SPEAKER:     Kathrin Bringmann, University of Wisconsin, Madison

TITLE:    Traces of Singular Moduli on Hilbert Modular Surfaces

ABSTRACT:
Suppose that $p\equiv 1\pmod 4$ is a prime, and that $\Op$ is the ring of integers of $K:=\Q(\sqrt{p})$. A classical result of Hirzebruch and Zagier asserts that certain generating functions for the intersection numbers of Hirzebruch-Zagier divisors on the Hilbert modular surface $(\h\times \h)/\SL_2(\Op)$ are weight $2$ holomorphic modular forms. Using recent work of Bruinier and Funke, we show that the generating functions of traces of singular moduli over these intersection points are often weakly holomorphic weight $2$ modular forms. For the singular moduli of $J_1(z)=j(z)-744$, we explicitly determine these generating functions using classical Weber functions, and we factorize their ``norms" as products of Hilbert class polynomials.