**Christelle Vincent** (University of Vermont)
- April 8,
**Chi-Yun Hsu** (UCLA)
- May 13,
**Zheng Liu** (UCSB)
- May 27,
**Bryden Cais** (University of Arizona)

**Previous years**: 2019 2018 2017 2016