×
Bhargava, Manjul; Shankar, Arul

Ternary cubic forms having bounded invariants, and the existence of a positive proportion of elliptic curves having rank 0. (English) Zbl 1317.11038

Ann. Math. (2) 181, No. 2, 587-621 (2015).
The authors have proved an asymptotic formula for the number of \(\mathrm{SL}_3(\mathbb{Z})\)-equivalence classes of integral ternary cubic forms which have bounded invariants.
As an application, they have obtained the average size of the 3-Selmer group of all elliptic curves when these are ordered by height. Specifically, they have shown that this is equal to 4. They also provide a proof of the fact that a positive proportion of all elliptic curves are of rank 0. Additionally, the authors have proven that a positive proportion of elliptic curves have analytic rank 0 as well.
Reviewer: Michael Th. Rassias (Princeton)

Cited in 2 Reviews
Cited in 24 Documents

MSC:

11E76 Forms of degree higher than two
11G05 Elliptic curves over global fields

Keywords:

cubic forms; invariants; elliptic curves; rank; Selmer group; Tate-Shafarevich group; Iwasawa main conjectures
PDF BibTeX XML Cite
\textit{M. Bhargava} and \textit{A. Shankar}, Ann. Math. (2) 181, No. 2, 587--621 (2015; Zbl 1317.11038)
Full Text: DOI arXiv
  • logo of hbz
  • logo of WorldCat

References:

[1] S. Y. An, S. Y. Kim, D. C. Marshall, S. H. Marshall, W. G. McCallum, and A. R. Perlis, ”Jacobians of genus one curves,” J. Number Theory, vol. 90, iss. 2, pp. 304-315, 2001. · Zbl 1066.14035
[2] S. Aronhold, ”Theorie der homogenen Funktionen dritten Grades von drei Veränderlichen,” J. reine Angew. Math., vol. 55, pp. 97-191, 1858. · ERAM 055.1455cj
[3] M. Artin, F. Rodriguez-Villegas, and J. Tate, ”On the Jacobians of plane cubics,” Adv. Math., vol. 198, iss. 1, pp. 366-382, 2005. · Zbl 1092.14054
[4] B. Bektemirov, B. Mazur, W. Stein, and M. Watkins, ”Average ranks of elliptic curves: tension between data and conjecture,” Bull. Amer. Math. Soc., vol. 44, iss. 2, pp. 233-254, 2007. · Zbl 1190.11032
[5] M. Bhargava, ”The density of discriminants of quintic rings and fields,” Ann. of Math., vol. 172, iss. 3, pp. 1559-1591, 2010. · Zbl 1220.11139
[6] M. Bhargava, The Ekedahl sieve and the density of squarefree values of invariant polynomials. · Zbl 1220.11139
[7] M. Bhargava and W. Ho, Coregular spaces and genus one curves. · Zbl 1342.14074
[8] M. Bhargava and A. Shankar, ”Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves,” Ann. of Math., vol. 181, iss. 1, pp. 191-242, 2015. · Zbl 1307.11071
[9] B. J. Birch and H. P. F. Swinnerton-Dyer, ”Notes on elliptic curves. I,” J. Reine Angew. Math., vol. 212, pp. 7-25, 1963. · Zbl 0118.27601
[10] A. Borel and Harish-Chandra, ”Arithmetic subgroups of algebraic groups,” Ann. of Math., vol. 75, pp. 485-535, 1962. · Zbl 0107.14804
[11] A. Brumer and K. Kramer, ”The rank of elliptic curves,” Duke Math. J., vol. 44, iss. 4, pp. 715-743, 1977. · Zbl 0376.14011
[12] J. W. S. Cassels, ”Arithmetic on curves of genus \(1\). IV. Proof of the Hauptvermutung,” J. Reine Angew. Math., vol. 211, pp. 95-112, 1962. · Zbl 0106.03706
[13] J. E. Cremona, T. A. Fisher, and M. Stoll, ”Minimisation and reduction of 2-, 3- and 4-coverings of elliptic curves,” Algebra Number Theory, vol. 4, iss. 6, pp. 763-820, 2010. · Zbl 1222.11073
[14] H. Darmon, F. Diamond, and R. Taylor, ”Fermat’s last theorem,” in Elliptic Curves, Modular Forms & Fermat’s Last Theorem, Cambridge, MA: Int. Press, 1997, pp. 2-140. · Zbl 0997.11504
[15] H. Davenport, ”On a principle of Lipschitz,” J. London Math. Soc., vol. 26, pp. 179-183, 1951. · Zbl 0042.27504
[16] A. J. de Jong, ”Counting elliptic surfaces over finite fields,” Mosc. Math. J., vol. 2, iss. 2, pp. 281-311, 2002. · Zbl 1031.11033
[17] T. Dokchitser and V. Dokchitser, ”On the Birch-Swinnerton-Dyer quotients modulo squares,” Ann. of Math., vol. 172, iss. 1, pp. 567-596, 2010. · Zbl 1223.11079
[18] W. Duke, ”Elliptic curves with no exceptional primes,” C. R. Acad. Sci. Paris Sér. I Math., vol. 325, iss. 8, pp. 813-818, 1997. · Zbl 1002.11049
[19] T. Fisher, ”Testing equivalence of ternary cubics,” in Algorithmic Number Theory, New York: Springer-Verlag, 2006, vol. 4076, pp. 333-345. · Zbl 1143.11325
[20] T. Fisher, ”The invariants of a genus one curve,” Proc. Lond. Math. Soc., vol. 97, iss. 3, pp. 753-782, 2008. · Zbl 1221.11135
[21] C. G. Gibson, Elementary Geometry of Algebraic Curves: An Undergraduate Introduction, Cambridge: Cambridge Univ. Press, 1998. · Zbl 0997.14500
[22] A. Kraus, ”Quelques remarques à propos des invariants \(c_4,\;c_6\) et \(\Delta\) d’une courbe elliptique,” Acta Arith., vol. 54, iss. 1, pp. 75-80, 1989. · Zbl 0628.14024
[23] S. Lang and A. Weil, ”Number of points of varieties in finite fields,” Amer. J. Math., vol. 76, pp. 819-827, 1954. · Zbl 0058.27202
[24] R. P. Langlands, ”The volume of the fundamental domain for some arithmetical subgroups of Chevalley groups,” in Algebraic Groups and Discontinuous Subgroups, Providence, RI: Amer. Math. Soc., 1966, pp. 143-148. · Zbl 0218.20041
[25] J. Nekovávr, Selmer Complexes, , 2006, vol. 310. · Zbl 1211.11120
[26] D. E. Rohrlich, ”Variation of the root number in families of elliptic curves,” Compositio Math., vol. 87, iss. 2, pp. 119-151, 1993. · Zbl 0791.11026
[27] J. H. Silverman, The Arithmetic of Elliptic Curves, New York: Springer-Verlag, 1986, vol. 106. · Zbl 0585.14026
[28] C. Skinner and E. Urban, ”The Iwasawa Main Conjectures for \( GL_2\),” Invent. Math., vol. 195, iss. 1, pp. 1-277, 2014. · Zbl 1301.11074
[29] S. Wong, ”On the density of elliptic curves,” Compositio Math., vol. 127, iss. 1, pp. 23-54, 2001. · Zbl 1003.11023
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.