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Monodromy Groups of Supersingular Abelian Surfaces over $\mathbb{Q}_p$

Moqing Chen

TL;DR

The paper develops a p-adic Hodge-theoretic framework to classify filtered $\varphi$-modules arising from supersingular abelian surfaces over $\mathbb{Q}_p$ (for $p\ge7$), and then computes the associated p-adic algebraic monodromy groups. It identifies four explicit families of objects and shows that generically the neutral monodromy component is $\mathbf{GL}_2\times_{\det}\mathbf{GL}_2$, with precise non-reductive and lower-rank possibilities occurring for special parameter values. A moduli space description is given via a coarse moduli $\mathcal{M}^{\mathrm{wa}}$ and a Grassmannian open $Y^{\mathrm{def}}$, connected through a projective GIT quotient to $\mathbb{P}^1$, along with a concrete parametrization by $\mathbb{P}^2(\mathbb{Q}_p)$. The results support a geometric view of how monodromy groups vary in p-adic families and relate to broader themes in p-adic geometry and Shimura variety dynamics.

Abstract

For primes $p\ge 7$, we give a parametrization of the filtered $\varphi$-modules attached to the $p$-adic Tate modules of abelian surfaces over $\mathbb{Q}_p$ with supersingular good reduction. We use this classification to determine the neutral components of the monodromy groups of the associated $p$-adic representations up to $\bar{\mathbb{Q}}_p$-isomorphism. Furthermore, we analyze the $p$-adic distribution of these groups in the moduli space of filtered $\varphi$-modules. In particular, we prove that the neutral components are generically isomorphic to $\mathbf{GL}_2 \times_{\det} \mathbf{GL}_2$.

Monodromy Groups of Supersingular Abelian Surfaces over $\mathbb{Q}_p$

TL;DR

The paper develops a p-adic Hodge-theoretic framework to classify filtered -modules arising from supersingular abelian surfaces over (for ), and then computes the associated p-adic algebraic monodromy groups. It identifies four explicit families of objects and shows that generically the neutral monodromy component is , with precise non-reductive and lower-rank possibilities occurring for special parameter values. A moduli space description is given via a coarse moduli and a Grassmannian open , connected through a projective GIT quotient to , along with a concrete parametrization by . The results support a geometric view of how monodromy groups vary in p-adic families and relate to broader themes in p-adic geometry and Shimura variety dynamics.

Abstract

For primes , we give a parametrization of the filtered -modules attached to the -adic Tate modules of abelian surfaces over with supersingular good reduction. We use this classification to determine the neutral components of the monodromy groups of the associated -adic representations up to -isomorphism. Furthermore, we analyze the -adic distribution of these groups in the moduli space of filtered -modules. In particular, we prove that the neutral components are generically isomorphic to .

Paper Structure

This paper contains 25 sections, 34 theorems, 127 equations.

Table of Contents

  1. Introduction
  2. Main classification result
  3. A moduli space description
  4. Distribution of Wintenberger types
  5. Strategy of proof
  6. Relation to previous work and broader motivation
  7. Preliminaries
  8. Crystalline representations and filtered $\varphi$-modules
  9. Hodge decomposition for admissible filtered $\varphi$-modules
  10. Filtered $\varphi$-modules arising from geometry
  11. Tools for computating monodromy groups
  12. Filtered $\varphi$-modules arising from supersingular abelian surfaces over $\mathbb{Q}_p$
  13. Supersingular $p$-Weil polynomials of degree $4$
  14. Linear-algebraic constraints
  15. The case $\chi_{\varphi}(X)=(X^2\pm p)^2$
  16. ...and 10 more sections

Key Result

Theorem A

Let $p\ge 7$. The objects in $\mathbf C$ are precisely the filtered $\varphi$-modules $D^{prod}_{\epsilon'}$, $D^{\epsilon,iso}_{\epsilon'}$, $D_{a'}^{\epsilon,\nu}$ or $D_{(a,b)}^{\epsilon,\mu}$ constructed in Definitions prodcase and canfam, with parameters satisfying the explicit arithmetric cond

Theorems & Definitions (67)

  • Theorem A: Theorem \ref{['mainthm0']}, Theorem \ref{['mainclass']}
  • Theorem B: Theorem \ref{['mainthm']}
  • Theorem C: Theorem \ref{['propmain']}
  • Theorem D: Theorem \ref{['distmono']}
  • Theorem E: Corollary \ref{['distwintype']}
  • Definition 2.1: fontaine1979modules
  • Theorem 2.2: wintenberger1984scindage
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5: volkov2005class
  • ...and 57 more