Graph laplacians and Drinfeld modular curves
The relationship between combinatorial laplacians and automorphic forms is an active area of current research with applications to a variety of problems arising in number theory, group theory, and coding theory. I will discuss certain combinatorial laplacians arising in the theory of Drinfeld modular curves, and their applications to estimating congruences between automorphic forms.
The basis problem
Modular forms arose as a natural object of study in number theory
due to connections with quadratic forms and theta series. The basis problem, studied in depth by Eichler and others, conversely asks what spaces of modular forms can be generated by theta series, especially theta series attached to
quaternion algebras. We will discuss a representation-theoretic approach to the basis problem using the Jacquet-Langlands correspondence. This approach both refines the solution to the basis problem by Hijikata-Pizer-Shemanske for elliptic modular forms and solves the basis problem for Hilbert modular forms. This has practical applications to computing spaces of modular forms and arithmetic applications to Eisenstein congruences.
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. Benedetti–Silhol, Finashin–Kharlamov 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.