Number Theory Seminar

Date: November 7, 2023

Time: 11:10AM - 12:00PM

Location: BLOC 302

Speaker: Matt Papanikolas, Texas A&M University

Title: Traces of singular moduli for Drinfeld modules

Abstract: In the 1990’s and 2000’s, Zagier proved remarkable results on traces of singular moduli that showed they satisfy various simple identities on average and that ultimately they constitute the Fourier coefficients of a particular half-integral weight modular form. In the present talk we will consider ways to phrase these same questions over the rational function field in one variable over a finite field. In this setting singular moduli are defined as values of the Drinfeld modular j-invariant on Heegner points in the Drinfeld upper half-space. Through work of Bae, Hsia, Wang, J.Yu, and J.-K. Yu, it is possible to define class polynomials over the rational function field whose roots are singular moduli but which also satisfy explicit connections with modular polynomials. Building on these constructions we devise average results for traces of singular moduli and recover formulas for Hurwitz class numbers that align with Zagier’s results. Joint with A. El-Guindy, R. Masri, and G. Zeng.