Ranks of quadratic twists of elliptic curves

Mark Watkins; Stephen Donnelly; Noam D. Elkies; Tom Fisher; Andrew Granville; Nicholas F. Rogers

doi:10.5802/pmb.9

Mark Watkins ; Stephen Donnelly ; Noam D. Elkies ; Tom Fisher ; Andrew Granville ; Nicholas F. Rogers

Publications mathématiques de Besançon. Algèbre et théorie des nombres, no. 2 (2014), pp. 63-98.

Résumé
Abstract

Nous donnons un compte rendu d’un projet de grande envergure sur les rangs des courbes elliptiques dans une famille de tordues quadratiques en nous focalisant sur les courbes associées aux nombres congruents. Afin d’exclure certaines courbes, nos méthodes incluent des tests sur les 2,4,8-groupes de Selmer, l’utilisation de la formule explicite de Guinand-Weil et également des 3-descentes dans quelques cas. Nous constatons que les tordues quadratiques de rang 6 sont assez répandues (bien que toujours assez difficile à trouver), alors que celles de rang 7 semblent bien plus rares. Nous décrivons aussi notre incapacité à obtenir des tordues quadratiques de rang 8 et expliquons en quoi nos résultats peuvent se comparer à certaines prédictions sur la croissance du rang en fonction du conducteur. Enfin, nous expliquons une heuristique due à Granville, qui, lorsqu’elle est interprétée judicieusement, pourrait prédire que le rang maximal pour cette famille est en effet égal à 7.

We report on a large-scale project to investigate the ranks of elliptic curves in a quadratic twist family, focussing on the congruent number curve. Our methods to exclude candidate curves include 2-Selmer, 4-Selmer, and 8-Selmer tests, the use of the Guinand-Weil explicit formula, and even 3-descent in a couple of cases. We find that rank 6 quadratic twists are reasonably common (though still quite difficult to find), while rank 7 twists seem much more rare. We also describe our inability to find a rank 8 twist, and discuss how our results here compare to some predictions of rank growth vis-à-vis conductor. Finally we explicate a heuristic of Granville, which when interpreted judiciously could predict that 7 is indeed the maximal rank in this quadratic twist family.

Reçu le : 2014-04-06
Publié le : 2015-04-20

Zbl

DOI : 10.5802/pmb.9

Classification : 11D25, 11D45, 11G05, 11G40, 14H52
Keywords: elliptic curves, quadratic twists, Selmer groups, explicit formula, Birch–Swinnerton-Dyer conjecture, algebraic rank
Mots clés : courbes elliptiques, tordues quadratiques, groupes de Selmer, formule explicite, conjecture de Birch–Swinnerton-Dyer, rang algébrique

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Mark Watkins; Stephen Donnelly; Noam D. Elkies; Tom Fisher; Andrew Granville; Nicholas F. Rogers. Ranks of quadratic twists of elliptic curves. Publications mathématiques de Besançon. Algèbre et théorie des nombres, no. 2 (2014), pp. 63-98. doi : 10.5802/pmb.9. https://pmb.centre-mersenne.org/articles/10.5802/pmb.9/

Bibliographie
Cité par

[1] J. Aguirre, F. Castañeda, J. C. Peral, High rank elliptic curves with torsion group Z/(2Z). Math. Comp. 73 (2004), 323–331 | DOI | Zbl

[2] B. J. Birch, H. P. F. Swinnerton-Dyer, Notes on elliptic curves. II. J. reine angew. Math. 218 (1965), 79–108. | DOI | MR | Zbl

[3] J. Bober, Conditionally bounding analytic ranks of elliptic curves. In ANTS X (2013), Proceedings of the Tenth Algorithmic Number Theory Symposium, edited by E. W. Howe and K. S. Kedlaya, The Open Book Series, Mathematical Sciences Publishers, 135–144. | DOI | Zbl

[4] A. R. Booker, Artin’s Conjecture, Turing’s Method, and the Riemann Hypothesis. Experiment. Math 15 (2006), 385–408. | DOI | MR | Zbl

[5] W. Bosma, J. Cannon, C. Playoust, The Magma algebra system. I. The user language. In Computational algebra and number theory, Proceedings of the 1st MAGMA Conference held at Queen Mary and Westfield College, London, August 23–27, 1993. Edited by J. Cannon and D. Holt, published by Elsevier Science B.V., Amsterdam (1997), 235–265. Cross-referenced as J. Symbolic Comput. 24 (1997), no. 3-4, 235–265. | DOI | MR | Zbl

[6] A. Bremner, J. W. S. Cassels, On the equation $Y^{2} = X (X^{2} + p)$ . Math. Comp. 42 (1984), no. 165, 257–264. | DOI | Zbl

[7] A. Brumer, The average rank of elliptic curves I. Inventiones math. 109 (1992), 445–472. | DOI | Zbl

[8] J. W. S. Cassels, Second descents for elliptic curves. J. reine angew. Math 494 (1998), 101–127. See also Arithmetic on curves of genus 1, IV. Proof of the Hauptvermutung. J. reine angew. Math 211 (1962), 95–112. | DOI | MR | Zbl

[9] H. Cohen, Advanced Topics in Computational Number Theory. Graduate Texts in Mathematics 193 (2000), Springer. | DOI | Zbl

[10] K. Conrad, Partial Euler products on the critical line. Canad. J. Math. 57 (2005), 267–297. | DOI | Zbl

[11] J. B. Conrey, J. P. Keating, M. O. Rubinstein, N. C. Snaith, On the frequency of vanishing of quadratic twists of modular $L$ -functions. In Number theory for the millennium, I (Urbana, IL, 2000), Papers from the conference held at the University of Illinois at Urbana–Champaign, Urbana, IL, May 21–26, 2000. Edited by M. A. Bennett, B. C. Berndt, N. Boston, H. G. Diamond, A. J. Hildebrand and W. Philipp, published by A K Peters, Ltd., Natick, MA (2002), 301–315. Preprint at | Zbl

[12] J. E. Cremona, B. Mazur, Visualizing elements in the Shafarevich-Tate group. Experiment. Math. 9, no. 1 (2000), 13–28. | DOI | MR | Zbl

[13] S. Donnelly, Computing the Cassels-Tate pairing on $Ш (E) [2]$ in Magma. Unpublished notes (2008).

[14] A. Dujella, A. S. Janfada, S. Salami, A search for high rank congruent number elliptic curves. J. Integer Seq. 12 (2009), 09.5.8. | Zbl

[15] N. D. Elkies, Letter to Prof D. B. Zagier (1988). Second part of the Appendix to [57]. | DOI

[16] N. D. Elkies, $Z^{28}$ in E(Q), etc. NMBRTHRY list posting, May 2006.

[17] N. D. Elkies, j=0, rank 15; also 3-rank 6 and 7 in real and imaginary quadratic fields, NMBRTHRY list posting, Dec 2009.

[18] N. D. Elkies, N. F. Rogers, Elliptic Curves $x^{3} + y^{3} = k$ of High Rank. In Algorithmic number theory, ANTS-VI (Burlington 2004), ed. by D. Buell, Springer LNCS 3076 (2004), 184–193. | DOI | Zbl

[19] N. D. Elkies, M. Watkins, Elliptic curves of large rank and small conductor. In Algorithmic number theory, ANTS-VI (Burlington 2004), ed. by D. Buell, Springer LNCS 3076 (2004), 42–56. | DOI | Zbl

[20] D. W. Farmer, S. M. Gonek, C. P. Hughes, The maximum size of $L$ -functions. J. reine Angew. Math. 609 (2007), 215–236. | DOI | Zbl

[21] T. A. Fisher, Higher descents on an elliptic curve with a rational 2-torsion point, preprint, | DOI | MR | Zbl

[22] F. Gouvêa, B. Mazur, The square-free sieve and the rank of elliptic curves. J. Amer. Math. Soc. 4 (1991), 1–23. | DOI | MR | Zbl

[23] J. E. Gower, S. S. Wagstaff Jr., Square form factorization. Math. Comp. 77 (2008), 551–588. | DOI | Zbl

[24] A. P. Guinand, Summation Formulae and Self-reciprocal Functions (III). Quarterly J. Math., 13 (1942), 30–39. | DOI | Zbl

[25] D. R. Heath-Brown, The size of Selmer groups for the congruent number problem, II. Inv. Math. 118, no. 1 (1994), 331–370. | DOI | Zbl

[26] T. Honda, Isogenies, rational points and section points of group varieties. Japan. J. Math., 30 (1960), 84–101. | DOI | MR | Zbl

[27] C. Hooley, On the Pellian equation and the class number of indefinite binary quadratic forms. J. reine Angew. Math., 353 (1984), 98–131. | DOI | MR | Zbl

[28] M.-A. Lacasse, Sur la répartition des unités dans les corps quadratiques réels. Master’s thesis, Université de Montréal, December 2011.

[29] S. Lang, Hyperbolic and Diophantine analysis, Bull. AMS, 14 (1986), 159–205. | DOI | Zbl

[30] A. F. Lavrik, On functional equations of Dirichlet functions. Mat. USSR Izv. 1, no. 2 (1967), 421–432. | DOI | Zbl

[31] S. McMath, F. Crabbe, D. Joyner, Continued Fractions and Parallel SQUFOF, Int. J. Pure Appl. Math. 34, no. 1 (2007), 19–38. | Zbl

[32] R. Martin, W. McMillen, An elliptic curve over Q with rank at least 24. NMBRTHRY Listserver, May 2000.

[33] K. Matsuno, Elliptic curves with large Tate-Shafarevich groups over a number field. Math. Res. Lett. 16 (2009), 449–461. | DOI | MR | Zbl

[34] F. Mertens, Ein Beitrag zur analytischen Zahlentheorie. (German) [A contribution to analytic number theory]. J. reine angew. Math. 78 (1874), 46–62. | DOI | MR | Zbl

[35] J.-F. Mestre, Construction d’une courbe elliptique de rang $\geq 12$ . (French) [Construction of an elliptic curve with rank at least 12]. C. R. Acad. Sci. Paris, 295, série I (1982), 643–644. | Zbl

[36] J.-F. Mestre, Formules explicities et minorations de conducteurs de variétés algébriques. (French) [Explicit formulas and lower bounds on conductors of algebraic varieties]. Compositio Math. 58 (1986), 209–232. | Zbl

[37] J.-F. Mestre, Courbes elliptiques de rang $\geq 11$ sur Q(t) (French). [Elliptic curves of rank at least 11 over $Q (t)$ ]. C. R. Acad. Sci. Paris, 313, série I (1991), 139–142. | Zbl

[38] J.-F. Mestre, Courbes elliptiques de rang $\geq 12$ sur Q(t) (French). [Elliptic curves of rank at least 12 over $Q (t)$ ]. C. R. Acad. Sci. Paris, 313, série I (1991), 171–174. | Zbl

[39] P. Monsky, appendix to [25].

[40] K.-i. Nagao, Examples of elliptic curves over $Q$ with rank $\geq 17$ . Proc. Japan Acad. Ser. A Math. Sci. 68, no. 9 (1992), 287–9. | DOI

[41] K.-i. Nagao, Construction of high-rank elliptic curves. Kobe J. Math. 11 (1994), 211–219. | Zbl

[42] A. Néron, Footnote in Œuvres de Henri Poincaré. Tome V, 1950, Gabay.

[43] S. Omar, Non-vanishing of Dirichlet L-functions at the central point. In Algorithmic Number Theory, ANTS-VIII (Calgary 2008), edited by A. J. van der Poorten and A. Stein. Springer LNCS 5011 (2008), 443–453. | DOI | Zbl

[44] R. Paley, N. Wiener, Fourier transforms in the Complex Domain. AMS Coll. Publ. 19, 1934. Twelfth printing 2000, | DOI | Zbl

[45] C. S. Rajan, On the size of the Shafarevich-Tate group of elliptic curves over function fields. Compositio Math. 105 (1997), no. 1, 29–41. | DOI | Zbl

[46] N. F. Rogers, Rank computations for the congruent number elliptic curves. Experiment. Math. 9, no. 4 (2000), 591–4. | DOI | MR | Zbl

[47] N. F. Rogers, Elliptic curves $x^{3} + y^{3} = k$ with high rank. Doctoral thesis, Harvard University (Cambridge, MA, USA), April 2004.

[48] K. Rubin, A. Silverberg, Ranks of elliptic curves in families of quadratic twists. Experiment. Math. 9 no. 4 (2000), 583–590. | DOI | MR | Zbl

[49] K. Rubin, A. Silverberg, Rank frequencies for quadratic twists of elliptic curves. Experiment. Math. 10, no. 4 (2001), 559–570. | DOI | MR | Zbl

[50] K. Rubin, A. Silverberg, Ranks of Elliptic Curves. Bull. AMS, 39 no. 4 (2002), 455–474. | DOI | Zbl

[51] M. O. Rubinstein, Elliptic curves of high rank and the Riemann zeta function. Experiment. Math 22. no. 4 (2013), 465–480. | DOI | Zbl

[52] E. F. Schaefer, M. Stoll, How to do a $p$ -descent on an elliptic curve. Trans. Amer. Math. Soc. 356 (2004), 1209–1231. | DOI | Zbl

[53] U. Schneiders, H. G. Zimmer, The rank of elliptic curves upon quadratic extension. In Computational number theory (Debrecen 1989), ed. A. Pethő, M. E. Pohst, H. C. Williams, and H. G. Zimmer, published by Walter de Gruyter & Co., Berlin (1991), 239–260. | DOI | Zbl

[54] D. Shanks, On Gauss and composition, II. In Number Theory and Applications (1989), edited by R. A. Mollin. Proceedings of the NATO Advanced Study Institute (Banff 1988), NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, 265. Kluwer Academic Publishers Group, Dordrecht, 1989, 179–204. See also [31].

[55] D. T. Tèĭt [J. T. Tate], I. R. Šafarevič, The rank of elliptic curves. Soviet Math. Dokl. 8 (1967), 917–920. [trans. J. W. S. Cassels from Dokl. Akad. Nauk SSSR 175 (1967), 770–773.]

[56] D. Ulmer, Elliptic curves with large rank over function fields. Ann. of Math. (2) 155 (2002), no. 1, 295–315. | DOI | MR | Zbl

[57] M. Watkins, A note on integral points on elliptic curves. J. Théor. Nombres Bordeaux, 18 (2006), no. 3, 707–719. | DOI | MR | Zbl

[58] M. Watkins, On elliptic curves and random matrix theory. J. Théor. Nombres Bordeaux, 20 (2008), no. 3, 829–845. | DOI | Zbl

[59] A. Weil, Sur les “formules explicites” de la théorie des nombres premiers. (French) [On the “explicit formulæ” in the theory of prime numbers]. In Festkrift tillaegnad Marcel Riesz, edited by C. W. K. Gleerup. Supplementary tome of Meddelanden Frän Lunds Univ. mat. Sem. [Comm. Sém. Math. Univ. Lund], 1952, 252–265. Also in Weil’s Œuvres Scientifiques [Collected Works], Vol. 2 (1952b), 48–61. | DOI

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