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.