I‑AIMS Institute for Artificial Intelligence
and Mathematical Sciences
/

Determining equations of families of cyclic curves

R. Sanjeewa, Tanush Shaska

I-AIMS paper 2008-3 · Published in Albanian J. Math. 2 (2008), 199–213 · 2008

In previous work we determined automorphism groups of cyclic algebraic curves defined over fields of any odd characteristic. In this paper we determine parametric equations of families of curves for each automorphism group for such curves.

MSC classes: Primary 14Hxx; Secondary 14H37, 14H10

PDF

References

  1. R. Brandt and H. Stichtenoth, Die Automorphismengruppen hyperelliptischer Kurven, Man. Math 55 (1986), 83–92.
  2. E. Bujulance, J. Gamboa, and G. Gromadzki, The full automorphism groups of hyperelliptic Riemann surfaces, Manuscripta Math. 79 (1993), no. 3-4, 267–282.
  3. W. Baily, On the automorphism group of a generic curve of genus > 2. J. Math. Kyoto Univ. 1 1961/1962 101–108; correction, 325.
  4. Y. Demirbas, Automorphism groups of hyperelliptic curves of genus 3 in characteristic 2, Computational aspects of algebraic curves, T. Shaska (Edt), Lect. Notes in Comp., World Scientific, 2005.
  5. Magaard, K.; Shaska, T.; Shpectorov, S.; Völklein, H.; The locus of curves with prescribed automorphism group. Communications in arithmetic fundamental groups (Kyoto, 1999/2001). Sürikaisekikenkyüsho Kökyüroku No. 1267 (2002), 112–141.
  6. Miller, G. A.; Blichfeldt, H. F.; Dickson, L. E.; Theory and applications of finite groups. (English) 2. ed. XVII + 390 p. New York, Stechert. Published: 1938
  7. P. G. Henn, Die Automorphismengruppen der algebraischen Funktionenkörper vom Geschlecht 3, PhD thesis, University of Heidelberg, (1976).
  8. P. Roquette, Abschtzung der Automorphismenanzahl von Funktionenkrpern bei Primzahlcharakteristik. Math. Z. 117 1970 157–163.
  9. A. Kontogeorgis, The Group of Automorphisms of Cyclic Extensions of Rational Function Fields, J. Algebra 216(2) (1999), 665-706.
  10. C. R. Valentini and L. M. Madan, A Hauptsatz of L. E. Dickson and Artin-Scheier extension, J. Reine Angew. Math. 318 (1980), 156-177.
  11. T. Shaska, Subvarieties of the Hyperelliptic Moduli Determined by Group Actions, Serdica Math. J. 32 (2006), 355-374.
  12. T. Shaska, Some special families of hyperelliptic curves. J. Algebra Appl. 3, 1 (2004), 75-89.
  13. T. Shaska, Determining the automorphism groups of hyperelliptic curves. Proceeding of the 2003 International Symposium on Symbolic Algebraic Computation, ACM Press, 2003, 248- 254.
  14. T. Shaska and J. Thompson, On the generic curve of genus 3. Affine algebraic geometry, 233–243, Contemp. Math., 369, Amer. Math. Soc., Providence, RI, 2005.
  15. T. Shaska and H. Völklein, Elliptic subfields and automorphisms of genus 2 function fields. Algebra, arithmetic and geometry with applications (West Lafayette, IN, 2000), 703–723, Springer, Berlin, 2004.
  16. E. Previato, T. Shaska, and S. Wijesiri, Thetanulls of cyclic curves of small genus, Albanian J. Math., Special issue on computational algebraic geometry, vol. 1, Nr. 4, 2007, pg. 265-282.
  17. T. Shaska, Some open problems in computational algebraic geometry, Albanian J. Math., Special issue on computational algebraic geometry, vol 1, Nr. 4, 2007, 309-321.
  18. T. Shaska and C. Shor, Codes over Fp2 and Fp×Fp, lattices, and theta functions, Advances in Coding Theory and Cryptology, vol 3. (2007), pg. 70-80.
  19. T. Shaska and Q. Wang, On the automorphism groups of some AG-codes based on Cab curves, Serdica Journal of Computing, 2007, vol. 1. pg. 193-206.
  20. T. Shaska and D. Sevilla, Hyperelliptic curves with reduced automorphism group A5, Appl. Algebra Engrg. Comm. Comput., (2007), vol. 1, pg. 3-20.
  21. J. Gutierrez and T. Shaska, Hyperelliptic curves with extra involutions, LMS J. of Comput. Math., 8 (2005), 102-115.
  22. T. Shaska and S. Zheng, A Maple package for hyperelliptic curves, Ed. I. Kotsieras, Maple conference, 2005, pg. 161-175.
  23. J. Gutierrez, D. Sevilla and T. Shaska, Hyperelliptic curves of genus 3 with prescribed automorphism group, Lect. Notes in Computing, vol 13. (2005), pg. 201-225.
  24. T. Shaska, Genus 2 curves covering elliptic curves, a computational approach, Lect. Notes in Computing, vol 13. (2005), pg. 151-195.
  25. T. Shaska, Genus 2 fields with degree 3 elliptic subfields, Forum. Math., vol. 16, 2, pg. 263-280, 2004.
  26. T. Shaska, Computational algebra and algebraic curves, ACM, SIGSAM Bulletin, Comm. Comp. Alg., Vol. 37, No. 4, 117-124, 2003.
  27. T. Shaska, Computational aspects of hyperelliptic curves, Computer mathematics. Proceedings of the sixth Asian symposium (ASCM 2003), Beijing, China, April 17-19, 2003. River Edge, NJ: World Scientific. Lect. Notes Ser. Comput. 10, 248-257 (2003).
  28. T. Shaska, Determining the automorphism group of a hyperelliptic curve, Proceedings of the 2003 International Symposium on Symbolic and Algebraic Computation, ACM Press, pg. 248 - 254, 2003.
  29. R. Sanjeewa and T. Shaska, Automorphism groups of cyclic curves, (preprint)
  30. H. Stichtenoth, ber die Automorphismengruppe eines algebraischen Funktionenkrpers von Primzahlcharakteristik. I. Eine Abschtzung der Ordnung der Automorphismengruppe. Arch. Math. (Basel) 24 (1973) 527–544.

All preprints