Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A characteristic p analogue of the André--Pink--Zannier conjecture

Yeuk Hay Joshua Lam, Ananth N. Shankar

TL;DR

The paper develops a characteristic $p$ analogue of the André--Pink--Zannier conjecture for ordinary points on Shimura varieties of Hodge type, proving that for points with big monodromy, any infinite prime-to-$p$ Hecke orbit has Zariski closure equal to a finite union of connected components. It also establishes an algebraic analogue of Hecke equidistribution in this setting and provides a function-field formulation over $F=ar{oldsymbol{F}}_p(t)$, together with an application to isogeny finiteness for Jacobians. The strategy combines global–local analysis of prime-to-$p$ Hecke correspondences, degree bounds for translates, and Tate-linearity obtained via Weyl-special points and Chai rigidity, supported by parabolicity results. The results yield a robust positive-characteristic framework for unlikely intersections in Shimura varieties and open avenues for broader equidistribution phenomena in this setting, with concrete consequences for isogeny classes of abelian varieties over function fields.

Abstract

We investigate the analogue of the André--Pink--Zannier conjecture in characteristic $p$. Precisely, we prove it for ordinary function field-valued points with big monodromy, in Shimura varieties of Hodge type. We also prove an algebraic characteristic $p$ analogue of Hecke-equidistribution (as formulated by Mazur) for Shimura varieties of Hodge type. We prove our main results by a global and local analysis of prime-to-$p$ Hecke correspondences, and by showing that Weyl special points are abundant in positive characterstic.

A characteristic p analogue of the André--Pink--Zannier conjecture

TL;DR

The paper develops a characteristic analogue of the André--Pink--Zannier conjecture for ordinary points on Shimura varieties of Hodge type, proving that for points with big monodromy, any infinite prime-to- Hecke orbit has Zariski closure equal to a finite union of connected components. It also establishes an algebraic analogue of Hecke equidistribution in this setting and provides a function-field formulation over , together with an application to isogeny finiteness for Jacobians. The strategy combines global–local analysis of prime-to- Hecke correspondences, degree bounds for translates, and Tate-linearity obtained via Weyl-special points and Chai rigidity, supported by parabolicity results. The results yield a robust positive-characteristic framework for unlikely intersections in Shimura varieties and open avenues for broader equidistribution phenomena in this setting, with concrete consequences for isogeny classes of abelian varieties over function fields.

Abstract

We investigate the analogue of the André--Pink--Zannier conjecture in characteristic . Precisely, we prove it for ordinary function field-valued points with big monodromy, in Shimura varieties of Hodge type. We also prove an algebraic characteristic analogue of Hecke-equidistribution (as formulated by Mazur) for Shimura varieties of Hodge type. We prove our main results by a global and local analysis of prime-to- Hecke correspondences, and by showing that Weyl special points are abundant in positive characterstic.
Paper Structure (20 sections, 29 theorems, 9 equations)

This paper contains 20 sections, 29 theorems, 9 equations.

Table of Contents

  1. Introduction
  2. An application: isogenous Jacobians
  3. Outline of proof of Theorem \ref{['thm: intro curve']}
  4. Prior and related work.
  5. Organization of the paper
  6. Acknowledgments
  7. Setup
  8. Shimura varieties
  9. Hecke correspondences
  10. Connected components
  11. Reduction to the case of maximal monodromy
  12. Galois-theoretic properties
  13. Stability under Hecke
  14. Degrees of subvarieties
  15. Hecke stable subvarieties
  16. ...and 5 more sections

Key Result

Theorem 1.2

Suppose that $S_K(G,X)$ intersects the ordinary locus of $\mathcal{A}_g$. Let $x\in S_K(G,X)(\overline{{\mathbb{F}}_p(t)})$ be an ordinary point with big monodromy. Let $\Sigma$ be an infinite subset of the prime-to-$p$ Hecke orbit of $x$. Then, the Zariski closure of $\Sigma$ is a finite union of c

Theorems & Definitions (71)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 1.4: Mazur
  • Conjecture 1.5: Algebraic equidistrbution of Hecke orbits
  • Theorem 1.6
  • Corollary 1.7
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • ...and 61 more