Faithfulness of Actions on Riemann-Roch Spaces

Köck, Bernhard ; Tait, Joseph

Canadian journal of mathematics, Tome 67 (2015) no. 4, pp. 848-869

Résumé

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.

Détail

DOI : 10.4153/CJM-2014-015-2

Mots-clés : 14H30, 30F30, 14L30, 14D15, 11R32, faithful actions, Riemann-Roch spaces, polydifferentials, hyperelliptic curves, equivariant deformation theory, Goppa codes, homology