Average ranks of elliptic curves: Tension between data and conjecture

Bektemirov, Baur; Mazur, Barry; Stein, William; Watkins, Mark

doi:10.1090/S0273-0979-07-01138-X

Authors: Baur Bektemirov, Barry Mazur, William Stein and Mark Watkins
Journal: Bull. Amer. Math. Soc. 44 (2007), 233-254
MSC (2000): Primary 11-02, 11D25, 11Y35, 11Y40
DOI: https://doi.org/10.1090/S0273-0979-07-01138-X
Published electronically: February 15, 2007
MathSciNet review: 2291676
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Rational points on elliptic curves are the gems of arithmetic: they are, to diophantine geometry, what units in rings of integers are to algebraic number theory, what algebraic cycles are to algebraic geometry. A rational point in just the right context, at one place in the theory, can inhibit and control--thanks to ideas of Kolyvagin--the existence of rational points and other mathematical structures elsewhere. Despite all that we know about these objects, the initial mystery and excitement that drew mathematicians to this arena in the first place remains in full force today.

We have a network of heuristics and conjectures regarding rational points, and we have massive data accumulated to exhibit instances of the phenomena. Generally, we would expect that our data support our conjectures, and if not, we lose faith in our conjectures. But here there is a somewhat more surprising interrelation between data and conjecture: they are not exactly in open conflict one with the other, but they are no great comfort to each other either. We discuss various aspects of this story, including recent heuristics and data that attempt to resolve this mystery. We shall try to convince the reader that, despite seeming discrepancy, data and conjecture are, in fact, in harmony.

[ABCRZ] B. Allombert, K. Belabas, H. Cohen, X. Roblot, and I. Zakharevitch, PARI/GP, computer software, pari.math.u-bordeaux.fr
[Bek04] J. D. Bekenstein, An alternative to the dark matter paradigm: relativistic MOND gravitation. Invited talk at the 28th Johns Hopkins Workshop on Current Problems in Particle Theory, June 2004, Johns Hopkins University, Baltimore. Published online in JHEP Proceedings of Science, arxiv.org/astro-ph/0412652
[Bha05] Manjul Bhargava, The density of discriminants of quartic rings and fields, Ann. of Math. (2) 162 (2005), no. 2, 1031–1063. MR 2183288, https://doi.org/10.4007/annals.2005.162.1031
[BSD] B. J. Birch and H. P. F. Swinnerton-Dyer, Notes on elliptic curves. I, J. Reine Angew. Math. 212 (1963), 7–25. MR 0146143, https://doi.org/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 0179168, https://doi.org/10.1515/crll.1965.218.79
[BCDT01] Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor, On the modularity of elliptic curves over 𝐐: wild 3-adic exercises, J. Amer. Math. Soc. 14 (2001), no. 4, 843–939. MR 1839918, https://doi.org/10.1090/S0894-0347-01-00370-8
[BM90] Armand Brumer and Oisín McGuinness, The behavior of the Mordell-Weil group of elliptic curves, Bull. Amer. Math. Soc. (N.S.) 23 (1990), no. 2, 375–382. MR 1044170, https://doi.org/10.1090/S0273-0979-1990-15937-3
[Cas62] J. W. S. Cassels, Arithmetic on curves of genus 1. IV. Proof of the Hauptvermutung, J. Reine Angew. Math. 211 (1962), 95–112. MR 0163915, https://doi.org/10.1515/crll.1962.211.95
[CDO] Henri Cohen, Francisco Diaz y Diaz, and Michel Olivier, Counting discriminants of number fields of degree up to four, Algorithmic number theory (Leiden, 2000) Lecture Notes in Comput. Sci., vol. 1838, Springer, Berlin, 2000, pp. 269–283. MR 1850611, https://doi.org/10.1007/10722028_15
[CKRS02] J. B. Conrey, J. P. Keating, M. O. Rubinstein, and N. C. Snaith, On the frequency of vanishing of quadratic twists of modular 𝐿-functions, Number theory for the millennium, I (Urbana, IL, 2000) A K Peters, Natick, MA, 2002, pp. 301–315. MR 1956231
[CKRS06] J. B. Conrey, J. P. Keating, M. O. Rubinstein, and N. C. Snaith, Random matrix theory and the Fourier coefficients of half-integral-weight forms, Experiment. Math. 15 (2006), no. 1, 67–82. MR 2229387
[CPRW] J. B. Conrey, A. Pokharel, M. O. Rubinstein, and M. Watkins, Secondary terms in the number of vanishings of quadratic twists of elliptic curve -functions (2005); arxiv.org/math.NT/0509059
[CRSW] J. B. Conrey, M. O. Rubinstein, N. C. Snaith, and M. Watkins, Discretisation for odd quadratic twists (2006); arxiv.org/math.NT/0509428
[Cre] J. E. Cremona, Elliptic curve tables, www.maths.nott.ac.uk/personal/jec/ftp/data
[Cre97] J. E. Cremona, Algorithms for modular elliptic curves, 2nd ed., Cambridge University Press, Cambridge, 1997. MR 1628193
[DFK04] Chantal David, Jack Fearnley, and Hershy Kisilevsky, On the vanishing of twisted 𝐿-functions of elliptic curves, Experiment. Math. 13 (2004), no. 2, 185–198. MR 2068892
[Duk97] William Duke, Elliptic curves with no exceptional primes, C. R. Acad. Sci. Paris Sér. I Math. 325 (1997), no. 8, 813–818 (English, with English and French summaries). MR 1485897, https://doi.org/10.1016/S0764-4442(97)80118-8
[Fal86] G. Faltings, Finiteness theorems for abelian varieties over number fields, Arithmetic Geometry (Storrs, CT, 1984), Springer, New York, 1986; translated from the German original [Invent. Math. 73 (1983), no. 3, 349-366; ibid. 75 (1984), no. 2, 381] by Edward Shipz, pp. 9-27. MR 0861971, MR 0718935 (85g:11026a), MR 0732554 (85g:11026b)
[Fer96] Stéfane Fermigier, Étude expérimentale du rang de familles de courbes elliptiques sur Q, Experiment. Math. 5 (1996), no. 2, 119–130 (French, with English and French summaries). MR 1418959
[Gol79] 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
[Gr] See, for instance, a book review of Stalking the Riemann Hypothesis: The Quest to Find the Hidden Law of Prime Numbers by Dan Rockmore, reviewed by S. W. Graham in the MAA Online book review column at www.maa.org/reviews/stalkingrh.html, where discrepancies in versions of the story are discussed.
[GT02] Andrew Granville and Thomas J. Tucker, It’s as easy as 𝑎𝑏𝑐, Notices Amer. Math. Soc. 49 (2002), no. 10, 1224–1231. MR 1930670
[HB93] D. R. Heath-Brown, The size of Selmer groups for the congruent number problem, Invent. Math. 111 (1993), no. 1, 171–195. MR 1193603, https://doi.org/10.1007/BF01231285
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, https://doi.org/10.1007/BF01231536
[Hel04] H. A. Helfgott, On the behaviour of root numbers in families of elliptic curves (2004), submitted; arxiv.org/math.NT/0408141
[KaSa99] Nicholas M. Katz and Peter Sarnak, Random matrices, Frobenius eigenvalues, and monodromy, American Mathematical Society Colloquium Publications, vol. 45, American Mathematical Society, Providence, RI, 1999. MR 1659828
[KeSn00] J. P. Keating and N. C. Snaith, Random matrix theory and 𝜁(1/2+𝑖𝑡), Comm. Math. Phys. 214 (2000), no. 1, 57–89. MR 1794265, https://doi.org/10.1007/s002200000261
J. P. Keating and N. C. Snaith, Random matrix theory and 𝐿-functions at 𝑠=1/2, Comm. Math. Phys. 214 (2000), no. 1, 91–110. MR 1794267, https://doi.org/10.1007/s002200000262
[Kol88] V. A. Kolyvagin, Finiteness of ${E}(\mathbf{Q})$ and ${{\mbox{ {\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n} \selectfont Sh}}}}({E},\mathbf{Q})$ for a subclass of Weil curves, Izv. Akad. Nauk SSSR Ser. Mat. 52 (1988), no. 3, 522-540, 670-671. MR 0954295 (89m:11056)
[KowA] E. Kowalski, Elliptic curves, rank in families and random matrices. To appear in the Proceedings of the Isaac Newton Institute workshop on random matrices and -functions (July 2004). Also related to the author's lecture at the AIM/Princeton workshop on the Birch and Swinnerton-Dyer conjecture (November 2003). See www.math.u-bordeaux1.fr/~kowalski/elliptic-curves-families.pdf
[KowB] E. Kowalski, On the rank of quadratic twists of elliptic curves over function fields. To appear in International J. Number Theory. Online at arxiv.org/math.NT/0503732
[Mal02] Gunter Malle, On the distribution of Galois groups, J. Number Theory 92 (2002), no. 2, 315–329. MR 1884706, https://doi.org/10.1006/jnth.2001.2713
[Mal04] Gunter Malle, On the distribution of Galois groups. II, Experiment. Math. 13 (2004), no. 2, 129–135. MR 2068887
[Mpl05] Matplotlib, computer software, matplotlib.sourceforge.net
[MSD74] B. Mazur and P. Swinnerton-Dyer, Arithmetic of Weil curves, Invent. Math. 25 (1974), 1–61. MR 0354674, https://doi.org/10.1007/BF01389997
[Meh04] Madan Lal Mehta, Random matrices, 3rd ed., Pure and Applied Mathematics (Amsterdam), vol. 142, Elsevier/Academic Press, Amsterdam, 2004. MR 2129906
[Mor22] L. J. Mordell, On the Rational Solutions of the Indeterminate Equations of the Third and Fourth Degrees, Proc. Cambridge Philos. Soc. 21 (1922-23), 179-192.
[RS02] Karl Rubin and Alice Silverberg, Ranks of elliptic curves, Bull. Amer. Math. Soc. (N.S.) 39 (2002), no. 4, 455–474. MR 1920278, https://doi.org/10.1090/S0273-0979-02-00952-7
[Sha85] D. Shanks, Solved and Unsolved Problems in Number Theory. Third edition, Chelsea, New York, 1985. MR 0798284 (86j:11001)
[Sil01] Alice Silverberg, Open questions in arithmetic algebraic geometry, Arithmetic algebraic geometry (Park City, UT, 1999) IAS/Park City Math. Ser., vol. 9, Amer. Math. Soc., Providence, RI, 2001, pp. 83–142. MR 1860041
[Sil92] Joseph H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1992. Corrected reprint of the 1986 original. MR 1329092
[SJ05] W. Stein and D. Joyner, Sage: System for algebra and geometry experimentation, Communications in Computer Algebra (SIGSAM Bulletin) (July 2005). Online at sage.sourceforge.net
[SW02] W. A. Stein and M. Watkins, A database of elliptic curves--first report, Algorithmic Number Theory (Sydney, 2002), Lecture Notes in Comput. Sci., vol. 2369, Springer, Berlin, 2002, pp. 267-275. Online from modular.ucsd.edu/papers/stein-watkins
[Tat75] J. Tate, Algorithm for determining the type of a singular fiber in an elliptic pencil, Modular functions of one variable, IV (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Springer, Berlin, 1975, pp. 33–52. Lecture Notes in Math., Vol. 476. MR 0393039
[Teg04] M. Tegmark et al., Cosmological parameters from SDSS and WMAP, Phys. Rev. D 69 (2004) 103501. Online at arxiv.org/astro-ph/0310723.
[Wal81] J.-L. Waldspurger, Sur les coefficients de Fourier des formes modulaires de poids demi-entier (French) [On the Fourier coefficients of modular forms of half-integral weight], J. Math. Pures Appl. (9) 60 (1981), no. 4, 375-484. MR 0646366 (83h:10061)
[Wat04] M. Watkins, Rank distribution in a family of cubic twists. To appear in Proceedings of the Issac Newton Institute Workshop on Elliptic Curves and Random Matrix Theory. Online at arxiv.org/math.NT/0412427
[Wat06] -, Some heuristics about elliptic curves, Preprint (2006). To be submitted to Experimental Mathematics; currently online at www.maths.bris.ac.uk/~mamjw/heur.ps
[Wil95] Andrew Wiles, Modular elliptic curves and Fermat’s last theorem, Ann. of Math. (2) 141 (1995), no. 3, 443–551. MR 1333035, https://doi.org/10.2307/2118559
[YouA] Matthew P. Young, Low-lying zeros of families of elliptic curves, J. Amer. Math. Soc. 19 (2006), no. 1, 205–250. MR 2169047, https://doi.org/10.1090/S0894-0347-05-00503-5
[YouB] M. P. Young, On the nonvanishing of elliptic curve L-functions at the central point (2005), to appear in Proc. London Math. Soc.; arxiv.org/math.NT/0508185, AIM preprint 2004-30.
[ZK87] D. Zagier and G. Kramarz, Numerical investigations related to the -series of certain elliptic curves, J. Indian Math. Soc. (N.S.) 52 (1987), 51-69 (1988). MR 0989230 (90d:11072)
[Zwi33] F. Zwicky, Die Rotverschiebung von extragalaktischen Nebeln (German) [The red shift of extragalactic nebulae], Helvetica Phys. Acta, vol. 6 (1933), 110-127.