Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 13 (2017), 089, 17 pages      arXiv:1607.08294
Contribution to the Special Issue on Modular Forms and String Theory in honor of Noriko Yui

A Universal Genus-Two Curve from Siegel Modular Forms

Andreas Malmendier a and Tony Shaska b
a) Department of Mathematics and Statistics, Utah State University, Logan, UT 84322, USA
b) Department of Mathematics and Statistics, Oakland University, Rochester, MI 48309, USA

Received July 18, 2017, in final form November 25, 2017; Published online November 30, 2017

Let $\mathfrak p$ be any point in the moduli space of genus-two curves $\mathcal M_2$ and $K$ its field of moduli. We provide a universal equation of a genus-two curve $\mathcal C_{\alpha, \beta}$ defined over $K(\alpha, \beta)$, corresponding to $\mathfrak p$, where $\alpha $ and $\beta$ satisfy a quadratic $\alpha^2+ b \beta^2= c$ such that $b$ and $c$ are given in terms of ratios of Siegel modular forms. The curve $\mathcal C_{\alpha, \beta}$ is defined over the field of moduli $K$ if and only if the quadratic has a $K$-rational point $(\alpha, \beta)$. We discover some interesting symmetries of the Weierstrass equation of $\mathcal C_{\alpha, \beta}$. This extends previous work of Mestre and others.

Key words: genus-two curves; Siegel modular forms.

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  1. Beshaj L., Hidalgo R., Malmendier A., Kruk S., Quispe S., Shaska T., Rational points on the moduli space of genus two, in Algebraic Curves and their Fibrations in Mathematical Physics and Arithmetic Geometry, Contemporary Math., Vol. 703, Amer. Math. Soc., Providence, RI, 2018, 87-120.
  2. Bolza O., On binary sextics with linear transformations into themselves, Amer. J. Math. 10 (1887), 47-70.
  3. Booker A.R., Sijsling J., Sutherland A.V., Voight J., Yasaki D., A database of genus-2 curves over the rational numbers, LMS J. Comput. Math. 19 (2016), suppl. A, 235-254, arXiv:1602.03715.
  4. Bröker R., Howe E.W., Lauter K.E., Stevenhagen P., Genus-2 curves and Jacobians with a given number of points, LMS J. Comput. Math. 18 (2015), 170-197, arXiv:1403.6911.
  5. Clebsch A., Gordan P., Theorie der Abelschen Functionen, Thesaurus Mathematicae, Vol. 7, Physica-Verlag, Würzburg, 1967.
  6. Freitag E., Siegelsche Modulfunktionen, Grundlehren der Mathematischen Wissenschaften, Vol. 254, Springer-Verlag, Berlin, 1983.
  7. Goren E.Z., Lauter K.E., Genus 2 curves with complex multiplication, Int. Math. Res. Not. 2012 (2012), 1068-1142, arXiv:1003.4759.
  8. Gritsenko V.A., Nikulin V.V., Igusa modular forms and ''the simplest'' Lorentzian Kac-Moody algebras, Sb. Math. 187 (1996), 1601-1641.
  9. Harris J., Morrison I., Moduli of curves, Graduate Texts in Mathematics, Vol. 187, Springer-Verlag, New York, 1998.
  10. Igusa J.I., Arithmetic variety of moduli for genus two, Ann. of Math. 72 (1960), 612-649.
  11. Igusa J.I., On Siegel modular forms of genus two, Amer. J. Math. 84 (1962), 175-200.
  12. Igusa J.I., Modular forms and projective invariants, Amer. J. Math. 89 (1967), 817-855.
  13. Igusa J.I., On the ring of modular forms of degree two over ${\bf Z}$, Amer. J. Math. 101 (1979), 149-183.
  14. Krishnamoorthy V., Shaska T., Völklein H., Invariants of binary forms, in Progress in Galois Theory, Dev. Math., Vol. 12, Springer, New York, 2005, 101-122, arXiv:1209.0446.
  15. Lauter K., Naehrig M., Yang T., Hilbert theta series and invariants of genus 2 curves, J. Number Theory 161 (2016), 146-174.
  16. Malmendier A., Morrison D.R., K3 surfaces, modular forms, and non-geometric heterotic compactifications, Lett. Math. Phys. 105 (2015), 1085-1118, arXiv:1406.4873.
  17. Malmendier A., Shaska T., The Satake sextic in F-theory, J. Geom. Phys. 120 (2017), 290-305, arXiv:1609.04341.
  18. Mestre J.F., Construction de courbes de genre $2$ à partir de leurs modules, in Effective Methods in Algebraic Geometry (Castiglioncello, 1990), Progr. Math., Vol. 94, Birkhäuser Boston, Boston, MA, 1991, 313-334.
  19. Shaska T., Völklein H., Elliptic subfields and automorphisms of genus 2 function fields, in Algebra, Arithmetic and Geometry with Applications (West Lafayette, IN, 2000), Springer, Berlin, 2004, 703-723, math.AG/0107142.

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