I‑AIMS Institute for Artificial Intelligence
and Mathematical Sciences
/

A universal genus-two curve from Siegel modular forms

Andreas Malmendier, Tanush Shaska

I-AIMS paper 2016-3 · Published in SIGMA 13 (2017), Paper 089, 17 pp. · 2017

Let 𝔭 be any point in the moduli space of genus-two curves ℳ₂ and K its field of moduli. We provide a universal equation of a genus-two curve C_(α,β), defined over K(α,β), corresponding to 𝔭, where α and β satisfy a quadratic α² + bβ² = c such that b and c are given in terms of ratios of Siegel modular forms. The curve C_(α,β) is defined over the field of moduli K if and only if the quadratic has a K-rational point (α,β). We discover some interesting symmetries of the Weierstrass equation of C_(α,β). This extends previous work of Mestre and others.

MSC classes: Primary 11F46; Secondary 14H10, 14H45

PDF

References

  1. L. Beshaj, R. Hidalgo, A. Malmendier, S. Kruk, S. Quispe, and T. Shaska, Rational points on the moduli space of genus two, Contemporary Math. (to appear) (2017), 87–120.
  2. Manjul Bhargava, John E. Cremona, Tom Fisher, Nick G. Jones, and Jonathan P. Keating, What is the probability that a random integral quadratic form in n variables has an integral zero?, Int. Math. Res. Not. IMRN (2016), no. 12, 3828–3848.
  3. Oskar Bolza, On binary sextics with linear transformations into themselves, Amer. J. Math. 10 (1887), no. 1, 47–70.
  4. A. Clebsch and P. Gordan, Theorie der Abelschen Functionen, Physica-Verlag, Würzburg, 1967, Thesaurus Mathematicae, (Neudrucke wichtiger mathematischer Werke), Band 7.
  5. E. Freitag, Siegelsche Modulfunktionen, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 254, Springer-Verlag, Berlin, 1983.
  6. V. A. Gritsenko and V. V. Nikulin, Igusa modular forms and “the simplest” Lorentzian Kac-Moody algebras, Mat. Sb. 187 (1996), no. 11, 27–66.
  7. Joe Harris and Ian Morrison, Moduli of curves, Graduate Texts in Mathematics, vol. 187, Springer-Verlag, New York, 1998.
  8. Jun-Ichi Igusa, Arithmetic variety of moduli for genus two, Ann. of Math. (2) 72 (1960), 612–649.
  9. ———, On Siegel modular forms of genus two, Amer. J. Math. 84 (1962), 175–200.
  10. ———, Modular forms and projective invariants, Amer. J. Math. 89 (1967), 817–855.
  11. ———, On the ring of modular forms of degree two over Z, Amer. J. Math. 101 (1979), no. 1, 149–183.
  12. V. Krishnamoorthy, T. Shaska, and H. Voelklein, Invariants of binary forms, Progress in Galois theory, Dev. Math., vol. 12, Springer, New York, 2005, pp. 101–122.
  13. A. Malmendier and T. Shaska, The satake sextic in elliptic fibrations on K3, Journal of Geometry and Physics (to appear) (2017), 1–20.
  14. Andreas Malmendier and David R. Morrison, K3 surfaces, modular forms, and non-geometric heterotic compactifications, Lett. Math. Phys. 105 (2015), no. 8, 1085–1118.
  15. Jean-François Mestre, Construction de courbes de genre 2 a partir de leurs modules` , E↵ective methods in algebraic geometry (Castiglioncello, 1990), Progr. Math., vol. 94, Birkhäuser Boston, Boston, MA, 1991, pp. 313–334.
  16. Jean-Pierre Serre, Spécialisation des éléments de Br2(Q(T1,· · · , Tn)), C. R. Acad. Sci. Paris Sér. I Math. 311 (1990), no. 7, 397–402.
  17. T. Shaska and L. Beshaj, Heights on algebraic curves, Advances on superelliptic curves and their applications, NATO Sci. Peace Secur. Ser. D Inf. Commun. Secur., vol. 41, IOS, Amsterdam, 2015, pp. 137–175.
  18. T. Shaska and H. Voelklein, Elliptic subfields and automorphisms of genus 2 function fields, Algebra, arithmetic and geometry with applications (West Lafayette, IN, 2000), Springer, Berlin, 2004, pp. 703–723.

All preprints