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Équations différentielles motiviques et au–delà
Algebraic specialisations of elliptic Gamma functions
IHP
Pierre Grisvard (ex salle 314)
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.