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On the Galois-module structure of polydifferentials of Artin-Schreier-Mumford curves, modular and integral representation theory

Aristides Kontogeorgis, Dimitra-Dionysia Stergiopoulou

TL;DR

The authors address the Galois-module structure of polydifferentials on Mumford curves in characteristic $p>0$ by translating holomorphic forms into harmonic cocycles and group cohomology. They establish explicit integral and modular decompositions for Artin-Schreier-Mumford curves, proving indecomposability of the $K[G]$-module of $1$-differentials and deriving detailed $K[G]$- and $K[A]$-module structures for higher polydifferentials. Two independent proofs are given for the main decompositions, one via derivations and Jordan-block analysis and another via Nakajima’s framework for weakly ramified covers, both corroborated by Köck’s projectivity theory. The results yield concrete decompositions with induced and skyscraper contributions, culminating in a precise $K[G]$-module description that highlights the injective hull structure of indecomposable summands and two distinguished non-projective components. This provides explicit, computable Galois-module structures for polydifferentials on maximal automorphism Mumford curves with potential applications to deformation theory and arithmetic geometry.

Abstract

We study the Galois-module structure of polydifferentials for Mumford curves, defined over a field of positive charactersitic, using the theory of harmonic cocycles. For the case of Artin-Schreier-Mumford curves the structure of holomorphic polydifferentials is explicitly computed.

On the Galois-module structure of polydifferentials of Artin-Schreier-Mumford curves, modular and integral representation theory

TL;DR

The authors address the Galois-module structure of polydifferentials on Mumford curves in characteristic by translating holomorphic forms into harmonic cocycles and group cohomology. They establish explicit integral and modular decompositions for Artin-Schreier-Mumford curves, proving indecomposability of the -module of -differentials and deriving detailed - and -module structures for higher polydifferentials. Two independent proofs are given for the main decompositions, one via derivations and Jordan-block analysis and another via Nakajima’s framework for weakly ramified covers, both corroborated by Köck’s projectivity theory. The results yield concrete decompositions with induced and skyscraper contributions, culminating in a precise -module description that highlights the injective hull structure of indecomposable summands and two distinguished non-projective components. This provides explicit, computable Galois-module structures for polydifferentials on maximal automorphism Mumford curves with potential applications to deformation theory and arithmetic geometry.

Abstract

We study the Galois-module structure of polydifferentials for Mumford curves, defined over a field of positive charactersitic, using the theory of harmonic cocycles. For the case of Artin-Schreier-Mumford curves the structure of holomorphic polydifferentials is explicitly computed.

Paper Structure

This paper contains 16 sections, 27 theorems, 113 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Invariants and direct factors
  4. Derivations and the group cohomology
  5. Polydifferentials on Mumford curves
  6. The space of $1$-differentials
  7. $G$-action on $1$-differentials
  8. Proof of Theorem \ref{['theoremnis1']}
  9. Computations on Artin-Schreier-Mumford curves continued.
  10. Computation of the space $H^1(\Gamma,P_n)^G$
  11. The $K[A]$-module structure of $\mathop{\mathrm{Der}}\nolimits(\Gamma, P_n)$
  12. Computations on cohomology
  13. First Proof of Theorem \ref{['mainFree']} (1)
  14. Second Proof of Theorem \ref{['mainFree']} (1)
  15. Using the theory of B. Köck. Study of the $K[A\times B]$-module structure
  16. ...and 1 more sections

Key Result

Proposition 2

The Artin-Schreier-Mumford curves are Mumford curves of the form $X_\Gamma$, where the group $\Gamma$ is, up to conjugacy, given by the commutator group $\Gamma=[A,B]$ of the cyclic subgroups $A,B \subset \mathrm{PGL}(2,K)$ of order $p$ generated by respectively, where $s\in K^{\times}$ and $|s|>1$. The groups $A$ and $B$ generate a discrete subgroup $N\subseteq\mathop{\mathrm{PGL}}\nolimits(2,K)

Figures (2)

  • Figure 1: Two lattice paths in blue color
  • Figure 2: $J_r \otimes J_{p-1}$ for $r<p-r$ (left) and $r>p-r$ (right).

Theorems & Definitions (54)

  • Definition 1
  • Proposition 2
  • Remark 3
  • Theorem 4
  • Theorem 5
  • Proposition 6
  • proof
  • Remark 7
  • Corollary 8
  • proof
  • ...and 44 more