Ranks of elliptic curves
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- by Karl Rubin and Alice Silverberg PDF
- Bull. Amer. Math. Soc. 39 (2002), 455-474 Request permission
Abstract:
This paper gives a general survey of ranks of elliptic curves over the field of rational numbers. The rank is a measure of the size of the set of rational points. The paper includes discussions of the Birch and Swinnerton-Dyer Conjecture, the Parity Conjecture, ranks in families of quadratic twists, and ways to search for elliptic curves of large rank.References
- A. O. L. Atkin and F. Morain, Elliptic curves and primality proving, Math. Comp. 61 (1993), no. 203, 29–68. MR 1199989, DOI 10.1090/S0025-5718-1993-1199989-X billing G. Billing, Beiträge zur arithmetischen Theorie der ebenen kubischen Kurven vom Geschlechte eins, Nova Acta Reg. Soc. Sc. Upsaliensis (4) 11 (1937), No. 1.
- 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
- B. J. Birch and H. P. F. Swinnerton-Dyer, Notes on elliptic curves. II, J. Reine Angew. Math. 218 (1965), 79–108. MR 179168, DOI 10.1515/crll.1965.218.79
- Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor, On the modularity of elliptic curves over $\mathbf Q$: wild 3-adic exercises, J. Amer. Math. Soc. 14 (2001), no. 4, 843–939. MR 1839918, DOI 10.1090/S0894-0347-01-00370-8
- Armand Brumer and Kenneth Kramer, The rank of elliptic curves, Duke Math. J. 44 (1977), no. 4, 715–743. MR 457453
- 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. W. S. Cassels, Arithmetic on an elliptic curve, Proc. Internat. Congr. Mathematicians (Stockholm, 1962) Inst. Mittag-Leffler, Djursholm, 1963, pp. 234–246. MR 0175891 random J. B. Conrey, J. P. Keating, M. O. Rubinstein, N. C. Snaith, On the frequency of vanishing of quadratic twists of modular $L$-functions, preprint. cornut C. Cornut, Mazur’s conjecture on higher Heegner points, Invent. Math. 148 (2002), 495–523.
- Noam D. Elkies, Heegner point computations, Algorithmic number theory (Ithaca, NY, 1994) Lecture Notes in Comput. Sci., vol. 877, Springer, Berlin, 1994, pp. 122–133. MR 1322717, DOI 10.1007/3-540-58691-1_{4}9 elkiesweb —, http://www.math.harvard.edu/~elkies/compnt.html.
- Stéfane Fermigier, Un exemple de courbe elliptique définie sur $\textbf {Q}$ de rang $\geq 19$, C. R. Acad. Sci. Paris Sér. I Math. 315 (1992), no. 6, 719–722 (French, with English and French summaries). MR 1183810
- Stéfane Fermigier, Une courbe elliptique définie sur $\mathbf Q$ de rang $\geq 22$, Acta Arith. 82 (1997), no. 4, 359–363 (French). MR 1483688, DOI 10.4064/aa-82-4-359-363
- Dorian Goldfeld, Conjectures on elliptic curves over quadratic fields, Number theory, Carbondale 1979 (Proc. Southern Illinois Conf., Southern Illinois Univ., Carbondale, Ill., 1979) Lecture Notes in Math., vol. 751, Springer, Berlin, 1979, pp. 108–118. MR 564926
- Dorian Goldfeld, Sur les produits partiels eulériens attachés aux courbes elliptiques, C. R. Acad. Sci. Paris Sér. I Math. 294 (1982), no. 14, 471–474 (French, with English summary). MR 679556 gk0 S. Goldwasser, J. Kilian, Almost all primes can be quickly certified, in Proc. 18th STOC (Berkeley, May 28–30, 1986), ACM, New York, 1986, 316–329.
- Shafi Goldwasser and Joe Kilian, Primality testing using elliptic curves, J. ACM 46 (1999), no. 4, 450–472. MR 1812127, DOI 10.1145/320211.320213
- F. Gouvêa and B. Mazur, The square-free sieve and the rank of elliptic curves, J. Amer. Math. Soc. 4 (1991), no. 1, 1–23. MR 1080648, DOI 10.1090/S0894-0347-1991-1080648-7
- Benedict H. Gross and Don B. Zagier, Heegner points and derivatives of $L$-series, Invent. Math. 84 (1986), no. 2, 225–320. MR 833192, DOI 10.1007/BF01388809
- Fritz J. Grunewald and Rainer Zimmert, Über einige rationale elliptische Kurven mit freiem Rang $\geq 8$, J. Reine Angew. Math. 296 (1977), 100–107 (German). MR 466147, DOI 10.1515/crll.1977.296.100
- Helmut Hasse, Mathematische Abhandlungen. Band 1, Walter de Gruyter, Berlin-New York, 1975 (German). Herausgegeben von Heinrich Wolfgang Leopoldt und Peter Roquette. MR 0465756
- Helmut Hasse, Mathematische Abhandlungen. Band 1, Walter de Gruyter, Berlin-New York, 1975 (German). Herausgegeben von Heinrich Wolfgang Leopoldt und Peter Roquette. MR 0465756
- D. R. Heath-Brown, The size of Selmer groups for the congruent number problem. II, Invent. Math. 118 (1994), no. 2, 331–370. With an appendix by P. Monsky. MR 1292115, DOI 10.1007/BF01231536
- Taira Honda, Isogenies, rational points and section points of group varieties, Jpn. J. Math. 30 (1960), 84–101. MR 155828, DOI 10.4099/jjm1924.30.0_{8}4
- Nicholas M. Katz and Peter Sarnak, Zeroes of zeta functions and symmetry, Bull. Amer. Math. Soc. (N.S.) 36 (1999), no. 1, 1–26. MR 1640151, DOI 10.1090/S0273-0979-99-00766-1
- Shoichi Kihara, On an infinite family of elliptic curves with rank $\geq 14$ over $\textbf {Q}$, Proc. Japan Acad. Ser. A Math. Sci. 73 (1997), no. 2, 32. MR 1442250
- Shoichi Kihara, On an elliptic curve over $\mathbf Q(t)$ of rank $\geq 14$, Proc. Japan Acad. Ser. A Math. Sci. 77 (2001), no. 4, 50–51. MR 1829378
- Neal Koblitz, Elliptic curve cryptosystems, Math. Comp. 48 (1987), no. 177, 203–209. MR 866109, DOI 10.1090/S0025-5718-1987-0866109-5
- Neal Koblitz, Introduction to elliptic curves and modular forms, 2nd ed., Graduate Texts in Mathematics, vol. 97, Springer-Verlag, New York, 1993. MR 1216136, DOI 10.1007/978-1-4612-0909-6
- V. A. Kolyvagin, Finiteness of $E(\textbf {Q})$ and SH$(E,\textbf {Q})$ for a subclass of Weil curves, Izv. Akad. Nauk SSSR Ser. Mat. 52 (1988), no. 3, 522–540, 670–671 (Russian); English transl., Math. USSR-Izv. 32 (1989), no. 3, 523–541. MR 954295, DOI 10.1070/IM1989v032n03ABEH000779
- V. A. Kolyvagin, Euler systems, The Grothendieck Festschrift, Vol. II, Progr. Math., vol. 87, Birkhäuser Boston, Boston, MA, 1990, pp. 435–483. MR 1106906
- Kenneth Kramer, Arithmetic of elliptic curves upon quadratic extension, Trans. Amer. Math. Soc. 264 (1981), no. 1, 121–135. MR 597871, DOI 10.1090/S0002-9947-1981-0597871-8
- Daniel Sion Kubert, Universal bounds on the torsion of elliptic curves, Proc. London Math. Soc. (3) 33 (1976), no. 2, 193–237. MR 434947, DOI 10.1112/plms/s3-33.2.193
- H. W. Lenstra Jr., Factoring integers with elliptic curves, Ann. of Math. (2) 126 (1987), no. 3, 649–673. MR 916721, DOI 10.2307/1971363 lutz E. Lutz, Sur l’equation $y^2=x^3-Ax-B$ dans les corps $p$-adic, J. Reine Angew. Math. 177 (1937), 238–247. MartinMc R. Martin, W. McMillen, posting to Number Theory server, March 16, 1998. MartinMc2 —, posting to Number Theory server, May 2, 2000.
- B. Mazur, Modular curves and the Eisenstein ideal, Inst. Hautes Études Sci. Publ. Math. 47 (1977), 33–186 (1978). With an appendix by Mazur and M. Rapoport. MR 488287, DOI 10.1007/BF02684339
- Jean-François Mestre, Construction d’une courbe elliptique de rang $\geq 12$, C. R. Acad. Sci. Paris Sér. I Math. 295 (1982), no. 12, 643–644 (French, with English summary). MR 688896
- Jean-François Mestre, Formules explicites et minorations de conducteurs de variétés algébriques, Compositio Math. 58 (1986), no. 2, 209–232 (French). MR 844410
- Jean-François Mestre, Courbes elliptiques de rang $\geq 11$ sur $\textbf {Q}(t)$, C. R. Acad. Sci. Paris Sér. I Math. 313 (1991), no. 3, 139–142 (French, with English summary). MR 1121576
- Jean-François Mestre, Courbes elliptiques de rang $\geq 12$ sur $\textbf {Q}(t)$, C. R. Acad. Sci. Paris Sér. I Math. 313 (1991), no. 4, 171–174 (French, with English summary). MR 1122689
- Jean-François Mestre, Un exemple de courbe elliptique sur $\textbf {Q}$ de rang $\geq 15$, C. R. Acad. Sci. Paris Sér. I Math. 314 (1992), no. 6, 453–455 (French, with English summary). MR 1154385
- Jean-François Mestre, Rang de courbes elliptiques d’invariant donné, C. R. Acad. Sci. Paris Sér. I Math. 314 (1992), no. 12, 919–922 (French, with English summary). MR 1168325
- Jean-François Mestre, Rang de certaines familles de courbes elliptiques d’invariant donné, C. R. Acad. Sci. Paris Sér. I Math. 327 (1998), no. 8, 763–764 (French, with English and French summaries). MR 1659978, DOI 10.1016/S0764-4442(98)80166-3 mestre6 —, Berkeley Number Theory Seminar, September 15, 2000.
- Victor S. Miller, Use of elliptic curves in cryptography, Advances in cryptology—CRYPTO ’85 (Santa Barbara, Calif., 1985) Lecture Notes in Comput. Sci., vol. 218, Springer, Berlin, 1986, pp. 417–426. MR 851432, DOI 10.1007/3-540-39799-X_{3}1 mordell L. J. Mordell, On the rational solutions of the indeterminate equations of the third and fourth degrees, Proc. Cambridge Philos. Soc. 21 (1922), 179–192.
- Koh-ichi Nagao, Examples of elliptic curves over $\textbf {Q}$ with rank $\geq 17$, Proc. Japan Acad. Ser. A Math. Sci. 68 (1992), no. 9, 287–289. MR 1202634
- Koh-ichi Nagao, An example of elliptic curve over $\textbf {Q}$ with rank $\ge 20$, Proc. Japan Acad. Ser. A Math. Sci. 69 (1993), no. 8, 291–293. MR 1249440
- Koh-ichi Nagao, An example of elliptic curve over $\textbf {Q}(T)$ with rank $\geq 13$, Proc. Japan Acad. Ser. A Math. Sci. 70 (1994), no. 5, 152–153. MR 1291171
- Koh-ichi Nagao and Tomonori Kouya, An example of elliptic curve over $\mathbf Q$ with rank $\geq 21$, Proc. Japan Acad. Ser. A Math. Sci. 70 (1994), no. 4, 104–105. MR 1276883 nagell T. Nagell, Solution de quelque problèmes dans la théorie arithmétique des cubiques planes du premier genre, Wid. Akad. Skrifter Oslo I (1935), No. 1, 1–25.
- Jan Nekovář, On the parity of ranks of Selmer groups. II, C. R. Acad. Sci. Paris Sér. I Math. 332 (2001), no. 2, 99–104 (English, with English and French summaries). MR 1813764, DOI 10.1016/S0764-4442(00)01808-5
- Sam Perlis, Maximal orders in rational cyclic algebras of composite degree, Trans. Amer. Math. Soc. 46 (1939), 82–96. MR 15, DOI 10.1090/S0002-9947-1939-0000015-X
- Saunders MacLane and O. F. G. Schilling, Infinite number fields with Noether ideal theories, Amer. J. Math. 61 (1939), 771–782. MR 19, DOI 10.2307/2371335
- David E. Penney and Carl Pomerance, A search for elliptic curves with large rank, Math. Comp. 28 (1974), 851–853. MR 376686, DOI 10.1090/S0025-5718-1974-0376686-X
- David E. Penney and Carl Pomerance, Three elliptic curves with rank at least seven, Math. Comput. 29 (1975), 965–967. MR 0376687, DOI 10.1090/S0025-5718-1975-0376687-2 poincare H. Poincaré, Sur les propriétés arithmétiques des courbes algébriques, J. Math. Pures Appl., Ser. 5, vol. 7 (1901), 161–233.
- Nicholas F. Rogers, Rank computations for the congruent number elliptic curves, Experiment. Math. 9 (2000), no. 4, 591–594. MR 1806294, DOI 10.1080/10586458.2000.10504662
- David E. Rohrlich, Galois theory, elliptic curves, and root numbers, Compositio Math. 100 (1996), no. 3, 311–349. MR 1387669 bama K. Rubin, Right triangles and elliptic curves, to appear in Bay Area Math Adventures, D. F. Hayes, ed., MAA.
- Karl Rubin and Alice Silverberg, Ranks of elliptic curves in families of quadratic twists, Experiment. Math. 9 (2000), no. 4, 583–590. MR 1806293, DOI 10.1080/10586458.2000.10504661 rs2 —, Rank frequencies for quadratic twists of elliptic curves, Exper. Math. 10 (2001), no. 4, 559–569. sil A. Silverberg, Open questions in arithmetic algebraic geometry, in Arithmetic Algebraic Geometry (Park City, UT, 1999), IAS/Park City Mathematics Series 9, AMS, Providence, RI (2001), 85–142.
- Joseph H. Silverman, Heights and the specialization map for families of abelian varieties, J. Reine Angew. Math. 342 (1983), 197–211. MR 703488, DOI 10.1515/crll.1983.342.197
- 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
- C. L. Stewart and J. Top, On ranks of twists of elliptic curves and power-free values of binary forms, J. Amer. Math. Soc. 8 (1995), no. 4, 943–973. MR 1290234, DOI 10.1090/S0894-0347-1995-1290234-5
- John T. Tate, The arithmetic of elliptic curves, Invent. Math. 23 (1974), 179–206. MR 419359, DOI 10.1007/BF01389745
- D. T. Tèĭt and I. R. Šafarevič, The rank of elliptic curves, Dokl. Akad. Nauk SSSR 175 (1967), 770–773 (Russian). MR 0237508
- Richard Taylor and Andrew Wiles, Ring-theoretic properties of certain Hecke algebras, Ann. of Math. (2) 141 (1995), no. 3, 553–572. MR 1333036, DOI 10.2307/2118560
- J. B. Tunnell, A classical Diophantine problem and modular forms of weight $3/2$, Invent. Math. 72 (1983), no. 2, 323–334. MR 700775, DOI 10.1007/BF01389327 vatsal V. Vatsal, Special values of anticyclotomic $L$-functions, preprint.
- André Weil, Number theory, Birkhäuser Boston, Inc., Boston, MA, 1984. An approach through history; From Hammurapi to Legendre. MR 734177, DOI 10.1007/978-0-8176-4571-7
- Andrew Wiles, Modular elliptic curves and Fermat’s last theorem, Ann. of Math. (2) 141 (1995), no. 3, 443–551. MR 1333035, DOI 10.2307/2118559
- P. Erdös and T. Grünwald, On polynomials with only real roots, Ann. of Math. (2) 40 (1939), 537–548. MR 7, DOI 10.2307/1968938
- P. Erdös and T. Grünwald, On polynomials with only real roots, Ann. of Math. (2) 40 (1939), 537–548. MR 7, DOI 10.2307/1968938
- Morgan Ward, Ring homomorphisms which are also lattice homomorphisms, Amer. J. Math. 61 (1939), 783–787. MR 10, DOI 10.2307/2371336
- Don Zagier, The Birch-Swinnerton-Dyer conjecture from a naive point of view, Arithmetic algebraic geometry (Texel, 1989) Progr. Math., vol. 89, Birkhäuser Boston, Boston, MA, 1991, pp. 377–389. MR 1085269, DOI 10.1007/978-1-4612-0457-2_{1}8
Additional Information
- Karl Rubin
- Affiliation: Department of Mathematics, Stanford University, Stanford, California 94305
- MR Author ID: 151435
- Email: rubin@math.stanford.edu
- Alice Silverberg
- Affiliation: Department of Mathematics, Ohio State University, Columbus, Ohio 43210
- MR Author ID: 213982
- Email: silver@math.ohio-state.edu
- Received by editor(s): January 5, 2002
- Received by editor(s) in revised form: February 20, 2002
- Published electronically: July 8, 2002
- Additional Notes: The authors thank the NSF (grants DMS-9800881 and DMS-9988869), the Alexander von Humboldt Foundation, and the Universität Erlangen-Nürnberg. Silverberg also thanks the NSA (grant MDA904-99-1-0007), MSRI, and AIM
- © Copyright 2002 American Mathematical Society
- Journal: Bull. Amer. Math. Soc. 39 (2002), 455-474
- MSC (2000): Primary 11G05; Secondary 11-02, 14G05, 11G40, 14H52
- DOI: https://doi.org/10.1090/S0273-0979-02-00952-7
- MathSciNet review: 1920278