Average rank of elliptic curves [after Manjul Bhargava and Arul Shankar]

Poonen, Bjorn

Séminaire Bourbaki volume 2011/2012 exposés 1043-1058, Astérisque, no. 352 (2013), Exposé no. 1049, 18 p.

MR Zbl

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     author = {Poonen, Bjorn},
     title = {Average rank of elliptic curves [after {Manjul} {Bhargava} and {Arul} {Shankar]}},
     booktitle = {S\'eminaire Bourbaki volume 2011/2012 expos\'es 1043-1058},
     series = {Ast\'erisque},
     note = {talk:1049},
     pages = {187--204},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {352},
     year = {2013},
     mrnumber = {3087347},
     zbl = {1295.11065},
     language = {en},
     url = {http://www.numdam.org/item/AST_2013__352__187_0/}
}

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Poonen, Bjorn. Average rank of elliptic curves [after Manjul Bhargava and Arul Shankar], dans Séminaire Bourbaki volume 2011/2012 exposés 1043-1058, Astérisque, no. 352 (2013), Exposé no. 1049, 18 p. http://www.numdam.org/item/AST_2013__352__187_0/

Bibliographie
Cité par

[1] E. Bekyel - "The density of elliptic curves having a global minimal Weierstrass equation", J. Number Theory 109 (2004), p. 41-58. | DOI | MR | Zbl

[2] M. Bhargava - "Higher composition laws. III. The parametrization of quartic rings", Ann. of Math. 159 (2004), p. 1329-1360. | DOI | MR | Zbl

[3] M. Bhargava, "The density of discriminants of quartic rings and fields", Ann. of Math. 162 (2005), p. 1031-1063. | DOI | Zbl

[4] M. Bhargava & A. Shankar - "Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves", to appear in Annals of Math. | Zbl

[5] M. Bhargava, "Ternary cubic forms having bounded invariants, and the existence of a positive proportion of elliptic curves having rank $0$ ", preprint arXiv:1007.0052vl. | DOI | Zbl

[6] B. J. Birch & W. Kuyk (eds.) - Modular functions of one variable. IV, Lecture Notes in Math., vol. 476, Springer, 1975. | Zbl

[7] B. J. Birch & H. P. F. Swinnerton-Dyer - "Notes on elliptic curves. I", J. reine angew. Math. 212 (1963), p. 7-25. | EuDML | Zbl

[8] B. J. Birch, "Notes on elliptic curves. II", J. reine angew. Math. 218 (1965), p. 79-108. | EuDML | Zbl

[9] C. Breuil, B. Conrad, F. Diamond & R. Taylor - "On the modularity of elliptic curves over $𝐐$ : wild $3$ -adic exercises", J. Amer. Math. Soc. 14 (2001), p. 843-939. | DOI | Zbl

[10] A. Brumer - "The average rank of elliptic curves. I", Invent. Math. 109 (1992), p. 445-472. | DOI | Zbl

[11] D. Bump, S. Friedberg & J. Hoffstein - "Nonvanishing theorems for $L$ -functions of modular forms and their derivatives", Invent. Math. 102 (1990), p. 543-618. | DOI | EuDML | Zbl

[12] J. W. S. Cassels - "Arithmetic on curves of genus 1. IV. Proof of the Hauptver-mutung", J. reine angew. Math. 211 (1962), p. 95-112. | EuDML | Zbl

[13] H. Cohen & H. W. Lenstra Jr. - "Heuristics on class groups of number fields", Lecture Notes in Math. 1068 (1984), p. 33-62. | Zbl

[14] J. E. Cremona - Algorithms for modular elliptic curves, Cambridge Univ. Press, 1997. | Zbl

[15] J. E. Cremona, "Elliptic curve data", http://www.warwick.ac.Uk/staff/J.E.Cremona/ftp/data/INDEX.html. | Zbl

[16] J. E. Cremona, T. A. Fisher, C. O'Neil, D. Simon & M. Stoll - "Explicit $n$ -descent on elliptic curves. I. Algebra", J. reine angew. Math. 615 (2008), p. 121-155. | Zbl

[17] C. Delaunay - "Heuristics on Tate-Shafarevitch groups of elliptic curves defined over $ℚ$ ", Experiment. Math. 10 (2001), p. 191-196. | DOI | EuDML | Zbl

[18] T. Dokchitser & V. Dokchitser - "On the Birch-Swinnerton-Dyer quotients modulo squares", Ann. of Math. 172 (2010), p. 567-596. | DOI | Zbl

[19] É. Fouvry - "Sur le comportement en moyenne du rang des courbes $y^{2} = x^{3} + k$ ", in Séminaire de théorie de nombres, Paris, 1990-91, Progr. Math., vol. 108, Birkhäuser, 1993, p. 61-84. | Zbl

[20] É. Fouvry & J. Pomykała - "Rang des courbes elliptiques et sommes d'exponentielles", Monatsh. Math. 116 (1993), p. 111-125. | DOI | EuDML | Zbl

[21] D. Goldfeld - "Conjectures on elliptic curves over quadratic fields", Lecture Notes in Math. 751 (1979), p. 108-118. | Zbl

[22] A. Granville - "ABC allows us to count squarefrees", Internat. Math. Res. Notices 1998 (1998). | DOI | Zbl

[23] B. H. Gross & D. B. Zagier - "Heegner points and derivatives of $L$ -series", Invent. Math. 84 (1986), p. 225-320. | DOI | EuDML | Zbl

[24] D. R. Heath-Brown - "The size of Selmer groups for the congruent number problem", Invent. Math. 111 (1993), p. 171-195. | DOI | EuDML | Zbl

[25] D. R. Heath-Brown, "The size of Selmer groups for the congruent number problem. II", Invent. Math. 118 (1994), p. 331-370. | DOI | EuDML | Zbl

[26] D. R. Heath-Brown, "The average analytic rank of elliptic curves", Duke Math. J. 122 (2004), p. 591-623. | DOI | Zbl

[27] A. J. De Jong - "Counting elliptic surfaces over finite fields", Mosc. Math. J. 2 (2002), p. 281-311. | DOI | Zbl

[28] D. M. Kane - "On the ranks of the $2$ -Selmer groups of twists of a given elliptic curve", preprint arXiv:1009.1365v3. | Zbl

[29] N. M. Katz & P. Sarnak - Random matrices, Frobenius eigenvalues, and monodromy, American Mathematical Society Colloquium Publications, vol. 45, Amer. Math. Soc., 1999. | Zbl

[30] N. M. Katz & P. Sarnak, "Zeroes of zeta functions and symmetry", Bull. Amer. Math. Soc. 36 (1999), p. 1-26. | DOI | Zbl

[31] V. A. Kolyvagin - "Finiteness of $E (𝐐)$ and $SH (E, 𝐐)$ for a subclass of Weil curves", Izv. Akad. Nauk SSSR Ser. Mat. 52 (1988), p. 522-540, 670-671.

[32] V. A. Kolyvagin, "Euler systems", in The Grothendieck Festschrift, Vol. II, Progr. Math., vol. 87, Birkhäuser, 1990, p. 435-483. | Zbl

[33] B. Mazur - "Modular curves and the Eisenstein ideal", Publ. Math. IHÉS 47 (1977), p. 33-186. | DOI | EuDML | Numdam | Zbl

[34] J.-F. Mestre - "Construction d'une courbe elliptique de rang $\geq 12$ ", C. R. Acad. Sci. Paris 295 (1982), p. 643-644. | Zbl

[35] P. Michel - "Rang moyen de familles de courbes elliptiques et lois de Sato-Tate", Monatsh. Math. 120 (1995), p. 127-136. | DOI | EuDML | Zbl

[36] P. Michel, "Le rang de familles de variétés abéliennes", J. Algebraic Geom. 6 (1997), p. 201-234. | Zbl

[37] L. J. Mordell - "On the rational solutions of the indeterminate equations of the third and fourth degrees", Proc. Cambridge Phil Soc. 21 (1922), p. 179-192. | JFM

[38] M. R. Murty & V. K. Murty - "Mean values of derivatives of modular $L$ -series", Ann. of Math. 133 (1991), p. 447-475. | DOI | Zbl

[39] H. Poincaré - "Sur les propriétés arithmétiques des courbes algébriques", J. Pures Appl. Math. 7 (1901), p. 161-234. | EuDML | JFM | Numdam

[40] B. Poonen - "Squarefree values of multivariable polynomials", Duke Math. J. 118 (2003), p. 353-373. | DOI | Zbl

[41] B. Poonen & E. Rains - "Random maximal isotropic subspaces and Selmer groups", J. Amer. Math. Soc. 25 (2012), p. 245-269. | DOI | Zbl

[42] J. H. Silverman - "The average rank of an algebraic family of elliptic curves", J. reine angew. Math. 504 (1998), p. 227-236. | Zbl

[43] C. Skinner & E. Urban - "The Iwasawa main conjectures for $G L_{2}$ ", preprint http://www.math.Columbia.edu/~urban/eurp/MC.pdf, 2010-11. | Zbl

[44] W. A. Stein - "Modular forms database", http://modular.math.Washington.edu/Tables/index.html.

[45] P. Swinnerton-Dyer - "The effect of twisting on the $2$ -Selmer group", Math. Proc. Cambridge Philos. Soc. 145 (2008), p. 513-526. | DOI | Zbl

[46] R. Wazir - "A bound for the average rank of a family of Abelian varieties", Boll. Unione Mat. Ital. Sez. B Artie. Ric. Mat. 7 (2004), p. 241-252. | EuDML | Zbl

[47] M. E. M. Wood - "Moduli spaces for rings and ideals", Ph.D. Thesis, Princeton University, 2009.

[48] M. P. Young - "Low-lying zeros of families of elliptic curves", J. Amer. Math. Soc. 19 (2006), p. 205-250. | DOI | Zbl

[49] G. Yu - "Average size of $2$ -Selmer groups of elliptic curves. II", Acta Arith. 117 (2005), p. 1-33. | DOI | EuDML | Zbl

[50] G. Yu, "Average size of $2$ -Selmer groups of elliptic curves. I", Trans. Amer. Math. Soc. 358 (2006), p. 1563-1584. | DOI | Zbl