Just-likely intersections on Hilbert modular surfaces

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 in $(x,y) \in C\times D$ with surfaces parameterized by $x$ and $y$ isogenous to each other is Zariski dense in $C\times D$, 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