Equidistribution for Tannakian monodromy groups

TL;DR

This paper establishes an equidistribution theorem for Tannakian monodromy groups using perverse sheaves and Tannakian categories, advancing understanding in algebraic geometry and number theory.

Contribution

It introduces a new approach to prove equidistribution for monodromy groups via Tannakian categories and perverse sheaves, extending previous results.

Findings

Proves generic unramifiedness of perverse sheaves on algebraic groups.

Establishes an equidistribution theorem for Tannakian monodromy groups.

Provides stratification and vanishing theorems for exponential sums.

Abstract

We prove that a perverse sheaf on a connected commutatitve algebraic group over a finite is generically unramified. This implies an equidistribution theorem for Tannakian monodromy groups in previously unavailable generality. We also prove a stratification theorem for exponential sums in families indexed by a scheme and the characters of a connected commutative algebraic group. Our method is based on Tannakian categories introduced by Gabber and Loeser. This method naturally yields fiber functors. We also prove vanishing theorems over a connected commutative algebraic group, classify the negligible sheaves, and prove relative weak propagation theorems for tori.