Higher descents on an elliptic curve with a rational 2-torsion point
HTML articles powered by AMS MathViewer
- by Tom Fisher PDF
- Math. Comp. 86 (2017), 2493-2518 Request permission
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.References
- B. J. Birch and H. P. F. Swinnerton-Dyer, Notes on elliptic curves. II, J. Reine Angew. Math. 218 (1965), 79â108. MR 179168, DOI 10.1515/crll.1965.218.79
- Wieb Bosma, John Cannon, and Catherine Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), no. 3-4, 235â265. Computational algebra and number theory (London, 1993). MR 1484478, DOI 10.1006/jsco.1996.0125
- A. Bremner and J. W. S. Cassels, On the equation $Y^{2}=X(X^{2}+p)$, Math. Comp. 42 (1984), no. 165, 257â264. MR 726003, DOI 10.1090/S0025-5718-1984-0726003-4
- J. W. S. Cassels, Arithmetic on curves of genus 1. VIII. On conjectures of Birch and Swinnerton-Dyer, J. Reine Angew. Math. 217 (1965), 180â199. MR 179169, DOI 10.1515/crll.1965.217.180
- J. W. S. Cassels, Lectures on elliptic curves, London Mathematical Society Student Texts, vol. 24, Cambridge University Press, Cambridge, 1991. MR 1144763, DOI 10.1017/CBO9781139172530
- J. W. S. Cassels, Second descents for elliptic curves, J. Reine Angew. Math. 494 (1998), 101â127. Dedicated to Martin Kneser on the occasion of his 70th birthday. MR 1604468, DOI 10.1515/crll.1998.001
- J. E. Cremona, Higher descents on elliptic curves, notes for a talk, 1997, http://homepages.warwick.ac.uk/~masgaj/papers/d2.ps
- J. E. Cremona and D. Rusin, Efficient solution of rational conics, Math. Comp. 72 (2003), no. 243, 1417â1441. MR 1972744, DOI 10.1090/S0025-5718-02-01480-1
- J. E. Cremona and T. A. Fisher, On the equivalence of binary quartics, J. Symbolic Comput. 44 (2009), no. 6, 673â682. MR 2509048, DOI 10.1016/j.jsc.2008.09.004
- 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, 763â820. MR 2728489, DOI 10.2140/ant.2010.4.763
- A. Dujella, High rank elliptic curves with prescribed torsion, http://web.math.pmf.unizg.hr/~duje/tors/tors.html
- Tom A. Fisher, The Cassels-Tate pairing and the Platonic solids, J. Number Theory 98 (2003), no. 1, 105â155. MR 1950441, DOI 10.1016/S0022-314X(02)00038-0
- 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. 125â138. MR 2467841, DOI 10.1007/978-3-540-79456-1_{8}
- J. R. Merriman, S. Siksek, and N. P. Smart, Explicit $4$-descents on an elliptic curve, Acta Arith. 77 (1996), no. 4, 385â404. MR 1414518, DOI 10.4064/aa-77-4-385-404
- S. Siksek, Descent on curves of genus $1$, PhD thesis, University of Exeter, 1995.
- Joseph H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986. MR 817210, DOI 10.1007/978-1-4757-1920-8
- Denis Simon, Solving quadratic equations using reduced unimodular quadratic forms, Math. Comp. 74 (2005), no. 251, 1531â1543. MR 2137016, DOI 10.1090/S0025-5718-05-01729-1
- D. Simon, Quadratic equations in dimensions 4, 5 and more, preprint 2005.
- S. Stamminger, Explicit 8-descent on elliptic curves, PhD thesis, International University Bremen, 2005.
- Peter Swinnerton-Dyer, $2^n$-descent on elliptic curves for all $n$, J. Lond. Math. Soc. (2) 87 (2013), no. 3, 707â723. MR 3073672, DOI 10.1112/jlms/jds063
- Mark Watkins, Stephen Donnelly, Noam D. Elkies, Tom Fisher, Andrew Granville, and Nicholas F. Rogers, Ranks of quadratic twists of elliptic curves, NumĂ©ro consacrĂ© au trimestre âMĂ©thodes arithmĂ©tiques et applicationsâ, automne 2013, Publ. Math. Besançon AlgĂšbre ThĂ©orie Nr., vol. 2014/2, Presses Univ. Franche-ComtĂ©, Besançon, 2015, pp. 63â98 (English, with English and French summaries). MR 3381037
- T. Womack, Explicit descent on elliptic curves, PhD thesis, University of Nottingham, 2003.
Additional Information
- Tom Fisher
- Affiliation: University of Cambridge, DPMMS, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, United Kingdom
- MR Author ID: 678544
- Email: T.A.Fisher@dpmms.cam.ac.uk
- Received by editor(s): September 15, 2015
- Received by editor(s) in revised form: March 16, 2016
- Published electronically: December 21, 2016
- © Copyright 2016 American Mathematical Society
- Journal: Math. Comp. 86 (2017), 2493-2518
- MSC (2010): Primary 11G05, 11Y50
- DOI: https://doi.org/10.1090/mcom/3163
- MathSciNet review: 3647969