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

Average ranks of elliptic curves: Tension between data and conjecture
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by Baur Bektemirov, Barry Mazur, William Stein and Mark Watkins PDF

Bull. Amer. Math. Soc. 44 (2007), 233-254

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.

References

PARI/GP

http://pari.math.u-bordeaux.fr

An alternative to the dark matter paradigm: relativistic MOND gravitation.

arxiv.org/astro-ph/0412652

Manjul Bhargava, The density of discriminants of quartic rings and fields, Ann. of Math. (2) 162 (2005), no. 2, 1031–1063. MR 2183288, DOI 10.4007/annals.2005.162.1031
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
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 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, DOI 10.1090/S0273-0979-1990-15937-3
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
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, DOI 10.1007/10722028_{1}5
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
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, DOI 10.1080/10586458.2006.10128953

Secondary terms in the number of vanishings of quadratic twists of elliptic curve $L$-functions

arxiv.org/math.NT/0509059

Discretisation for odd quadratic twists

arxiv.org/math.NT/0509428

Elliptic curve tables

http://www.maths.nott.ac.uk/personal/jec/ftp/data

J. E. Cremona, Algorithms for modular elliptic curves, 2nd ed., Cambridge University Press, Cambridge, 1997. MR 1628193
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, DOI 10.1080/10586458.2004.10504532
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, DOI 10.1016/S0764-4442(97)80118-8
Gerd Faltings, Finiteness theorems for abelian varieties over number fields, Arithmetic geometry (Storrs, Conn., 1984) Springer, New York, 1986, pp. 9–27. Translated from the German original [Invent. Math. 73 (1983), no. 3, 349–366; MR0718935; ibid. 75 (1984), no. 2, 381; MR0732554] by Edward Shipz. MR 861971
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, DOI 10.1080/10586458.1996.10504583
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

Stalking the Riemann Hypothesis: The Quest to Find the Hidden Law of Prime Numbers

www.maa.org/reviews/stalkingrh.html

Andrew Granville and Thomas J. Tucker, It’s as easy as $abc$, Notices Amer. Math. Soc. 49 (2002), no. 10, 1224–1231. MR 1930670
D. R. Heath-Brown, The size of Selmer groups for the congruent number problem, Invent. Math. 111 (1993), no. 1, 171–195. MR 1193603, DOI 10.1007/BF01231285

On the behaviour of root numbers in families of elliptic curves

arxiv.org/math.NT/0408141

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, DOI 10.1090/coll/045
J. P. Keating and N. C. Snaith, Random matrix theory and $\zeta (1/2+it)$, Comm. Math. Phys. 214 (2000), no. 1, 57–89. MR 1794265, DOI 10.1007/s002200000261
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

Elliptic curves, rank in families and random matrices

www.math.u-bordeaux1.fr/~kowalski/elliptic-curves-families.pdf

On the rank of quadratic twists of elliptic curves over function fields

arxiv.org/math.NT/0503732

Gunter Malle, On the distribution of Galois groups, J. Number Theory 92 (2002), no. 2, 315–329. MR 1884706, DOI 10.1006/jnth.2001.2713
Gunter Malle, On the distribution of Galois groups. II, Experiment. Math. 13 (2004), no. 2, 129–135. MR 2068887, DOI 10.1080/10586458.2004.10504527

Matplotlib

matplotlib.sourceforge.net

B. Mazur and P. Swinnerton-Dyer, Arithmetic of Weil curves, Invent. Math. 25 (1974), 1–61. MR 354674, DOI 10.1007/BF01389997
Madan Lal Mehta, Random matrices, 3rd ed., Pure and Applied Mathematics (Amsterdam), vol. 142, Elsevier/Academic Press, Amsterdam, 2004. MR 2129906

On the Rational Solutions of the Indeterminate Equations of the Third and Fourth Degrees

21

Karl Rubin and Alice Silverberg, Ranks of elliptic curves, Bull. Amer. Math. Soc. (N.S.) 39 (2002), no. 4, 455–474. MR 1920278, DOI 10.1090/S0273-0979-02-00952-7
Daniel Shanks, Solved and unsolved problems in number theory, 3rd ed., Chelsea Publishing Co., New York, 1985. MR 798284
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, DOI 10.1007/bf03026854
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, DOI 10.1007/978-1-4757-4252-7

Sage: System for algebra and geometry experimentation

sage.sourceforge.net

A database of elliptic curves—first report

modular.ucsd.edu/papers/stein-watkins

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) Lecture Notes in Math., Vol. 476, Springer, Berlin, 1975, pp. 33–52. MR 0393039

Cosmological parameters from SDSS and WMAP,

69

arxiv.org/astro-ph/0310723

J.-L. Waldspurger, Sur les coefficients de Fourier des formes modulaires de poids demi-entier, J. Math. Pures Appl. (9) 60 (1981), no. 4, 375–484 (French). MR 646366

Rank distribution in a family of cubic twists

arxiv.org/math.NT/0412427

Some heuristics about elliptic curves

www.maths.bris.ac.uk/~mamjw/heur.ps

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
Matthew P. Young, Low-lying zeros of families of elliptic curves, J. Amer. Math. Soc. 19 (2006), no. 1, 205–250. MR 2169047, DOI 10.1090/S0894-0347-05-00503-5

On the nonvanishing of elliptic curve L-functions at the central point

arxiv.org/math.NT/0508185

D. Zagier and G. Kramarz, Numerical investigations related to the $L$-series of certain elliptic curves, J. Indian Math. Soc. (N.S.) 52 (1987), 51–69 (1988). MR 989230

Die Rotverschiebung von extragalaktischen Nebeln