Just-likely intersections on Hilbert modular surfaces

TL;DR

This paper proves a density result for points on certain Hilbert modular surfaces in positive characteristic, linking intersection theory, isogenies, and height computations in algebraic geometry.

Contribution

It establishes a Zariski density result for points corresponding to isogenous abelian surfaces on Hilbert modular surfaces, confirming a case of the just-likely intersection conjecture.

Findings

Zariski density of points with isogenous abelian surfaces

Computation of Faltings height change under p-power isogenies

Validation of a case of the just-likely intersection conjecture

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.