Dorian Goldfeld

Affiliation: Columbia University

Title Of Talk: A Generalization of Dedekind's eta function for Hecke groups over a real quadratic field \n

Abstract: Let $D\equiv 1 \pmod{4}$ be a discriminant of a real quadratic field. For $z$ in the upper half plane we consider the Hecke group $H(\sqrt{D})$ generated by the transformations $$z \mapsto -\frac{1}{z}, \qquad z \mapsto z + \sqrt{D}.$$ The fundamental domain for this Hecke group has infinite volume. For $q = e^{2\pi iz}$ we construct a certain generalization of the $q$-product for the Dedekind eta function and show that this infinite product is a (non square integrable) holomorphic modular form for the Hecke group $H(\sqrt{D})$ which vanishes at the cusp at $\infty.$ We also show that these modular forms have Fourier expansions where the Fourier coefficients have exponential growth. This is joint work with Debmalya Basak, Winston Heap, Nicolas Robles, Alexandru Zaharescu.

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