The elliptic Gamma function was introduced in 1997 by Ruijsenaars and has been mainly studied by mathematical physicists.
Defined by a double infinite product, it can be considered as a kind of analog for SL3(Z) of the Jacobi theta function.
Together with its higher dimensional avatars, they form a nice hierarchy of meromorphic multivariable functions enjoying symmetries governed by SLn, n>2.
We will present joint work of P.C. with Nicolas Bergeron and Luis Garcia, as well as on-going PhD work of Pierre Morain.
In this body of work, we specialize elliptic Gamma functions at certain explicit points belonging to a number field K with exactly one complex embedding.
We then conjecture that the resulting complex numbers are indeed specific algebraic units lying above K, thereby proposing that elliptic Gamma functions might play a central role towards a solution of Hilbert’s 12th problem for such ground fields, encompassing Complex Multiplication.
We provide a great wealth of numerical evidence for this conjecture, as well as some theoretical results.
Vladimir Rubtsov, Vasily Golyshev, Ilia Gaiur
