A Tannakian Category and a Horizontal Equidistribution Conjecture for Exponential Sums

Abstract

We construct a Tannakian category of perverse sheaves on the additive group over the integers, with sheaf convolution as the tensor operation. A class of number-theoretic equidistribution problems, including the equidistribution of Gauss sums and Kloosterman sums and the equidistribution of roots of polynomial congruences, can be viewed as problems about the equidistribution of conjugacy classes in the Tannakian group of this category. Using this correspondence, we conjecture answers to these equidistribution questions.