# Bounds for the stalks of perverse sheaves in characteristic p and a conjecture of Shende and Tsimerman

## TL;DR

This paper establishes bounds on the stalks of perverse sheaves in characteristic p using intersection multiplicities, proving a conjecture related to Betti numbers and equidistribution in algebraic geometry.

## Contribution

It provides a characteristic p analogue of Massey's result and proves a conjecture of Shende and Tsimerman, linking perverse sheaves, intersection theory, and number theory.

## Key findings

- Bounds on stalk dimensions of perverse sheaves in characteristic p
- Proof of Shende and Tsimerman's conjecture on Betti numbers
- Implications for the Michel-Venkatesh mixing conjecture

## Abstract

We prove a characteristic p analogue of a result of Massey which bounds the dimensions of the stalks of a perverse sheaf in terms of certain intersection multiplicities of the characteristic cycle of that sheaf. This uses the construction of the characteristic cycle of a perverse sheaf in characteristic p by Saito. We apply this to prove a conjecture of Shende and Tsimerman on the Betti numbers of the intersections of two translates of theta loci in a hyperelliptic Jacobian. This implies a function field analogue of the Michel-Venkatesh mixing conjecture about the equidistribution of CM points on a product of two modular curves.