Unlikely Intersections in Finite Characteristic

TL;DR

This paper uses Honda-Tate theory and additive combinatorics to explore unlikely intersections in finite characteristic, proposing conjectures about abelian varieties and providing counterexamples to related questions.

Contribution

It introduces a heuristic argument against certain unlikely intersection conjectures and offers a negative answer to a question about Shimura varieties using additive combinatorics.

Findings

Heuristic argument based on Honda-Tate theory against many unlikely intersection conjectures.

Conjecture that every abelian variety of dimension 4 is isogenous to a Jacobian.

Counterexample to a question of Chai and Oort involving Shimura varieties.

Abstract

We present a heuristic argument based on Honda-Tate theory against many conjectures in `unlikely intersections' over the algebraic closure of a finite field; notably, we conjecture that every abelian variety of dimension 4 is isogenous to a Jacobian. Using methods of additive combinatorics, we are able to give a negative answer to a related question of Chai and Oort where the ambient Shimura Variety is a power of the modular curve.