How many rational points does a random curve have?
HTML articles powered by AMS MathViewer
- by Wei Ho
- Bull. Amer. Math. Soc. 51 (2014), 27-52
- DOI: https://doi.org/10.1090/S0273-0979-2013-01433-2
- Published electronically: September 30, 2013
- PDF | Request permission
Previous version: Original version posted September 18, 2013
Corrected version: Current version corrects publisher's error in rendering author's corrections.
Abstract:
A large part of modern arithmetic geometry is dedicated to or motivated by the study of rational points on varieties. For an elliptic curve over ${\mathbb {Q}}$, the set of rational points forms a finitely generated abelian group. The ranks of these groups, when ranging over all elliptic curves, are conjectured to be evenly distributed between rank $0$ and rank $1$, with higher ranks being negligible. We will describe these conjectures and discuss some results on bounds for average rank, highlighting recent work of Bhargava and Shankar.References
- Baur Bektemirov, Barry Mazur, William Stein, and Mark Watkins, Average ranks of elliptic curves: tension between data and conjecture, Bull. Amer. Math. Soc. (N.S.) 44 (2007), no. 2, 233–254. MR 2291676, DOI 10.1090/S0273-0979-07-01138-X
- Manjul Bhargava, Higher composition laws. I. A new view on Gauss composition, and quadratic generalizations, Ann. of Math. (2) 159 (2004), no. 1, 217–250. MR 2051392, DOI 10.4007/annals.2004.159.217
- Manjul Bhargava, Higher composition laws. II. On cubic analogues of Gauss composition, Ann. of Math. (2) 159 (2004), no. 2, 865–886. MR 2081442, DOI 10.4007/annals.2004.159.865
- Manjul Bhargava, Higher composition laws. III. The parametrization of quartic rings, Ann. of Math. (2) 159 (2004), no. 3, 1329–1360. MR 2113024, DOI 10.4007/annals.2004.159.1329
- 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
- Manjul Bhargava, Higher composition laws. IV. The parametrization of quintic rings, Ann. of Math. (2) 167 (2008), no. 1, 53–94. MR 2373152, DOI 10.4007/annals.2008.167.53
- Manjul Bhargava, The density of discriminants of quintic rings and fields, Ann. of Math. (2) 172 (2010), no. 3, 1559–1591. MR 2745272, DOI 10.4007/annals.2010.172.1559
- Manjul Bhargava, The geometric squarefree sieve and unramified nonabelian extensions of quadratic fields, preprint, 2011.
- Manjul Bhargava, Most hyperelliptic curves over $\mathbb {Q}$ have no rational points, http://arxiv.org/abs/1308.0395.
- Manjul Bhargava and Benedict H. Gross, The average size of the 2-Selmer group of Jacobians of hyperelliptic curves having a rational Weierstrass point, 2012, http://arxiv.org/abs/1208.1007.
- Manjul Bhargava and Wei Ho, On the average sizes of Selmer groups in families of elliptic curves, preprint, 2012.
- Manjul Bhargava and Wei Ho, Coregular spaces and genus one curves, 2013, http://arxiv.org/abs/1306.4424.
- Manjul Bhargava and Arul Shankar, Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves, 2010, http://arxiv.org/abs/1006.1002.
- Manjul Bhargava and Arul Shankar, Ternary cubic forms having bounded invariants, and the existence of a positive proportion of elliptic curves having rank 0, 2010, http://arxiv.org/abs/1007.0052.
- Manjul Bhargava, Daniel Kane, Hendrik Lenstra, Bjorn Poonen, and Eric Rains, Modeling the distribution of ranks, Selmer groups, and Shafarevich-Tate groups of elliptic curves, 2013, http://arxiv.org/abs/1304.3971.
- 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
- Enrico Bombieri, The Mordell conjecture revisited, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 17 (1990), no. 4, 615–640. MR 1093712
- 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, The average rank of elliptic curves. I, Invent. Math. 109 (1992), no. 3, 445–472. MR 1176198, DOI 10.1007/BF01232033
- 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
- Claude Chabauty, Sur les points rationnels des courbes algébriques de genre supérieur à l’unité, C. R. Acad. Sci. Paris 212 (1941), 882–885 (French). MR 4484
- H. Cohen and H. W. Lenstra Jr., Heuristics on class groups of number fields, Number theory, Noordwijkerhout 1983 (Noordwijkerhout, 1983) Lecture Notes in Math., vol. 1068, Springer, Berlin, 1984, pp. 33–62. MR 756082, DOI 10.1007/BFb0099440
- H. Cohen and J. Martinet, Class groups of number fields: numerical heuristics, Math. Comp. 48 (1987), no. 177, 123–137. MR 866103, DOI 10.1090/S0025-5718-1987-0866103-4
- Robert F. Coleman, Effective Chabauty, Duke Math. J. 52 (1985), no. 3, 765–770. MR 808103, DOI 10.1215/S0012-7094-85-05240-8
- 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
- John Cremona, The elliptic curve database for conductors to 130000, Algorithmic number theory, Lecture Notes in Comput. Sci., vol. 4076, Springer, Berlin, 2006, pp. 11–29. MR 2282912, DOI 10.1007/11792086_{2}
- John Cremona, mwrank program, 2012, http://homepages.warwick.ac.uk/~masgaj/mwrank/.
- J. E. Cremona and D. Rusin, Efficient solution of rational conics, Math. Comp. 72 (2003), no. 243, 1417–1441. MR 1972744, DOI 10.1090/S0025-5718-02-01480-1
- 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, 763–820. MR 2728489, DOI 10.2140/ant.2010.4.763
- Boris Datskovsky and David J. Wright, Density of discriminants of cubic extensions, J. Reine Angew. Math. 386 (1988), 116–138. MR 936994, DOI 10.1515/crll.1988.386.116
- H. Davenport, On a principle of Lipschitz, J. London Math. Soc. 26 (1951), 179–183. MR 43821, DOI 10.1112/jlms/s1-26.3.179
- H. Davenport, On the class-number of binary cubic forms. I, J. London Math. Soc. 26 (1951), 183–192. MR 43822, DOI 10.1112/jlms/s1-26.3.183
- H. Davenport, On the class-number of binary cubic forms. II, J. London Math. Soc. 26 (1951), 192–198. MR 43823, DOI 10.1112/jlms/s1-26.3.192
- H. Davenport, Corrigendum: “On a principle of Lipschitz“, J. London Math. Soc. 39 (1964), 580. MR 166155, DOI 10.1112/jlms/s1-39.1.580-t
- H. Davenport and H. Heilbronn, On the density of discriminants of cubic fields, Bull. London Math. Soc. 1 (1969), 345–348. MR 254010, DOI 10.1112/blms/1.3.345
- Christophe Delaunay, Heuristics on Tate-Shafarevitch groups of elliptic curves defined over $\Bbb Q$, Experiment. Math. 10 (2001), no. 2, 191–196. MR 1837670
- Christophe Delaunay, Heuristics on class groups and on Tate-Shafarevich groups: the magic of the Cohen-Lenstra heuristics, Ranks of Elliptic Curves and Random Matrix Theory, London Math. Soc. Lecture Note Ser., vol. 341, Cambridge Univ. Press, Cambridge, 2007, pp. 323–340.
- B. N. Delone and D. K. Faddeev, The theory of irrationalities of the third degree, Translations of Mathematical Monographs, Vol. 10, American Mathematical Society, Providence, RI, 1964. MR 160744
- Tim Dokchitser and Vladimir Dokchitser, On the Birch-Swinnerton-Dyer quotients modulo squares, Ann. of Math. (2) 172 (2010), no. 1, 567–596. MR 2680426, DOI 10.4007/annals.2010.172.567
- Torsten Ekedahl, An infinite version of the Chinese remainder theorem, Comment. Math. Univ. St. Paul. 40 (1991), no. 1, 53–59. MR 1104780
- Noam Elkies, Three lectures on elliptic surfaces and curves of high rank, 2007, http://arxiv.org/abs/0709.2908.
- G. Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math. 73 (1983), no. 3, 349–366 (German). MR 718935, DOI 10.1007/BF01388432
- Gerd Faltings, Diophantine approximation on abelian varieties, Ann. of Math. (2) 133 (1991), no. 3, 549–576. MR 1109353, DOI 10.2307/2944319
- Carl Friedrich Gauss, Disquisitiones arithmeticae, 1801.
- 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
- 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
- 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
- D. R. Heath-Brown, The average analytic rank of elliptic curves, Duke Math. J. 122 (2004), no. 3, 591–623. MR 2057019, DOI 10.1215/S0012-7094-04-12235-3
- A. J. de Jong, Counting elliptic surfaces over finite fields, Mosc. Math. J. 2 (2002), no. 2, 281–311. Dedicated to Yuri I. Manin on the occasion of his 65th birthday. MR 1944508, DOI 10.17323/1609-4514-2002-2-2-281-311
- Anthony C. Kable and Akihiko Yukie, Prehomogeneous vector spaces and field extensions. II, Invent. Math. 130 (1997), no. 2, 315–344. MR 1474160, DOI 10.1007/s002220050187
- Anthony C. Kable and Akihiko Yukie, The mean value of the product of class numbers of paired quadratic fields. I, Tohoku Math. J. (2) 54 (2002), no. 4, 513–565. MR 1936267
- Daniel M. Kane, On the ranks of the 2-Selmer groups of twists of a given elliptic curve, 2012, http://arxiv.org/abs/1009.1365.
- 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
- Zev Klagsbrun, Barry Mazur, and Karl Rubin, Selmer ranks of quadratic twists of elliptic curves, 2011, http://arxiv.org/abs/1111.2321.
- 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
- 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
- B. Mazur and K. Rubin, Ranks of twists of elliptic curves and Hilbert’s tenth problem, Invent. Math. 181 (2010), no. 3, 541–575. MR 2660452, DOI 10.1007/s00222-010-0252-0
- F. Mertens, Ueber einige asymptotische Gesetze der Zahlentheorie, J. Reine Angew. Math. 77 (1874), 289–338.
- Louis J. Mordell, On the rational solutions of the indeterminate equation of the third and fourth degrees, Proc. Cambridge Philos. Soc. 21 (1922), 179–192.
- Bjorn Poonen, Squarefree values of multivariable polynomials, Duke Math. J. 118 (2003), no. 2, 353–373. MR 1980998, DOI 10.1215/S0012-7094-03-11826-8
- Bjorn Poonen, Bertini theorems over finite fields, Ann. of Math. (2) 160 (2004), no. 3, 1099–1127. MR 2144974, DOI 10.4007/annals.2004.160.1099
- Bjorn Poonen, Average rank of elliptic curves, Séminaire Bourbaki, 2011-2012, 64ème année no. 1049.
- Bjorn Poonen and Eric Rains, Random maximal isotropic subspaces and Selmer groups, J. Amer. Math. Soc. 25 (2012), no. 1, 245–269. MR 2833483, DOI 10.1090/S0894-0347-2011-00710-8
- Bjorn Poonen and Michael Stoll, Chabauty’s method proves that most odd degree hyperelliptic curves have only one rational point, in preparation.
- Mikio Sato and Takuro Shintani, On zeta functions associated with prehomogeneous vector spaces, Ann. of Math. (2) 100 (1974), 131–170. MR 344230, DOI 10.2307/1970844
- Arul Shankar and Xiaoheng Wang, Average size of the 2-Selmer group of jacobians of monic even hyperelliptic curves, 2013, http://arxiv.org/abs/1307.3531.
- Carl Ludwig Siegel, The average measure of quadratic forms with given determinant and signature, Ann. of Math. (2) 45 (1944), 667–685. MR 12642, DOI 10.2307/1969296
- Carl Ludwig Siegel, Über einige Anwendungen diophantischer Approximationen (1929), Gesammelte Abhandlungen. Bände I, II, III, Springer-Verlag, Berlin, 1966, pp. 209–266.
- A. Silverberg, The distribution of ranks in families of quadratic twists of elliptic curves, Ranks of elliptic curves and random matrix theory, London Math. Soc. Lecture Note Ser., vol. 341, Cambridge Univ. Press, Cambridge, 2007, pp. 171–176. MR 2322342, DOI 10.1017/CBO9780511735158.008
- 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
- Christopher Skinner and Eric Urban, Vanishing of $L$-functions and ranks of Selmer groups, International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 473–500. MR 2275606
- Christopher Skinner and Eric Urban, The Iwasawa main conjectures for GL(2), preprint. Available at http://www.math.columbia.edu/~urban/eurp/MC.pdf, 2010.
- William A. Stein and Mark 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. MR 2041090, DOI 10.1007/3-540-45455-1_{2}2
- W. A. Stein et al., Sage Mathematics Software (Version 5.3), The Sage Development Team, 2012, http://www.sagemath.org.
- Peter Swinnerton-Dyer, The effect of twisting on the 2-Selmer group, Math. Proc. Cambridge Philos. Soc. 145 (2008), no. 3, 513–526. MR 2464773, DOI 10.1017/S0305004108001588
- Takashi Taniguchi, A mean value theorem for the square of class number times regulator of quadratic extensions, Ann. Inst. Fourier (Grenoble) 58 (2008), no. 2, 625–670 (English, with English and French summaries). MR 2410385
- Takashi Taniguchi and Frank Thorne, Secondary terms in counting functions for cubic fields, 2011, http://arxiv.org/abs/1102.2914.
- 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
- Jack A. Thorne, The Arithmetic of Simple Singularities, ProQuest LLC, Ann Arbor, MI, 2012. Thesis (Ph.D.)–Harvard University. MR 3054927
- Paul Vojta, Siegel’s theorem in the compact case, Ann. of Math. (2) 133 (1991), no. 3, 509–548. MR 1109352, DOI 10.2307/2944318
- Mark Watkins, Rank distribution in a family of cubic twists, Ranks of elliptic curves and random matrix theory, London Math. Soc. Lecture Note Ser., vol. 341, Cambridge Univ. Press, Cambridge, 2007, pp. 237–246. MR 2322349, DOI 10.1017/CBO9780511735158.015
- Mark Watkins, Some heuristics about elliptic curves, Experiment. Math. 17 (2008), no. 1, 105–125. MR 2410120
- 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
- David J. Wright and Akihiko Yukie, Prehomogeneous vector spaces and field extensions, Invent. Math. 110 (1992), no. 2, 283–314. MR 1185585, DOI 10.1007/BF01231334
- 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
- Gang Yu, Average size of 2-Selmer groups of elliptic curves. II, Acta Arith. 117 (2005), no. 1, 1–33. MR 2110501, DOI 10.4064/aa117-1-1
- Gang Yu, Average size of 2-Selmer groups of elliptic curves. I, Trans. Amer. Math. Soc. 358 (2006), no. 4, 1563–1584. MR 2186986, DOI 10.1090/S0002-9947-05-03806-7
- Akihiko Yukie, Shintani zeta functions, London Mathematical Society Lecture Note Series, vol. 183, Cambridge University Press, Cambridge, 1993. MR 1267735
- Akihiko Yukie, Prehomogeneous vector spaces and field extensions. III, J. Number Theory 67 (1997), no. 1, 115–137. MR 1485429, DOI 10.1006/jnth.1997.2182
- 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
Bibliographic Information
- Wei Ho
- Affiliation: Department of Mathematics, Columbia University, New York, New York 10027
- MR Author ID: 770878
- Email: who@math.columbia.edu
- Received by editor(s): May 23, 2013
- Published electronically: September 30, 2013
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Bull. Amer. Math. Soc. 51 (2014), 27-52
- MSC (2010): Primary 11G05, 14H52; Secondary 11G30, 14H25
- DOI: https://doi.org/10.1090/S0273-0979-2013-01433-2
- MathSciNet review: 3119821