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 p. Specifically, let C, D be two proper curves inside a mod p Hilbert modular surface associated to a real quadratic field split at p. Suppose that the curves are generically ordinary, and that at least one of them is ample. Then, the set of points in \((x,y) \in C\times D\) $(x,y)\in C\times D$ with abelian surfaces parameterized by x and y isogenous to each other is Zariski dense in \(C\times D\) $C\times D$ , 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.