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

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