The Galois Structure of the spaces of polydifferentials on the Drinfeld Curve
The Galois Structure of the spaces of polydifferentials on the Drinfeld Curve
Let C be a smooth projective curve over an algebraically closed field F equipped with the action of a finite group G. When p=char(F) divides the order of G, the long-standing problem of computing the induced representation of G on the space M = H^0(C,Ω^{⊗m}_C) of globally holomorphic polydifferentials remains unsolved in general. In this paper, we study the case of the group G=SL2(F_q) (where q is a power of p) acting on the Drinfeld curve C which is the projective plane curve given by the equation XY^q−X^qY−Z^{q+1}=0. When q=p, we fully decompose M as a direct sum of indecomposable F[G]-modules. For arbitrary q, we give a partial decomposition in terms of an explicit F-basis of M. Finally, in the appendix, we compute the a-number and p-rank of the Drinfeld curve.
Galois module structure, polydifferentials, Drinfeld curve, Green correspondence, p-rank, a-number
Koeck, Bernhard
84d11519-7828-43a6-852b-0c1b80edeef9
Marchment, Denver-James Logan
2f9e2454-9f8c-4640-9c7e-59afece3f22c
22 June 2026
Koeck, Bernhard
84d11519-7828-43a6-852b-0c1b80edeef9
Marchment, Denver-James Logan
2f9e2454-9f8c-4640-9c7e-59afece3f22c
Koeck, Bernhard and Marchment, Denver-James Logan
(2026)
The Galois Structure of the spaces of polydifferentials on the Drinfeld Curve.
arXiv.
Abstract
Let C be a smooth projective curve over an algebraically closed field F equipped with the action of a finite group G. When p=char(F) divides the order of G, the long-standing problem of computing the induced representation of G on the space M = H^0(C,Ω^{⊗m}_C) of globally holomorphic polydifferentials remains unsolved in general. In this paper, we study the case of the group G=SL2(F_q) (where q is a power of p) acting on the Drinfeld curve C which is the projective plane curve given by the equation XY^q−X^qY−Z^{q+1}=0. When q=p, we fully decompose M as a direct sum of indecomposable F[G]-modules. For arbitrary q, we give a partial decomposition in terms of an explicit F-basis of M. Finally, in the appendix, we compute the a-number and p-rank of the Drinfeld curve.
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Published date: 22 June 2026
Keywords:
Galois module structure, polydifferentials, Drinfeld curve, Green correspondence, p-rank, a-number
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Local EPrints ID: 512676
URI: http://eprints.soton.ac.uk/id/eprint/512676
ISSN: 2331-8422
PURE UUID: 168092e3-725c-492c-8910-f99730df02e7
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Date deposited: 14 Jul 2026 16:55
Last modified: 09 Aug 2026 00:53
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Author:
Denver-James Logan Marchment
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