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Just-likely intersections on Hilbert modular surfaces

Asvin G., Qiao He, Ananth N. Shankar

TL;DR

The paper establishes a just-likely intersection density result for Hilbert modular surfaces in characteristic $p$, proving that two generically ordinary curves parameterizing abelian surfaces with real multiplication yield a Zariski-dense set of pairs whose parameterized abelian surfaces are isogenous via powers of a split Frobenius. The authors develop a local product structure and a partial Frobenius framework to control local intersection numbers, and they connect these local intersections to global counts through intersections with the non-ordinary locus. A key technical advance is an exact formula for the change of Faltings height under $p$-power inseparable isogenies of abelian surfaces with $\mathcal O_F$-action, which drives the global intersection estimates when passing along $\pi_1$ and $\pi_2$-orbits. Together, these results yield density statements for isogeny relations on $C\times D$ and demonstrate exponential height growth under inseparable isogenies, highlighting deep links between Frobenius dynamics, intersection theory, and height theory 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.

Just-likely intersections on Hilbert modular surfaces

TL;DR

The paper establishes a just-likely intersection density result for Hilbert modular surfaces in characteristic , proving that two generically ordinary curves parameterizing abelian surfaces with real multiplication yield a Zariski-dense set of pairs whose parameterized abelian surfaces are isogenous via powers of a split Frobenius. The authors develop a local product structure and a partial Frobenius framework to control local intersection numbers, and they connect these local intersections to global counts through intersections with the non-ordinary locus. A key technical advance is an exact formula for the change of Faltings height under -power inseparable isogenies of abelian surfaces with -action, which drives the global intersection estimates when passing along and -orbits. Together, these results yield density statements for isogeny relations on and demonstrate exponential height growth under inseparable isogenies, highlighting deep links between Frobenius dynamics, intersection theory, and height theory 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.
Paper Structure (9 sections, 14 theorems, 17 equations)

This paper contains 9 sections, 14 theorems, 17 equations.

Table of Contents

  1. Introduction
  2. Outline of proof
  3. Change of Faltings height under $p$-power isogeny
  4. Background material
  5. Partial Frobenius structure and coordinates
  6. Local intersection on varieties with a split Frobenius
  7. Change of height under $p$-power isogenies
  8. The non-ordinary locus.
  9. Just-likely intersections on Hilbert Modular Surfaces

Key Result

Theorem 1.1

Let $F$ denote a real quadratic field, and let $p$ denote a rational prime that splits in $F$. Let $\mathcal{H}$ denote the mod $p$ Hilbert modular surface associated to $\mathcal{O}_F$, and let $C,D\subset \mathcal{H}$ denote two generically ordinary, proper curves in $\mathcal{H}$, at least one of

Theorems & Definitions (31)

  • Conjecture 1
  • Theorem 1.1
  • Definition 1
  • Lemma 2.1
  • proof
  • Definition 2
  • Proposition 2.2
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • ...and 21 more