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Higher descents on an elliptic curve with a rational 2-torsion point

Author: Tom Fisher
Journal: Math. Comp. 86 (2017), 2493-2518
MSC (2010): Primary 11G05, 11Y50
Published electronically: December 21, 2016
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Abstract: Let $ E$ be an elliptic curve over a number field $ K$. Descent calculations on $ E$ can be used to find upper bounds for the rank of the Mordell-Weil group, and to compute covering curves that assist in the search for generators of this group. The general method of $ 4$-descent, developed in the PhD theses of Siksek, Womack and Stamminger, has been implemented in Magma (when $ K=\mathbb{Q}$) and works well for elliptic curves with sufficiently small discriminant. By extending work of Bremner and Cassels, we describe the improvements that can be made when $ E$ has a rational $ 2$-torsion point. In particular, when $ E$ has full rational $ 2$-torsion, we describe a method for $ 8$-descent that is practical for elliptic curves $ E/\mathbb{Q}$ with large discriminant.

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  • [BSD] B. J. Birch and H. P. F. Swinnerton-Dyer, Notes on elliptic curves. II, J. Reine Angew. Math. 218 (1965), 79-108. MR 0179168
  • [BCP] 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,
  • [BC] A. Bremner and J. W. S. Cassels, On the equation $ Y^{2}=X(X^{2}+p)$, Math. Comp. 42 (1984), no. 165, 257-264. MR 726003,
  • [C1] J. W. S. Cassels, Arithmetic on curves of genus 1. VIII. On conjectures of Birch and Swinnerton-Dyer, J. Reine Angew. Math. 217 (1965), 180-199. MR 0179169
  • [C2] J. W. S. Cassels, Lectures on elliptic curves, London Mathematical Society Student Texts, vol. 24, Cambridge University Press, Cambridge, 1991. MR 1144763
  • [C3] J. W. S. Cassels, Second descents for elliptic curves, J. Reine Angew. Math. 494 (1998), 101-127. MR 1604468,
  • [Cr] J. E. Cremona, Higher descents on elliptic curves, notes for a talk, 1997,
  • [CR] J. E. Cremona and D. Rusin, Efficient solution of rational conics, Math. Comp. 72 (2003), no. 243, 1417-1441. MR 1972744,
  • [CF] J. E. Cremona and T. A. Fisher, On the equivalence of binary quartics, J. Symbolic Comput. 44 (2009), no. 6, 673-682. MR 2509048,
  • [CFS] 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,
  • [D] A. Dujella, High rank elliptic curves with prescribed torsion,
  • [F1] Tom A. Fisher, The Cassels-Tate pairing and the Platonic solids, J. Number Theory 98 (2003), no. 1, 105-155. MR 1950441,
  • [F2] 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,
  • [MSS] 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
  • [Sik] S. Siksek, Descent on curves of genus $ 1$, PhD thesis, University of Exeter, 1995.
  • [Sil] Joseph H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986. MR 817210
  • [S1] Denis Simon, Solving quadratic equations using reduced unimodular quadratic forms, Math. Comp. 74 (2005), no. 251, 1531-1543 (electronic). MR 2137016,
  • [S2] D. Simon, Quadratic equations in dimensions 4, 5 and more, preprint 2005.
  • [St] S. Stamminger, Explicit 8-descent on elliptic curves, PhD thesis, International University Bremen, 2005.
  • [SD] Peter Swinnerton-Dyer, $ 2^n$-descent on elliptic curves for all $ n$, J. Lond. Math. Soc. (2) 87 (2013), no. 3, 707-723. MR 3073672,
  • [W+] Mark Watkins, Stephen Donnelly, Noam D. Elkies, Tom Fisher, Andrew Granville, and Nicholas F. Rogers, Ranks of quadratic twists of elliptic curves, Numéro consacré au trimestre ``Méthodes arithmétiques et applications'', automne 2013, Publ. Math. Besançon Algèbre Théorie Nr., vol. 2014/2, Presses Univ. Franche-Comté, Besançon, 2015, pp. 63-98 (English, with English and French summaries). MR 3381037
  • [Wo] T. Womack, Explicit descent on elliptic curves, PhD thesis, University of Nottingham, 2003.

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Additional Information

Tom Fisher
Affiliation: University of Cambridge, DPMMS, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, United Kingdom

Received by editor(s): September 15, 2015
Received by editor(s) in revised form: March 16, 2016
Published electronically: December 21, 2016
Article copyright: © Copyright 2016 American Mathematical Society

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