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Holomorphic differentials of alternating four covers

Frauke M. Bleher, Margarita Bustos Gonzalez

TL;DR

This work addresses the problem of determining the exact $kG$-module structure of the space of holomorphic differentials on a smooth projective curve $X$ over a field $k$ of characteristic two, when a group $G\cong A_4$ acts faithfully and the $H$-subcover is totally ramified with $X/H\cong \mathbb{P}^1_k$. By classifying alternating four covers in characteristic two and analyzing ramification via Artin-Schreier and Kummer theory, the authors prove that $H^0(X,\Omega_X)$ decomposes into an explicit and infinite family of indecomposable $kG$-modules, with multiplicities governed by lower ramification data and the action of an order-$3$ element on branch generators. The main results include precise descriptions of summands like $N_{2l_y,\ast,i}$, $M_{3,1,i}$, $S_i$, and $B_{6l,\phi^3}$, and corollaries for Harbater-Katz-Gabber $A_4$-covers, thereby providing a complete picture of the module structure in this tame setting. The appendix complements the geometric results with a detailed account of string and band modules for $kH$ and $kG$, along with induction/restriction relations, forming a robust algebraic framework for these decompositions.

Abstract

Suppose $k$ is an algebraically closed field of characteristic two, let $A_4$ be an alternating group on four letters, and let $H$ be the unique Sylow two-subgroup of $A_4$. Let $X$ be a smooth projective irreducible curve over $k$ with a faithful $A_4$-action such that the quotient curve $X/H$ is a projective line and the $H$-cover $X\to X/H$ is totally ramified, in the sense that it is ramified and every branch point is totally ramified. Under these assumptions, we determine the precise $kA_4$-module structure of the space of holomorphic differentials of $X$ over $k$. We show that there are infinitely many different isomorphism classes of indecomposable $kA_4$-modules that can occur as direct summands, and we give precise formulas for the multiplicities with which they occur.

Holomorphic differentials of alternating four covers

TL;DR

This work addresses the problem of determining the exact -module structure of the space of holomorphic differentials on a smooth projective curve over a field of characteristic two, when a group acts faithfully and the -subcover is totally ramified with . By classifying alternating four covers in characteristic two and analyzing ramification via Artin-Schreier and Kummer theory, the authors prove that decomposes into an explicit and infinite family of indecomposable -modules, with multiplicities governed by lower ramification data and the action of an order- element on branch generators. The main results include precise descriptions of summands like , , , and , and corollaries for Harbater-Katz-Gabber -covers, thereby providing a complete picture of the module structure in this tame setting. The appendix complements the geometric results with a detailed account of string and band modules for and , along with induction/restriction relations, forming a robust algebraic framework for these decompositions.

Abstract

Suppose is an algebraically closed field of characteristic two, let be an alternating group on four letters, and let be the unique Sylow two-subgroup of . Let be a smooth projective irreducible curve over with a faithful -action such that the quotient curve is a projective line and the -cover is totally ramified, in the sense that it is ramified and every branch point is totally ramified. Under these assumptions, we determine the precise -module structure of the space of holomorphic differentials of over . We show that there are infinitely many different isomorphism classes of indecomposable -modules that can occur as direct summands, and we give precise formulas for the multiplicities with which they occur.
Paper Structure (14 sections, 14 theorems, 223 equations)

This paper contains 14 sections, 14 theorems, 223 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Totally ramified Klein four covers in characteristic two
  4. Cyclic covers of degree three in characteristic two
  5. Holomorphic differentials of alternating four covers in characteristic two
  6. Examples
  7. Example 1
  8. Example 2
  9. Example 3
  10. Appendix: Indecomposable modules in characteristic two
  11. Basic string algebras over $k$
  12. Indecomposable $kH$-modules
  13. Indecomposable $kG$-modules
  14. Induction and restriction of modules for $kH$ and $kG$

Key Result

Theorem 1.1

Let $G$ be an alternating four group with unique Klein four Sylow $2$-subgroup $H$. Suppose $X\to X/G$ is a $G$-cover of smooth projective curves such that the $H$-subcover $X\to X/H$ is totally ramified and $X/H=\mathbb{P}^1_k$. Then the $kG$-module structure of $\mathrm{H}^0(X,\Omega_{X/k})$ is fu More precisely, the isomorphism classes of the indecomposable $kG$-modules that actually occur as d

Theorems & Definitions (44)

  • Theorem 1.1
  • Remark 2.3
  • Remark 2.4
  • Remark 2.6
  • Remark 2.8
  • Remark 2.9
  • Remark 2.10
  • Lemma 2.11
  • proof
  • Theorem 2.12
  • ...and 34 more