I‑AIMS Institute for Artificial Intelligence
and Mathematical Sciences
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Theta functions and algebraic curves with automorphisms

Tanush Shaska, G. S. Wijesiri

I-AIMS paper 2009-1 · Published in NATO Sci. Peace Secur. Ser. D Inf. Commun. Secur. 24 (2009), 193–237 · 2009

Let 𝒳 be an irreducible, smooth, projective curve of genus g ≥ 2 defined over the complex field ℂ. Then there is a covering π: 𝒳 → ℙ¹, where ℙ¹ denotes the projective line. The problem of expressing branch points of the covering π in terms of the transcendentals (period matrix, thetanulls, e.g.) is classical. It goes back to Riemann, Jacobi, Picard and Rosenhein. Many mathematicians, including Picard and Thomae, have offered partial treatments for this problem. In this work, we address the problem for cyclic curves of genus 2, 3, and 4 and find relations among theta functions for curves with automorphisms. We consider curves of genus g > 1 admitting an automorphism σ such that 𝒳^σ has genus zero and σ generates a normal subgroup of the automorphism group Aut(𝒳) of 𝒳.

To characterize the locus of cyclic curves by analytic conditions on its Abelian coordinates, in other words, theta functions, we use some classical formulas, recent results of Hurwitz spaces, and symbolic computations, especially for genera 2 and 3. For hyperelliptic curves, we use Thomae’s formula to invert the period map and discover relations among the classical thetanulls of cyclic curves. For non hyperelliptic curves, we write the equations in terms of thetanulls.

Fast genus 2 curve arithmetic in the Jacobian of the curve is used in cryptography and is based on inverting the moduli map for genus 2 curves and on some other relations on theta functions. We determine similar formulas and relations for genus 3 hyperelliptic curves and offer an algorithm for how this can be done for higher genus curves. It is still to be determined whether our formulas for g = 3 can be used in cryptographic applications as in g = 2.

MSC classes: Primary 14H42; Secondary 14H30, 14G50

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References

  1. A.Nakayashiki, On the Thomae formula for ZN curves. Publications of the Research Institute for Mathematical Sciences, 33(6):987–1015, 1997.
  2. H.F. Baker, Abelian Function, Abel’s theorem and the allied theory of theta functions, Cambridge University Press, 1897.
  3. V.Z. Enolski and T. Grava, Thomae type formulae for singular ZN curves,Letters in Mathematical Physics, 76:187–214, 2006.
  4. H.M. Farkas, Generalizations of the λ function, Proceedings of the Hirzebruch 65 Conference on Algebraic Geometry, 9:231–239, 1996.
  5. J. Quer G. Cardona, Field of moduli and field of definition for curves of genus 2, computational aspects of algebraic curves, Lecture Notes Series on Computing, 13:71–83, 2005.
  6. http://www.albmath.org/algcurves/, Algebraic curves and their applications.
  7. D. Sevilla J. Gutierrez and T. Shaska, Hyperelliptic curves of genus 3 with prescribed automorphism group,Lecture Notes in Computer Science, 13:109–123, 2005.
  8. A. Krazer, Lehrbuch der thetafunctionen, Chelsea, 1970.
  9. K. Magaard and T. Shaska, Equations of genus 4 algebraic curves with automorphisms, work in progress.
  10. K. Magaard, T. Shaska, S. Shpectorov, and H. Völklein, The locus of curves with prescribed automorphism group, S¯urikaisekikenky¯usho K¯oky¯uroku, (1267):112–141, 2002, Communications in arithmetic fundamental groups (Kyoto, 1999/2001).
  11. David Mumford, Tata lectures on theta. II, volume 43 of Progress in Mathematics, Birkhäuser Boston Inc., Boston, MA, 1984, Jacobian theta functions and differential equations, With the collaboration of C. Musili, M. Nori, E. Previato, M. Stillman and H. Umemura.
  12. David Mumford, Tata lectures on theta. II, Modern Birkhäuser Classics. Birkhäuser Boston Inc., Boston, MA, 2007, Jacobian theta functions and differential equations, With the collaboration of C. Musili, M. Nori, E. Previato, M. Stillman and H. Umemura, Reprint of the 1984 original.
  13. P.Gaudry, Fast genus 2 arithmetic based on theta functions, Journal of Mathematical Cryptology, 1(3):243–265, 2007.
  14. E. Picard, Sur des fonctions de deux variables indépendantes analogues aux fonctions modulaires, Acta Mathematica, 2(1):114–135, 1883.
  15. E. Previato, T. Shaska, and G. S. Wijesiri, Thetanulls of cyclic curves of small genus, Albanian Journal of Mathematics, 1(4):265–282, 2007.
  16. H.E. Rauch and H.M.Farkas, Theta functions with applications to Riemann surfaces, Williams and Wilkins, 1974.
  17. T. Shaska and H. Völklein, Elliptic subfields and automorphisms of genus two fields, Algebra, Arithmetic and Geometry with Applications., pages 687–707, 2004.
  18. Algebraic curves and their applications http://www.albmath.org/algcurves/
  19. H. Shiga, On the representation of the Picard modular function by θ constants. i, ii, Publications of the Research Institute for Mathematical Sciences, 24(3):311–360, 1988.

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