The canonical representation of the Drinfeld curve
The canonical representation of the Drinfeld curve
We compute the decomposition of the canonical representation arising from the action of the group SL2 (Fq) on the Drinfeld curve over the algebraic closure of the finite field Fq for q a prime power. We first solve the problem for q = p a prime number, where methods from a recent paper by Bleher, Chinburg, and Kontogeorgis apply because SL2 (Fp) has cyclic Sylow p-subgroups. The computations simplify drastically compared to the general case treated in the paper because in addition of being cyclic, the Sylow p-subgroups have order p and their normaliser is p-hypo-elementary. This allows us to use the Green correspondence in the case of trivial intersection. Secondly, we solve the problem for q a general prime power, by computing a concrete basis for the space of global holomorphic differentials and studying the action of SL2 (Fq) on it.
University of Southampton
Laurent, Lucas, Pierre
e5e6f530-0471-4d62-9fb6-7eddae0ac54d
Laurent, Lucas, Pierre
e5e6f530-0471-4d62-9fb6-7eddae0ac54d
Koeck, Bernhard
84d11519-7828-43a6-852b-0c1b80edeef9
Laurent, Lucas, Pierre
(2021)
The canonical representation of the Drinfeld curve.
University of Southampton, Doctoral Thesis, 40pp.
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Thesis
(Doctoral)
Abstract
We compute the decomposition of the canonical representation arising from the action of the group SL2 (Fq) on the Drinfeld curve over the algebraic closure of the finite field Fq for q a prime power. We first solve the problem for q = p a prime number, where methods from a recent paper by Bleher, Chinburg, and Kontogeorgis apply because SL2 (Fp) has cyclic Sylow p-subgroups. The computations simplify drastically compared to the general case treated in the paper because in addition of being cyclic, the Sylow p-subgroups have order p and their normaliser is p-hypo-elementary. This allows us to use the Green correspondence in the case of trivial intersection. Secondly, we solve the problem for q a general prime power, by computing a concrete basis for the space of global holomorphic differentials and studying the action of SL2 (Fq) on it.
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Submitted date: September 2021
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Local EPrints ID: 457262
URI: http://eprints.soton.ac.uk/id/eprint/457262
PURE UUID: 0a57f49a-222c-4932-9a56-33212fd35ea9
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Date deposited: 30 May 2022 16:37
Last modified: 17 Mar 2024 02:53
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Author:
Lucas, Pierre Laurent
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