Just-likely intersections on Hilbert modular surfaces

Abstract

In this paper, we prove an intersection-theoretic result pertaining to curves in certain Hilbert modular surfaces in positive characteristic. Specifically, we show that given two appropriate curves C,D parameterizing abelian surfaces with real multiplication, the set of points (x,y) in the product CxD with surfaces parameterized by x and y isogenous to each other is Zariski dense in C x D, thereby proving a case of a just-likely intersection conjecture. We also compute the change in Faltings height under appropriate p-power isogenies of abelian surfaces with real multiplication over characteristic p global fields.