Local solubility and height bounds for coverings of elliptic curves

Fisher, T.; Sills, G.

doi:10.1090/S0025-5718-2012-02587-7

Local solubility and height bounds for coverings of elliptic curves
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by T. A. Fisher and G. F. Sills PDF

Math. Comp. 81 (2012), 1635-1662 Request permission

Abstract:

We study genus one curves that arise as $2$-, $3$- and $4$-coverings of elliptic curves. We describe efficient algorithms for testing local solubility and modify the classical formulae for the covering maps so that they work in all characteristics. These ingredients are then combined to give explicit bounds relating the height of a rational point on one of the covering curves to the height of its image on the elliptic curve. We use our results to improve the existing methods for searching for rational points on elliptic curves.

References

Amod Agashe and William Stein, Visibility of Shafarevich-Tate groups of abelian varieties, J. Number Theory 97 (2002), no. 1, 171–185. MR 1939144, DOI 10.1006/jnth.2002.2810
Sang Yook An, Seog Young Kim, David C. Marshall, Susan H. Marshall, William G. McCallum, and Alexander R. Perlis, Jacobians of genus one curves, J. Number Theory 90 (2001), no. 2, 304–315. MR 1858080, DOI 10.1006/jnth.2000.2632
Michael Artin, Fernando Rodriguez-Villegas, and John Tate, On the Jacobians of plane cubics, Adv. Math. 198 (2005), no. 1, 366–382. MR 2183258, DOI 10.1016/j.aim.2005.06.004
B. J. Birch and H. P. F. Swinnerton-Dyer, Notes on elliptic curves. I, J. Reine Angew. Math. 212 (1963), 7–25. MR 146143, DOI 10.1515/crll.1963.212.7
Wieb Bosma, John Cannon, and Catherine Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), no. 3-4, 235–265. Computational algebra and number theory (London, 1993). MR 1484478, DOI 10.1006/jsco.1996.0125
Nils Bruin, Some ternary Diophantine equations of signature $(n,n,2)$, Discovering mathematics with Magma, Algorithms Comput. Math., vol. 19, Springer, Berlin, 2006, pp. 63–91. MR 2278923, DOI 10.1007/978-3-540-37634-7_{3}
J. W. S. Cassels, Arithmetic on curves of genus $1$. IV. Proof of the Hauptvermutung, J. Reine Angew. Math. 211 (1962), 95–112. MR 163915, DOI 10.1515/crll.1962.211.95
J. E. Cremona, Algorithms for modular elliptic curves, 2nd ed., Cambridge University Press, Cambridge, 1997. MR 1628193
B.M. Creutz, Explicit second $p$-descent on elliptic curves, PhD thesis, Jacobs University Bremen, 2010.
J. E. Cremona, T. A. Fisher, C. O’Neil, D. Simon, and M. Stoll, Explicit $n$-descent on elliptic curves. I. Algebra, J. Reine Angew. Math. 615 (2008), 121–155. MR 2384334, DOI 10.1515/CRELLE.2008.012
John E. Cremona, Tom A. Fisher, and Michael Stoll, Minimisation and reduction of 2-, 3- and 4-coverings of elliptic curves, Algebra Number Theory 4 (2010), no. 6, 763–820. MR 2728489, DOI 10.2140/ant.2010.4.763
J. E. Cremona, M. Prickett, and Samir Siksek, Height difference bounds for elliptic curves over number fields, J. Number Theory 116 (2006), no. 1, 42–68. MR 2197860, DOI 10.1016/j.jnt.2005.03.001
S. Donnelly, Computing the Cassels-Tate pairing, in preparation.
Noam D. Elkies, Rational points near curves and small nonzero $|x^3-y^2|$ via lattice reduction, Algorithmic number theory (Leiden, 2000) Lecture Notes in Comput. Sci., vol. 1838, Springer, Berlin, 2000, pp. 33–63. MR 1850598, DOI 10.1007/10722028_{2}
Tom Fisher, The invariants of a genus one curve, Proc. Lond. Math. Soc. (3) 97 (2008), no. 3, 753–782. MR 2448246, DOI 10.1112/plms/pdn021
Tom Fisher, Some improvements to 4-descent on an elliptic curve, Algorithmic number theory, Lecture Notes in Comput. Sci., vol. 5011, Springer, Berlin, 2008, pp. 125–138. MR 2467841, DOI 10.1007/978-3-540-79456-1_{8}
T.A. Fisher, The Hessian of a genus one curve, to appear in Proc. Lond. Math. Soc.
T.A. Fisher, Higher descents on an elliptic curve with a rational $2$-torsion point, in preparation.
T.A. Fisher and G.F. Sills, Local solubility and height bounds for coverings of elliptic curves, longer version of this paper, arXiv:1103.4944v1 [math.NT]
Marc Hindry and Joseph H. Silverman, Diophantine geometry, Graduate Texts in Mathematics, vol. 201, Springer-Verlag, New York, 2000. An introduction. MR 1745599, DOI 10.1007/978-1-4612-1210-2
J. R. Merriman, S. Siksek, and N. P. Smart, Explicit $4$-descents on an elliptic curve, Acta Arith. 77 (1996), no. 4, 385–404. MR 1414518, DOI 10.4064/aa-77-4-385-404
M.M. Sadek, Models of genus one curves, PhD thesis, University of Cambridge, 2009.
Edward F. Schaefer and Michael Stoll, How to do a $p$-descent on an elliptic curve, Trans. Amer. Math. Soc. 356 (2004), no. 3, 1209–1231. MR 2021618, DOI 10.1090/S0002-9947-03-03366-X
S. Siksek, Descent on curves of genus one, PhD thesis, University of Exeter, 1995. http://www.warwick.ac.uk/staff/S.Siksek/papers/phdnew.pdf
Samir Siksek, Infinite descent on elliptic curves, Rocky Mountain J. Math. 25 (1995), no. 4, 1501–1538. MR 1371352, DOI 10.1216/rmjm/1181072159
G.F. Sills, Height bounds for $n$-coverings, PhD thesis, University of Cambridge, 2010.
S. Stamminger, Explicit 8-descent on elliptic curves, PhD thesis, International University Bremen, 2005.
M. Stoll, Descent on elliptic curves, lecture notes, arXiv:math/0611694v1 [math.NT]
T. Womack, Explicit descent on elliptic curves, PhD thesis, University of Nottingham, 2003. http://www.warwick.ac.uk/staff/J.E.Cremona/