Uniform stratified vanishing and equidistribution on $\mathbb{G}_m^d$

Abstract

We prove a uniform stratified generic vanishing theorem for perverse sheaves on $\mathbb{G}_m^d$ over finite fields, with constants depending only on the dimension and on the complexity à la Sawin. As a corollary, we prove an equidistribution theorem for $\mathbb{G}_m^d$ extending Katz's theorem for $\mathbb{G}_m$. The corollary applies to sequences of perverse sheaves of bounded complexity with common tannakian monodromy group, over finite fields of varying characteristic. As the cardinalities of the fields tend to infinity, the Frobenius conjugacy classes attached to the multiplicative characters equidistribute in the space of conjugacy classes of a maximal compact subgroup.