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

Ruofan Jiang

TL;DR

The paper develops a characteristic p analogue of two central conjectures—Mumford–Tate and André–Oort—for ordinary mod p Shimura varieties of Hodge type, introducing a unified framework built on Tate-linearity, Ax–Schanuel principles, and Serre–Tate theory. It proves these conjectures for products of GSpin Shimura varieties by reducing to an Ax–Schanuel-type conjecture and leveraging monodromy and endomorphism data via Kuga–Satake (Kuga–Satake) constructions, parabolicity results, and compatible coefficient systems. A key conceptual advance is the Tate-linear perspective in characteristic p, which ties local Serre–Tate geometry to global monodromy and endomorphism constraints, allowing a modular reduction to lower-dimensional cases and a detailed analysis of two-factor interactions. The results provide a robust mod p analogue of unlikely intersections in a Hodge-type setting, with a geometric squeeze theorem and rigidity principles that yield a full AO description on ordinary strata, and they set the stage for further extensions beyond GSpin products. Overall, the work bridges p-adic monodromy, deformation theory, and unlikely intersections in a novel characteristic p framework, offering new tools for understanding mod p Shimura subvarieties and their special factors.

Abstract

We formulate characteristic $p$ analogues of the Mumford--Tate and the André--Oort conjectures for ordinary mod $p$ Shimura varieties of Hodge type, and set up general frameworks for studying them. We prove the two conjectures for (subvarieties of) arbitrary products of GSpin Shimura varieties, by reducing them, via a notion of linearity for mod $p$ Shimura varieties, to a third conjecture of Ax--Schanuel type. Along the way, we solve Chai's Tate-linear conjecture for products of GSpin Shimura varieties, and reveal an intimate relation among the four conjectures mentioned above. Our proof uses Crew's parabolicity conjecture which is recently proven by D'Addezio.

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

TL;DR

The paper develops a characteristic p analogue of two central conjectures—Mumford–Tate and André–Oort—for ordinary mod p Shimura varieties of Hodge type, introducing a unified framework built on Tate-linearity, Ax–Schanuel principles, and Serre–Tate theory. It proves these conjectures for products of GSpin Shimura varieties by reducing to an Ax–Schanuel-type conjecture and leveraging monodromy and endomorphism data via Kuga–Satake (Kuga–Satake) constructions, parabolicity results, and compatible coefficient systems. A key conceptual advance is the Tate-linear perspective in characteristic p, which ties local Serre–Tate geometry to global monodromy and endomorphism constraints, allowing a modular reduction to lower-dimensional cases and a detailed analysis of two-factor interactions. The results provide a robust mod p analogue of unlikely intersections in a Hodge-type setting, with a geometric squeeze theorem and rigidity principles that yield a full AO description on ordinary strata, and they set the stage for further extensions beyond GSpin products. Overall, the work bridges p-adic monodromy, deformation theory, and unlikely intersections in a novel characteristic p framework, offering new tools for understanding mod p Shimura subvarieties and their special factors.

Abstract

We formulate characteristic analogues of the Mumford--Tate and the André--Oort conjectures for ordinary mod Shimura varieties of Hodge type, and set up general frameworks for studying them. We prove the two conjectures for (subvarieties of) arbitrary products of GSpin Shimura varieties, by reducing them, via a notion of linearity for mod Shimura varieties, to a third conjecture of Ax--Schanuel type. Along the way, we solve Chai's Tate-linear conjecture for products of GSpin Shimura varieties, and reveal an intimate relation among the four conjectures mentioned above. Our proof uses Crew's parabolicity conjecture which is recently proven by D'Addezio.
Paper Structure (75 sections, 55 theorems, 111 equations)

This paper contains 75 sections, 55 theorems, 111 equations.

Table of Contents

  1. Introduction
  2. Main results I: the Mumford--Tate and the André--Oort conjectures
  3. Main results II: the Tate-linear and the Ax--Schanuel conjectures
  4. Linearity in char 0
  5. Linearity in char $p$
  6. Proof strategies for Theorem \ref{['Mainthm:MTGSpin']} and Theorem \ref{['MainThm:genTlGSpin']}
  7. Comparison with the Mumford--Tate conjecture in characteristic 0
  8. Proof strategies for Theorem \ref{['Mainthm:AOGSpin']}
  9. Organization of the paper
  10. Notations and conventions
  11. Preliminaries
  12. Shimura varieties of Hodge type and special subvarieties
  13. Shimura varieties of Hodge type and integral canonical models
  14. Special subvarieties and naïve integral models
  15. GSpin Shimura varieties
  16. ...and 60 more sections

Key Result

Theorem 1.3

Conjecture conj:MTforAg is true if $f_0$ factors through the naïve integral model $\mathscr{S}_{\mathbf{I},\mathbb{F}_q}$.

Theorems & Definitions (115)

  • Conjecture 1.1: Characteristic $p$ analogue of the Mumford--Tate conjecture for ordinary Shimura varieties of Hodge type
  • Conjecture 1.2: Characteristic $p$ analogue of the André--Oort conjecture for ordinary Shimura varieties of Hodge type
  • Theorem 1.3: $=$Theorem \ref{['thm:MTTllinear2']}. Characteristic $p$ Mumford--Tate conjecture for ordinary strata of products of GSpin Shimura varieties
  • Theorem 1.4: Characteristic $p$ André--Oort conjecture for ordinary GSpin Shimura varieties
  • Example 1.5
  • Theorem 1.6: Characteristic $p$ André--Oort conjecture for products of ordinary modular curves
  • Theorem 1.7: $=$ Theorem \ref{['thm:AOprod']}. Characteristic $p$ André--Oort conjecture for ordinary strata of products of GSpin Shimura varieties
  • Conjecture 1.8: Tate-linear conjecture, see Ch03
  • Conjecture 1.9: Characteristic $p$ analogue of the Ax--Schanuel conjecture for ordinary Shimura varieties of Hodge type
  • Theorem 1.10: $=$Proposition \ref{['prop:MTimpliesTl']}
  • ...and 105 more