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

Bibliographie

[BM] [BM] Bertin, J. and Mézard, A., Déformations formelles des revêtements sauvagement ramifiés de courbes algébriques. Invent. Math. 141(2000), no. 1, 195–238. Google Scholar | DOI

[Bor] [Bor] Borne, N., Cohomology of G-sheaves in positive characteristic. Adv. Math. 201(2006), no. 2, 454–515. Google Scholar | DOI

[Bro] [Bro] Broughton, S. A., The homology and higher representations of the automorphism group of a Riemann surface. Trans. Amer. Math. Soc. 300(1987), no. 1, 153–158. Google Scholar | DOI

[CW] [CW] Chevalley, C. and Weil, A., Über das Verhalten der integrale 1. Gattung bei automorphismen des Funktionenkörpers. Abh. Math. Sem. Univ. Hamb. 10(1934), no. 1, 358–361. Google Scholar | DOI

[FGM+] [FGM+] Friedlander, H., Garton, D., Malmskog, Beth, Pries, R., and Weir, C., The a-numbers of Jacobians of Suzuki curves. Proc. Amer. Math. Soc. 141(2013), no. 9, 3019–3028. Google Scholar | DOI

[FK] [FK] Farkas, H. M. and Kra, I., Riemann surfaces. Graduate Texts in Mathematics, 71, Springer-Verlag, New York, 1980. Google Scholar

[FW] [FW] Fischbacher-Weitz, H. B., Equivariant Riemann-Roch theorems for curves over perfect fields. Ph.D.Thesis, University of Southampton, 2008. Google Scholar

[FWK] [FWK] Fischbacher-Weitz, H. and Köck, B., Equivariant Riemann-Roch theorems for curves over perfect fields. Manuscripta Math. 128(2009), no. 1, 89–105. Google Scholar | DOI

[GJK] [GJK] Glass, D., Joyner, D., and Ksir, A., Codes from Riemann-Roch spaces for y2 = xp −x over GF(). Int. J. Inf. Coding Theory 1(2010), no. 3, 298–312. Google Scholar | DOI

[GK] [GK] Giulietti, M. and Korchmáros, G., On automorphism groups of certain Goppa codes. Des. Codes Cryptogr. 47(2008), no. 1–3, 177–190. Google Scholar | DOI

[Har] [Har] Hartshorne, R., Algebraic geometry. Graduate Texts in Mathematics, 52, Springer-Verlag, New York, 1977. Google Scholar

[Hor] [Hor] Hortsch, R., On the canonical representation of curves in positive characteristic. New York J. Math. 18(2012), 911–924. Google Scholar

[HS] [HS] Haas, A. and Susskind, P., The geometry of the hyperelliptic involution in genus two. Proc. Amer. Math. Soc. 105(1989), no. 1, 159–165. Google Scholar | DOI

[JK] [JK] Joyner, D. and Ksir, A., Automorphism groups of some AG codes. IEEE Trans. Inform. Theory 52(2006), no. 7, 3325–3329. Google Scholar | DOI

[Kan] [Kan] Kani, E., The Galois-module structure of the space of holomorphic differentials of a curve. J. Reine Angew. Math. 367(1986), 187–206. Google Scholar | DOI

[Kar] [Kar] Karanikolopoulos, S., On holomorphic polydifferentials in positive characteristic. Math. Nachr. 285(2012), no. 7, 852–877. Google Scholar | DOI

[KaKo] [KaKo] Karanikolopoulos, S. and Kontogeorgis, A., Representation of cyclic groups in positive characteristic and Weierstrass semigroups. J. Number Theory 133(2013), no. 1, 158–175. Google Scholar | DOI

[Köc] [Köc] Köck, B., Galois structure of Zariski cohomology for weakly ramified covers of curves. Amer. J. Math. 126(2004), no. 5, 1085–1107. Google Scholar | DOI

[KöKo] [KöKo] Köck, B. and Kontogeorgis, A., Quadratic differentials and equivariant deformation theory of curves. Ann. Inst. Fourier (Grenoble) 62(2012), no. 3, 1015–1043. Google Scholar | DOI

[Kon] [Kon] Kontogeorgis, A., Polydifferentials and the deformation functor of curves with automorphisms. J.Pure Appl. Algebra 210(2007), no. 2, 551–558. Google Scholar | DOI

[Lew] [Lew] Lewittes, J., Automophisms of compact Riemann surfaces. Amer. J. Math. 85(1963), 734–752. Google Scholar | DOI

[Liu] [Liu] Liu, Q., Algebraic geometry and arithmetic curves. Oxford Graduate Texts in Mathematics, 6, Oxford Science Publications, Oxford University Press, Oxford, 2002. Google Scholar

[Nak1] [Nak1] Nakajima, S., Action of an automorphism of order p on cohomology groups of an algebraic curve. J. Pure Appl. Algebra 42(1986), no. 1, 85–94. Google Scholar | DOI

[Nak2] [Nak2] Nakajima, S., Galois module structure of cohomology groups for tamely ramified coverings of algebraic varieties. J. Number Theory 22(1986), no. 1, 115–123. Google Scholar | DOI

[Ser] [Ser] Serre, J.-P., Local fields. Graduate Texts in Mathematics, 67, Springer-Verlag, New York, 1979. Google Scholar

[Sil] [Sil] Silverman, J. H., The arithmetic of elliptic curves. Second ed., Graduate Texts in Mathematics, 106, Springer, Dordrecht, 2009. Google Scholar

[Sti] [Sti] Stichtenoth, H., Algebraic function fields and codes. Second ed., Graduate Texts in Mathematics, 254, Springer-Verlag, Berlin, 2009. Google Scholar

[VM] [VM] Valentini, R. C. and Madan, M. L., Automorphisms and holomorphic differentials in characteristic p. J. Number Theory 13(1981), no. 1, 106–115. Google Scholar | DOI