Period computations for covers of elliptic curves

Rubinstein-Salzedo, Simon

doi:10.1090/S0025-5718-2014-02797-X

Period computations for covers of elliptic curves
HTML articles powered by AMS MathViewer

by Simon Rubinstein-Salzedo PDF

Math. Comp. 83 (2014), 2455-2470 Request permission

Abstract:

In this article, we construct algebraic equations for a curve $C$ and a map $f$ to an elliptic curve $E$, with pre-specified branching data. We do this by determining certain relations that the periods of $C$ and $E$ must satisfy and using these relations to approximate the coefficients to high precision. We then conjecture which algebraic numbers the coefficients are, and then we prove this conjecture to be correct.

References

A. Baker, Linear forms in the logarithms of algebraic numbers. I, II, III, Mathematika 13 (1966), 204–216; ibid. 14 (1967), 102–107; ibid. 14 (1967), 220–228. MR 220680, DOI 10.1112/s0025579300003843
D. H. Bailey and J. M. Borwein, High-precision numerical integration: progress and challenges, J. Symbolic Comput. 46 (2011), no. 7, 741–754. MR 2795208, DOI 10.1016/j.jsc.2010.08.010
Sybilla Beckmann, Ramified primes in the field of moduli of branched coverings of curves, J. Algebra 125 (1989), no. 1, 236–255. MR 1012673, DOI 10.1016/0021-8693(89)90303-7
Jonathan M. Borwein and Petr Lisoněk, Applications of integer relation algorithms, Discrete Math. 217 (2000), no. 1-3, 65–82 (English, with English and French summaries). Formal power series and algebraic combinatorics (Vienna, 1997). MR 1766260, DOI 10.1016/S0012-365X(99)00256-3
Enrico Bombieri and Alfred J. van der Poorten, Continued fractions of algebraic numbers, Computational algebra and number theory (Sydney, 1992) Math. Appl., vol. 325, Kluwer Acad. Publ., Dordrecht, 1995, pp. 137–152. MR 1344927
Jean-Marc Couveignes and Louis Granboulan, Dessins from a geometric point of view, The Grothendieck theory of dessins d’enfants (Luminy, 1993) London Math. Soc. Lecture Note Ser., vol. 200, Cambridge Univ. Press, Cambridge, 1994, pp. 79–113. MR 1305394
Pierre Dèbes and Jean-Claude Douai, Algebraic covers: field of moduli versus field of definition, Ann. Sci. École Norm. Sup. (4) 30 (1997), no. 3, 303–338 (English, with English and French summaries). MR 1443489, DOI 10.1016/S0012-9593(97)89922-3
Simon Donaldson, Riemann surfaces, Oxford Graduate Texts in Mathematics, vol. 22, Oxford University Press, Oxford, 2011. MR 2856237, DOI 10.1093/acprof:oso/9780198526391.001.0001
Frank Herrlich and Gabriela Schmithüsen, Dessins d’enfants and origami curves, Handbook of Teichmüller theory. Vol. II, IRMA Lect. Math. Theor. Phys., vol. 13, Eur. Math. Soc., Zürich, 2009, pp. 767–809. MR 2516744, DOI 10.4171/055-1/19
Fredrik Johansson et al., mpmath: a Python library for arbitrary-precision floating-point arithmetic (version 0.14), February 2010, http://code.google.com/p/mpmath/.
Maxim Kontsevich and Don Zagier, Periods, Mathematics unlimited—2001 and beyond, Springer, Berlin, 2001, pp. 771–808. MR 1852188
A. K. Lenstra, H. W. Lenstra Jr., and L. Lovász, Factoring polynomials with rational coefficients, Math. Ann. 261 (1982), no. 4, 515–534. MR 682664, DOI 10.1007/BF01457454
Curtis T. McMullen, Teichmüller curves in genus two: the decagon and beyond, J. Reine Angew. Math. 582 (2005), 173–199. MR 2139715, DOI 10.1515/crll.2005.2005.582.173
PARI/GP, version 2.3.5, Bordeaux, 2010, http://pari.math.u-bordeaux.fr/.
David P. Roberts, An $ABC$ construction of number fields, Number theory, CRM Proc. Lecture Notes, vol. 36, Amer. Math. Soc., Providence, RI, 2004, pp. 237–267. MR 2076600, DOI 10.1090/crmp/036/19
Simon Rubinstein-Salzedo, Controlling ramification in number fields, 2012, Ph.D. thesis, http://www.math.dartmouth.edu/~simon/thesis.pdf.
Simon Rubinstein-Salzedo, Covers of elliptic curves with unique, totally ramified branch points, Mathematische Nachrichten, 2013, DOI 10.1002/mana.201200285.
W. A. Stein et al, Sage Mathematics Software (Version 4.5.1), The Sage Development Team, 2010, http://www.sagemath.org.
SymPy Development Team, SymPy: Python library for symbolic mathematics, 2012.
Wolfram Research, Inc., Champaign, Illinois, Mathematica, Version 6.0.1.0, 2007.
Anton Zorich, Flat surfaces, Frontiers in number theory, physics, and geometry. I, Springer, Berlin, 2006, pp. 437–583. MR 2261104, DOI 10.1007/978-3-540-31347-2_{1}3