Cited By
View all- Bartoli DMontanucci MZini G(2018)AG codes and AG quantum codes from the GGS curveDesigns, Codes and Cryptography10.1007/s10623-017-0450-586:10(2315-2344)Online publication date: 1-Oct-2018
Twists of genus three curves over finite fields
Authors: 
Stephen Meagher, Jaap TopAuthors Info & Claims
Pages 347 - 368
Abstract
In this article we recall how to describe the twists of a curve over a finite field and we show how to compute the number of rational points on such a twist by methods of linear algebra. We illustrate this in the case of plane quartic curves with at least 16 automorphisms. In particular we treat the twists of the Dyck-Fermat and Klein quartics. Our methods show how in special cases non-Abelian cohomology can be explicitly computed. They also show how questions which appear difficult from a function field perspective can be resolved by using the theory of the Jacobian variety.
References
[1]
Some genus 3 curves with many points. In: Lecture Notes in Comput. Sci., vol. 2369. Springer-Verlag, Berlin. pp. 163-171.
[2]
Buhler, J., Schoen, C. and Top, J., Cycles, L-functions and triple products of elliptic curves. J. Reine Angew. Math. v492. 93-133.
[3]
Ciani, Edgardo, I vari tipi possibili di quartiche piane più volte omologiche armoniche. Rend. Circ. Mat. Palermo. v13. 347-373.
[4]
Ciani, E. and Wieleitner, H., Projektive Spezialisierungen von Kurven vierter und dritter Ordnung. In: Pascal, E. (Ed.), Repertorium der höheren Mathematik, II: Geometrie, erste Hälfte, Grundlagen und ebene Geometrie, Teubner Verlag, Leipzig/Berlin.
[5]
Dyck, Walther, Notiz über eine reguläre Riemann'sche Fläche vom Geschlechte drei und die zugehörige "Normalcurve" vierter Ordnung. Math. Ann. v17. 510-516.
[6]
Edge, W.L., A plane quartic curve with twelve undulations. Edinburgh Math. Notes. v35. 10-13.
[7]
Elkies, Noam D., The Klein quartic in number theory. In: Math. Sci. Res. Inst. Publ., vol. 35. Cambridge University Press, Cambridge. pp. 51-101.
[8]
Kantor, S., Theorie der endlichen Gruppen von eindeutigen Transformationen in der Ebene. 1895. Mayer & Müller, Berlin.
[9]
Klassen, Matthew J. and Schaefer, Edward F., Arithmetic and geometry of the curve y3+1=x4. Acta Arith. v74. 241-257.
[10]
Klein, Felix, Ueber die Transformation siebenter Ordnung der elliptischen Functionen. Math. Ann. v14. 428-471.
[11]
Milne, J.S., Abelian varieties. In: Arithmetic Geometry, Springer-Verlag, New York. pp. 103-150.
[12]
Milne, J.S., Jacobian varieties. In: Arithmetic Geometry, Springer-Verlag, New York. pp. 167-212.
[13]
Mumford, David, Geometric Invariant Theory. 1965. Ergeb. Math. Grenzgeb., 1965.Springer-Verlag, Berlin/New York.
[14]
Neumann, Peter M., A lemma that is not Burnside's. Math. Sci. v4. 133-141.
[15]
Rubin, Karl, Modularity of mod 5 representations. In: Modular Forms and Fermat's Last Theorem, Springer-Verlag, New York. pp. 463-474.
[16]
Serre, Jean-Pierre, Local Fields. 1979. Grad. Texts in Math., 1979.Springer-Verlag, New York.
[17]
Serre, Jean-Pierre, Sur le nombre des points rationnels d'une courbe algébrique sur un corps fini. C. R. Acad. Sci. Paris Sér. I Math. v296. 397-402.
[18]
Serre, Jean-Pierre, Lectures on Curves over Finite Fields. 1985. Harvard University.
[19]
Serre, Jean-Pierre, Cohomologie Galoisienne. 1997. Lecture Notes in Math., 1997.cinquième éd. Springer-Verlag, Berlin.
[20]
Jaap Top, Hecke L-series related with algebraic cycles or with Siegel modular forms, PhD thesis, University of Utrecht, 1989.
[21]
A.M. Vermeulen, Weierstrass points of weight two on curves of genus three, PhD thesis, University of Amsterdam, 1983.
[22]
Wiman, A., Zur Theorie der endlichen Gruppen von birationalen Transformationen in der Ebene. Math. Ann. v48. 195-240.
[23]
Wright, E.M., Burnside's lemma: a historical note. J. Combin. Theory Ser. B. v30. 89-90.
- Twists of genus three curves over finite fields
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Published: 01 September 2010
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View all- Bartoli DMontanucci MZini G(2018)AG codes and AG quantum codes from the GGS curveDesigns, Codes and Cryptography10.1007/s10623-017-0450-586:10(2315-2344)Online publication date: 1-Oct-2018
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