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Faithfulness of Actions on Riemann-Roch Spaces

Published online by Cambridge University Press:  20 November 2018

Bernhard Köck  and
Joseph Tait
Bernhard Köck
Affiliation:
Mathematical Sciences, University of Southampton, Southampton SO17 1TJ, United Kingdom e-mail: B.Koeck@soton.ac.uk, Joe.Tait@soton.ac.uk
Joseph Tait
Affiliation:
Mathematical Sciences, University of Southampton, Southampton SO17 1TJ, United Kingdom e-mail: B.Koeck@soton.ac.uk, Joe.Tait@soton.ac.uk
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Abstract

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Given a faithful action of a finite group $G$ on an algebraic curve $X$ of genus $gx\,\ge \,2$, we give explicit criteria for the induced action of $G$ on the Riemann–Roch space ${{H}^{0}}\left( X,\,{{\mathcal{O}}_{X}}\left( D \right) \right)$ to be faithful, where $D$ is a $G$-invariant divisor on $X$ of degree at least ${{2}_{gX}}\,-\,2$. This leads to a concise answer to the question of when the action of $G$ on the space ${{H}^{0}}\left( X,\,\Omega _{X}^{\otimes m} \right)$ of global holomorphic polydifferentials of order $m$ is faithful. If $X$ is hyperelliptic, we provide an explicit basis of ${{H}^{0}}\left( X,\,\Omega _{X}^{\otimes m} \right)$. Finally, we give applications in deformation theory and in coding theory and discuss the analogous problem for the action of $G$ on the first homology ${{H}_{1}}\left( X,\,\mathbb{Z}/m\mathbb{Z} \right)$ if $X$ is a Riemann surface.

Keywords

14H3030F3014L3014D1511R32faithful actionsRiemann-Roch spacespolydifferentialshyperelliptic curvesequivariant deformation theoryGoppa codeshomology
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 67 , Issue 4 , 01 August 2015 , pp. 848 - 869
Copyright
Copyright © Canadian Mathematical Society 2015

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