The canonical representation of the Drinfeld curve

If \(C\) is a smooth projective curve over an algebraically closed field \(\mathbb{F}\) and \(G\) is a subgroup of automorphisms of \(C\), then \(G\) acts linearly on the \(\mathbb{F}\)-vector space of holomorphic differentials \(H^0\big(C,\Omega_C\big)\) by pulling back differentials. In other word...

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Bibliographic Details
Published inarXiv.org
Main Authors Lucas, Laurent, Köck, Bernhard
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 15.08.2024
Subjects
Automorphisms
Decomposition
Representations
Subgroups
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Summary:If \(C\) is a smooth projective curve over an algebraically closed field \(\mathbb{F}\) and \(G\) is a subgroup of automorphisms of \(C\), then \(G\) acts linearly on the \(\mathbb{F}\)-vector space of holomorphic differentials \(H^0\big(C,\Omega_C\big)\) by pulling back differentials. In other words, \(H^0\big(C,\Omega_C\big)\) is a representation of \(G\) over the field \(\mathbb{F}\), called \(\textit{the canonical representation}\) 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 \(C \longrightarrow C/G\) is wild. In this paper, we compute this decomposition for \(C\) the Drinfeld curve \({XY^q-X^qY-Z^{q+1}=0}\), \(\mathbb{F}=\bar{\mathbb{F}}_q\), and \({G=SL_2\big(\mathbb{F}_q\big)}\) where \(q\) is a prime power.
ISSN:2331-8422