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$p$-adic monodromy and mod $p$ unlikely intersections, II

Ruofan Jiang

TL;DR

The paper addresses how char p phenomena govern the interplay between monodromy and unlikely intersections for ordinary abelian varieties in characteristic p. It proves MT_p ⇔ logAL_p and MT_p ⇒ geoAO_p, situating these results within a Tate-conjecture framework and extending them via both representation-theoretic and algebraization techniques. A central innovation is the crystalline Hodge loci and Tate-linear loci, which connect p-adic Hodge theory, formal torus geometry, and rigid-analytic/algebraic methods to achieve logAL_p results and MT_p in new settings, including abelian fourfolds of p-adic Mumford type. The work also adapts classical methods (Commelin, Pink, Tankeev, Moonen–Zarhin) to the mod p/p-adic context and develops a Tate-linear algebraization program that leverages p-adic Hodge theory to prove MT_p and geoAO_p in broader families, with potential applications to Picard rank jumping and S-integrality problems in forthcoming work.

Abstract

We study ordinary abelian schemes in characteristic $p$ and their moduli spaces from the perspective of char $p$ Mumford--Tate, log Ax--Lindemann, and geometric André--Oort conjectures (abbreviated as $\MTT_p$, $\mathrm{logAL}_p$ and geoAO$_p$). In this paper, we achieve multiple goals: (\textbf{A}) establish the implication $\mathrm{MT}_p\Leftrightarrow \mathrm{logAL}_p \Rightarrow \mathrm{geoAO_p}$, and show that they all follow from the Tate conjecture for abelian varieties. The equivalence $\mathrm{MT}_p\Leftrightarrow \mathrm{logAL}_p$ is exploited from both sides, which enables us to \noindent(\textbf{B}) develop a representation theory approach to $\mathrm{logAL}_p$ and $\mathrm{geoAO_p}$ by first establishing many cases of MT$_p$ via classical techniques, and (\textbf{C}) develop an algebraization approach to $\MTT_p$ that transcends the limitation of classical methods. In particular, we introduce ``crystalline Hodge loci'', a rigid analytic geometric object that encodes the essential information needed for proving $\mathrm{logAL}_p$, while being very approachable via (integral and relative) $p$-adic Hodge theory. This enables us to prove $\mathrm{logAL}_p$ for compact Tate-linear curves with unramified $p$-adic monodromy. As an application, we establish $\MTT_p$ for many abelian fourfolds of $p$-adic Mumford type.

$p$-adic monodromy and mod $p$ unlikely intersections, II

TL;DR

The paper addresses how char p phenomena govern the interplay between monodromy and unlikely intersections for ordinary abelian varieties in characteristic p. It proves MT_p ⇔ logAL_p and MT_p ⇒ geoAO_p, situating these results within a Tate-conjecture framework and extending them via both representation-theoretic and algebraization techniques. A central innovation is the crystalline Hodge loci and Tate-linear loci, which connect p-adic Hodge theory, formal torus geometry, and rigid-analytic/algebraic methods to achieve logAL_p results and MT_p in new settings, including abelian fourfolds of p-adic Mumford type. The work also adapts classical methods (Commelin, Pink, Tankeev, Moonen–Zarhin) to the mod p/p-adic context and develops a Tate-linear algebraization program that leverages p-adic Hodge theory to prove MT_p and geoAO_p in broader families, with potential applications to Picard rank jumping and S-integrality problems in forthcoming work.

Abstract

We study ordinary abelian schemes in characteristic and their moduli spaces from the perspective of char Mumford--Tate, log Ax--Lindemann, and geometric André--Oort conjectures (abbreviated as , and geoAO). In this paper, we achieve multiple goals: (\textbf{A}) establish the implication , and show that they all follow from the Tate conjecture for abelian varieties. The equivalence is exploited from both sides, which enables us to \noindent(\textbf{B}) develop a representation theory approach to and by first establishing many cases of MT via classical techniques, and (\textbf{C}) develop an algebraization approach to that transcends the limitation of classical methods. In particular, we introduce ``crystalline Hodge loci'', a rigid analytic geometric object that encodes the essential information needed for proving , while being very approachable via (integral and relative) -adic Hodge theory. This enables us to prove for compact Tate-linear curves with unramified -adic monodromy. As an application, we establish for many abelian fourfolds of -adic Mumford type.

Paper Structure

This paper contains 62 sections, 63 theorems, 58 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main results
  3. Main results in category $\mathrm{(a)}$
  4. Main results in category $\mathrm{(b)}$
  5. Main results in category $\mathrm{(c)}$
  6. Tate-linear subvarieties and the crystalline Hodge loci
  7. Tate-linear loci
  8. Geometry of the Tate-linear loci
  9. Organization of the paper
  10. Preliminaries
  11. Shimura varieties of Hodge type and special subvarieties
  12. Special subvarieties and naïve integral models
  13. Special subvarieties in the mod $p$ fiber
  14. Local coordinates on mixed characteristic Shimura varieties
  15. Canonical coordinates
  16. ...and 47 more sections

Key Result

Theorem 1.7

The following are true:

Figures (1)

  • Figure 1: A web of connections

Theorems & Definitions (138)

  • Definition 1.1
  • Conjecture 1.2: Char $p$ Mumford--Tate conjecture
  • Definition 1.3
  • Conjecture 1.4: Tate-linear conjecture
  • Conjecture 1.5: Char $p$ log Ax--Lindemann conjecture
  • Conjecture 1.6: Char $p$ geometric André--Oort conjecture
  • Theorem 1.7: $\subseteq$ Theorem \ref{['thm:logALimpliesMT']}+Theorem \ref{['thm:MTimpliesAO']}+Proposition \ref{['prop:relationbetweencycleconjectures']}
  • Corollary 1.8
  • Corollary 1.8
  • Theorem 1.9: $=$ Theorem \ref{['modpCommelin']}
  • ...and 128 more