Unlikely Intersections in Finite Characteristic

Ananth Shankar, Jacob Tsimerman

Published 2016-10-11Version 1

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, contrary to a conjecture of Oort. Using methods of additive combinatorics, we are able to disprove a related conjecture of Oort where the ambient Shimura Variety is a power of the modular curve.