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Splitting of almost ordinary abelian surfaces in families and the $S$-integrality conjectures

Ruofan Jiang

TL;DR

The paper proves that for a non-isotrivial almost ordinary abelian surface $A$ over a global function field of odd characteristic $p$, absent global real multiplication, there exist infinitely many places where its reduction gains endomorphisms isomorphic to $\mathbb{Z}[x]/(x^2-m)$ with $m\in\boldsymbol{Δ}$ and $(\tfrac{m}{p})=1$, implying non-simplicity at infinitely many places. The approach is to reinterpret endomorphism splittings as intersections with special divisors $Z(m)$ on the Siegel moduli space $\mathcal{A}_2$, relate global intersections to coefficients of Eisenstein series, and analyze local intersections via explicit deformation theory of $p$-divisible groups, including a robust algebraization step for formal special endomorphisms in the almost ordinary and boundary settings. A major technical advance is the incorporation of logarithmic geometry and toroidal compactifications to handle boundary contributions, enabling precise local-global decay estimates that yield the infinitude result. As an application, the authors extend the $S$-integrality philosophy from elliptic curves to abelian surfaces over global function fields, providing a framework for unlikely intersections in this Shimura-variety context. The work thus unifies splitting questions for reductions, $S$-integrality phenomena, and arithmetic intersections via a blend of $p$-adic Hodge theory, modularity of special divisors, and algebraization of formal endomorphisms.

Abstract

Let $A$ be a non-isotrivial almost ordinary abelian surface with possibly bad reductions over a global function field of odd characteristic $p$. Suppose $Δ$ is an infinite set of positive integers, such that $\left(\frac{m}{p}\right)=1$ for $\forall m\in Δ$. If $A$ does not admit any global real multiplication, we prove the existence of infinitely many places modulo which the reduction of $A$ has endomorphism ring containing $\mathbb{Z}[x]/(x^2-m)$ for some $m\in Δ$. This implies that there are infinitely many places modulo which $A$ is not simple, generalizing the main result of arXiv:1812.11679 to the non-ordinary case. As an another application, we also generalize the $S$-integrality theorem for elliptic curves over number fields, as proved in arXiv:math/0509485, to the setting of abelian surfaces over global function fields.

Splitting of almost ordinary abelian surfaces in families and the $S$-integrality conjectures

TL;DR

The paper proves that for a non-isotrivial almost ordinary abelian surface over a global function field of odd characteristic , absent global real multiplication, there exist infinitely many places where its reduction gains endomorphisms isomorphic to with and , implying non-simplicity at infinitely many places. The approach is to reinterpret endomorphism splittings as intersections with special divisors on the Siegel moduli space , relate global intersections to coefficients of Eisenstein series, and analyze local intersections via explicit deformation theory of -divisible groups, including a robust algebraization step for formal special endomorphisms in the almost ordinary and boundary settings. A major technical advance is the incorporation of logarithmic geometry and toroidal compactifications to handle boundary contributions, enabling precise local-global decay estimates that yield the infinitude result. As an application, the authors extend the -integrality philosophy from elliptic curves to abelian surfaces over global function fields, providing a framework for unlikely intersections in this Shimura-variety context. The work thus unifies splitting questions for reductions, -integrality phenomena, and arithmetic intersections via a blend of -adic Hodge theory, modularity of special divisors, and algebraization of formal endomorphisms.

Abstract

Let be a non-isotrivial almost ordinary abelian surface with possibly bad reductions over a global function field of odd characteristic . Suppose is an infinite set of positive integers, such that for . If does not admit any global real multiplication, we prove the existence of infinitely many places modulo which the reduction of has endomorphism ring containing for some . This implies that there are infinitely many places modulo which is not simple, generalizing the main result of arXiv:1812.11679 to the non-ordinary case. As an another application, we also generalize the -integrality theorem for elliptic curves over number fields, as proved in arXiv:math/0509485, to the setting of abelian surfaces over global function fields.
Paper Structure (52 sections, 49 theorems, 101 equations, 1 table)

This paper contains 52 sections, 49 theorems, 101 equations, 1 table.

Table of Contents

  1. Introduction
  2. Real multiplication and splitting of reductions
  3. $S$-integrality conjectures
  4. Outline of the proof
  5. Algebraization of formal special endomorphisms
  6. Organization of the paper
  7. Notation
  8. Preliminaries
  9. $\mathcal{A}_2$ as GSpin Shimura variety
  10. Siegel moduli of abelian surfaces
  11. The mod $p$ fiber
  12. Deformation theory
  13. Explicit deformation theory of p-divisible groups
  14. Explicit computations for $\widehat{\mathbf{L}}_{{\mathrm{cris}},P}$
  15. Horizontal extension of special endomorphisms
  16. ...and 37 more sections

Key Result

Theorem 1.1

Let $\Delta$ be an infinite set of positive integers such that $\left(\frac{m}{p}\right)=1$ for $\forall m\in \Delta$. If $A$ does not admit any global real multiplication, then there are infinitely many places modulo which the endomorphism ring of the reduction of $A$ contains $\mathbb{Z}[x]/(x^2-m

Theorems & Definitions (71)

  • Theorem 1.1: Theorem \ref{['T:interMain']}
  • Example 1.2
  • Theorem 1.3: Theorem \ref{['T:maintheorem']}
  • Conjecture 1.4: $S$-integrality conjecture for abelian varieties, see Conj 1.2 of loc.cit
  • Theorem 1.7: Theorem \ref{['T:unlikelyint']}
  • Theorem 1.8
  • Theorem 1.9: Theorem \ref{['T: algebraicC']}
  • Definition 1
  • Remark 2.1
  • Remark 2.2
  • ...and 61 more