^{}Department of Mathematical Sciences, Sharif University of Technology, Tehran, Iran.

Receive Date: 13 July 2016,
Revise Date: 11 December 2016,
Accept Date: 15 December 2016

Abstract

It is shown that the knowledge of a surjective morphism $X\to Y$ of complex curves can be effectively used to make explicit calculations. The method is demonstrated by the calculation of $j(n\tau)$ (for some small $n$) in terms of $j(\tau)$ for the elliptic curve with period lattice $(1,\tau)$, the period matrix for the Jacobian of a family of genus-$2$ curves complementing the classic calculations of Bolza and explicit general formulae for branched covers of an elliptic curve with exactly one ramification point.

C. Birkenhake and H. Lange, Complex Abelian Varieties, Grundlehren Math. Wiss. 302, Springer-Verlag, 2nd edition, Berlin-Heidelberg, 2004.

O. Bolza, On binary sextics with linear transformations into themselves, Amer. J. Math. 10 (1887), no. 1, 47--70.

H. Cohen, A Course in Computational Algebraic Number Theory, Grad. Texts in Math. 138, Springer-Verlag, Berlin-Heidelberg, 1993.

J. Dixmier, On the projective invariants of quartic plane curves, Adv. Math. 64 (1987), no. 3, 279--304.

E. Girondo and G. González-Diez, On complex curves and complex surfaces defined over number fields, in: Teichmüller Theory and Moduli Problem, pp. 247--280, Ramanujan Math. Soc. Lect. Notes Ser. 10, Ramanujan Math. Soc. Mysore, 2010.

J.I. Igusa, Arithmetic variety of moduli for genus two, Ann. of Math. (2) 72 (1960), no. 3, 612--649.

J.I. Igusa, On Siegel modular forms of genus two, Amer. J. Math. 84 (1962), no. 1, 175--200.

A. Kamalinejad and M. Shahshahani, On computations with dessins d'enfants, Math. Comp. 86 (2017), no. 303, 419--436.

V. Krishnamoorthy, T. Shaska and H. Völklein, Invariants of binary forms, in: Progress in Galois Theory, pp. 101--122, Dev. Math. 12, Springer, New York, 2005.

J. Milnor, Dynamics in One Complex Variable, Ann. of Math. Stud. 160, Princeton Univ. Press, 3rd edition, Princeton, 2006.

S. Rubinstein-Salzedo, Covers of elliptic curves with unique, totally ramified branch points, Math. Nachr. 286 (2013), no. 14-15, 1530--1536.

S. Rubinstein-Salzedo, Period computations for covers of elliptic curves, Math. Comp. 83 (2014), no. 289, 2455--2470.

T. Shaska and J. Thompson, On the generic curve of genus 3, in: Affine Algebraic Geometry, pp. 233--243, Contemp. Math. 369, Amer. Math. Soc. Providence, RI, 2005.

T. Shioda, On the graded ring of invariants of binary octavics, Amer. J. Math. 89 (1967), no. 4, 1022--1046.

T. Shioda and N. Mitani, Singular abelian surfaces and binary quadratic forms, in: Classification of Algebraic Varieties and Compact Complex Manifolds, pp. 259--287, Lecture Notes in Math. 412, Springer, Berlin, 1974.

B. Sturmfels, Algorithms in Invariant Theory, Texts Monogr. Symbol. Comput., Springer-Verlag, 2nd edition, Wien, 2008.