The Galois module structure of holomorphic poly-differentials and Riemann-Roch spaces

Frauke M. Bleher; Adam Wood

doi:10.1016/j.jalgebra.2023.05.010

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The Galois module structure of holomorphic poly-differentials and Riemann-Roch spaces

Journal article

Peer reviewed

The Galois module structure of holomorphic poly-differentials and Riemann-Roch spaces

Frauke M. Bleher and Adam Wood

Journal of algebra, Vol.631, pp.756-803

10/2023

DOI: 10.1016/j.jalgebra.2023.05.010

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Abstract

Suppose X is a smooth projective geometrically irreducible curve over a perfect field k of positive characteristic p. Let G be a finite group acting faithfully on X over k such that G has non-trivial, cyclic Sylow p-subgroups. If E is a G-invariant Weil divisor on X with deg(E)>2g(X)−2, we prove that the decomposition of H0(X,OX(E)) into a direct sum of indecomposable kG-modules is uniquely determined by the class of E modulo G-invariant principal divisors, together with the ramification data of the cover X→X/G. The latter is given by the lower ramification groups and the fundamental characters of the closed points of X that are ramified in the cover. As a consequence, we obtain that if m>1 and g(X)≥2, then the kG-module structure of H0(X,ΩX⊗m) is uniquely determined by the class of a canonical divisor on X/G modulo principal divisors, together with the ramification data of X→X/G. This extends to arbitrary m>1 the m=1 case treated by the first author with T. Chinburg and A. Kontogeorgis. We discuss applications to the tangent space of the global deformation functor associated to (X,G) and to congruences between prime level cusp forms in characteristic 0. In particular, we complete the description of the precise kPSL(2,Fℓ)-module structure of all prime level ℓ cusp forms of even weight in characteristic p=3.

Cyclic Sylow subgroups

Galois module structure

Holomorphic poly-differentials

Riemann-Roch spaces

Details

Title: Subtitle: The Galois module structure of holomorphic poly-differentials and Riemann-Roch spaces
Creators: Frauke M. Bleher - University of Iowa
Adam Wood - St. Olaf College
Resource Type: Journal article
Publication Details: Journal of algebra, Vol.631, pp.756-803
Publisher: Elsevier Inc
DOI: 10.1016/j.jalgebra.2023.05.010
ISSN: 0021-8693
eISSN: 1090-266X
Grant note: DOI: 10.13039/100000001, name: National Science Foundation, award: DMS-1801328
Language: English
Electronic publication date: 05/2023
Date published: 10/2023
Academic Unit: Mathematics
Record Identifier: 9984419460002771

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