The canonical representation of the Drinfeld curve
The canonical representation of the Drinfeld curve
If C is a smooth projective curve over an algebraically closed field F and G is a subgroup of automorphisms of C, then G acts linearly on the F-vector space of holomorphic differentials H_0(C,Ω_C) by pulling back differentials. In other words, H_0(C,Ω_C) is a representation of G over the field F, called the canonical representation of C. Computing its decomposition as a direct sum of indecomposable representations is still an open problem when the ramification of the cover of curves C⟶C/G is wild. In this paper, we compute this decomposition for C the Drinfeld curve XY^q−X^qY−Z^q+1=0, F=\bar{F}q, and G=SL2(F_q) where q is a prime power.
Koeck, Bernhard
84d11519-7828-43a6-852b-0c1b80edeef9
Laurent, Lucas Pierre
e5e6f530-0471-4d62-9fb6-7eddae0ac54d
11 August 2021
Koeck, Bernhard
84d11519-7828-43a6-852b-0c1b80edeef9
Laurent, Lucas Pierre
e5e6f530-0471-4d62-9fb6-7eddae0ac54d
[Unknown type: UNSPECIFIED]
Abstract
If C is a smooth projective curve over an algebraically closed field F and G is a subgroup of automorphisms of C, then G acts linearly on the F-vector space of holomorphic differentials H_0(C,Ω_C) by pulling back differentials. In other words, H_0(C,Ω_C) is a representation of G over the field F, called the canonical representation of C. Computing its decomposition as a direct sum of indecomposable representations is still an open problem when the ramification of the cover of curves C⟶C/G is wild. In this paper, we compute this decomposition for C the Drinfeld curve XY^q−X^qY−Z^q+1=0, F=\bar{F}q, and G=SL2(F_q) where q is a prime power.
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Accepted/In Press date: 11 August 2021
Published date: 11 August 2021
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Local EPrints ID: 470353
URI: http://eprints.soton.ac.uk/id/eprint/470353
ISSN: 2331-8422
PURE UUID: b8f7de57-38a0-4e45-a635-8539ec5bdffc
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Date deposited: 06 Oct 2022 17:12
Last modified: 17 Mar 2024 02:53
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Author:
Lucas Pierre Laurent
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