## Introduction to Modular Forms and Congruence Primes

### Huixi Li – Clemson University

Modular forms are interesting objects in number theory. In this presentation I will first go over the proof of the Lagrange’s four squares theorem using elliptic modular forms. Second I will introduce Hilbert modular forms and Siegel modular forms, with the motivation of generalizing the four squares theorem to totally real fields. Finally I will talk about our recent result on congruence primes for Hilbert Siegel eigenforms. This is joint work with Jim Brown.

## How to count lines on a cubic surface arithmetically

### Jesse Kass – University of South Carolina

Salmon and Cayley proved the celebrated 19th century result that a smooth cubic surface over the complex numbers contains exactly 27 lines.  By contrast, the count over the real numbers depends on the surface, and these possible counts were classified by Segre.  BenedettiSilhol, FinashinKharlamov and Okonek–Teleman made the striking observation that Segre’s work shows a certain signed count is always 3.  In my talk, I will explain how to extend this result to an arbitrary field.  Although I will not use any homotopy, I will draw motivation from A1-homotopy theory.  This is joint work with Kirsten Wickelgren.

## Fermat’s Theorem on Sums of Two Squares

### Huixi Li – Clemson University

Fermat’s theorem on sums of two squares states that an odd prime can be written as the sum of two integer squares if and only if it is congruent to 1 modulo 4. In this presentation I will talk about several proofs of this theorem, and I will compute the ratio of the sum of the bigger terms and the sum of the smaller terms in the representation of such primes as the sum of two squares of positive integers. The result verifies some conjectures of Zhiwei Sun.