Elliptic units for complex cubic fields

Abstract

This is joint work with Nicolas Bergeron and Pierre Charollois. The elliptic Gamma function — a generalization of the q-Gamma function, which is itself the q-analog of the ordinary Gamma function — is a meromorphic special function in several variables that mathematical physicists have shown to satisfy modular functional equations under SL(3,Z). In this talk I will present evidence (numerical and theoretical) that products of values of this function are often algebraic numbers that satisfy explicit reciprocity laws and are related to derivatives of Hecke L-functions of cubic fields at s=0. We will discuss the relation to Stark's conjectures and will see that this function conjecturally allows to extend the theory of complex multiplication to complex cubic fields as envisioned by Hilbert's 12th problem.