  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.

Algebraic Geometry;Number Theory