Philadelphia Area Number Theory Seminar

Wednesday, October 2, 2019 - 2:15pm

Cihan Karabulut

William Paterson University

In 1975, Cohen generalized Hurwitz class number using Dirichlet’s class number formula to a number H(r, n) which is closely related to the value of a certain Dirichlet L-series at 1 − r and showed that for r ≥ 2 the generating function P∞ n=0 H(r, n)q n is a modular form of weight r + 1/2 on Γ0(4). In this talk, I will begin by describing Hurwitz class number and class number relations and then proceed to discuss Cohen’s result. I will then discuss a family of modular forms on Γ0(N) which were constructed by Ueno in a similar way as Cohen’s construction where the numbers H(r, n) are replaced with zeta functions of binary Hermitian forms evaluated at integral arguments. Finally, I will discuss some new work (joint with Jorge Fl´orez and An Hoa Vu) showing that the generating series considered by Ueno are in fact Eisenstein series and as a consequence we obtain an explicit formula for the special values of zeta functions associated with binary Hermitian forms.