Talk 7: On Eisenstein's Jugendtraum for complex cubic fields (online talk)

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 try to convince you that this function, and some avatars of it, can be used to extend the theory of complex multiplication to complex cubic fields. In an on-going joint work with Pierre Charollois and Luis Garcia, we indeed propose a conjectural solution to Hilbert's 12th problem for (cubic) almost totally real fields and we give a lot of numerical evidences that support this conjecture.