Finite hypergeometric functions

Abstract

The ideas of this thesis are based on an article written by F. Beukers and A. Mellit. They have shown that, when defined over Q, finite hypergeometric sums correspond to point counting on projective varieties over finite fields. In my thesis we look at what happens if the hypergeometric sums are not defined over Q anymore.

In order to do such calculations we use a conjectural link to classical analytic hypergeometric functions. As a consequence of the work done we have found conjectural values for symmetric products of Gauss sums, even when the latter are not defined a priori.