In the early 2000s Ruijsenaars and Felder-Varchenko have introduced the elliptic gamma function, a remarkable multivariable meromorphic q-series that comes from mathematical physics. It satisfies modular functional equations under the group SL3(ℤ) which make it a higher-dimensional analogue of the Jacobi theta function. In this work, we unveil the place that this function and its avatars play in number theory. Our main thesis is that these functions play the role of modular units in extending the theory of complex multiplication to complex cubic fields. In other words we propose a conjectural solution to Hilbert’s 12th problem for complex cubic fields, following a line of research actually initiated by G. Eisenstein. We give a lot of numerical evidence that support this conjecture, and relate it to the Stark conjecture by proving an analogue of the Kronecker limit formula in this cubic setting.
This is joint work with Nicolas Bergeron and Luis Garcia.
This video is part of the Institute for Advanced Study‘s Number theory seminar.
