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Hauteur asymptotique des points de Heegner

Published online by Cambridge University Press:  20 November 2018

Guillaume Ricotta  and
Thomas Vidick
Guillaume Ricotta
Affiliation:
Université de Bordeaux 1, Laboratoire A2X, Bâtiment A33, Bureau 301 M, 351 cours de la Liberation, 33405 Talence cedex, France courriel:, [email protected]
Thomas Vidick
Affiliation:
École Normale Supérieure, 75005 Paris, France courriel:, [email protected]
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Abstract

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Geometric intuition suggests that the Néron–Tate height of Heegner points on a rational elliptic curve $E$ should be asymptotically governed by the degree of its modular parametrisation. In this paper, we show that this geometric intuition asymptotically holds on average over a subset of discriminants. We also study the asymptotic behaviour of traces of Heegner points on average over a subset of discriminants and find a difference according to the rank of the elliptic curve. By the Gross–Zagier formulae, such heights are related to the special value at the critical point for either the derivative of the Rankin–Selberg convolution of $E$ with a certain weight one theta series attached to the principal ideal class of an imaginary quadratic field or the twisted $L$-function of $E$ by a quadratic Dirichlet character. Asymptotic formulae for the first moments associated with these $L$-series and $L$-functions are proved, and experimental results are discussed. The appendix contains some conjectural applications of our results to the problem of the discretisation of odd quadratic twists of elliptic curves.

Keywords

Primary: 11G50secondary: 11M41
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 60 , Issue 6 , 01 December 2008 , pp. 1406 - 1436
Copyright
Copyright © Canadian Mathematical Society 2008

References

[AbUl] Abbes, A. et Ullmo, E., À propos de la conjecture deManin pour les courbes elliptiques modulaires. CompositioMath. 103(1996), no. 3, 269–286.Google Scholar
[BiSw] Birchet, B. J. et Swinnerton-Dyer, H. P. F., Notes on elliptic curves. I. II. J. Reine Angew. Math. 212(1963), 7–25; 218(1965), 79–108.Google Scholar
[Bo] Bombieri, E., Le grand crible dans la Théorie Analytique des Nombres, Astérisque No. 18, Société Mathématique de France, Paris, 1974.Google Scholar
[CKRS] Conrey, J B., Keating, J P., Rubinstein, M. O., et Snaith, N. C., On the frequency of vanishing of quadratic twists of modular L-functions. Dans: Number Theory for the Millenium, I. A.K. Peters, Natick,MA, 2002, pp. 301–315.Google Scholar
[CRSW] Conrey, J. B., Rubinstein, M. O., Snaith, N. C., et M. Watkins, Discretisation for odd quadratic twists. http://www.arXiv.org/math.NT/0509428.Google Scholar
[Cr] Cremona, J. E., Elliptic Curve Data. disponible à http://www.warwick.ac.uk/˜ masgaj/ftp/data/INDEX.html Google Scholar
[DaGr] Darmon, H. et Green, P., Elliptic curves and class fields of real quadratic fields: algorithms and evidence. Experiment. Math. 11(2002), 37–55.Google Scholar
[De] Delaunay, C., Formes modulaires et invariants de courbes elliptiques définies sur Q. Thèse de doctorat, Université Bordeaux I, 2002, http://igd.univ-lyon1.fr/˜ delaunay/.Google Scholar
[De2] Delaunay, C., Moments of the orders of Tate-Shafarevich groups. Int. J. Number Theory 1(2005), no. 2, 243–264.Google Scholar
[Ed] Edixhoven, B., On theManin constants of modular elliptic curves. Dans: Arithmetic Algebraic Geometry. Progr. Math. 89. Birkhäuser Boston, Boston, MA, 1991, pp. 25–39.Google Scholar
[Gr] Gross, B. H., Heegner Points on X0(N). Dans: Modular Forms. Halsted Press, 1984, pp. 87–105.Google Scholar
[GrZa] Gross, B. H. et Zagier, D., Heegner points and derivatives of L-series. Invent. Math. 84(1986), 225–320.Google Scholar
[He] Heath-Brown, D. R., A mean value estimate for real character sums. Acta Arith. 72(1995), no. 3, 235–275.Google Scholar
[Iw] Iwaniec, H., On the order of vanishing of modular L-functions at the critical point. Sém. Théor. Nombres Bordeaux 2(1990), no. 2, 365–375.Google Scholar
[IwKo] Iwaniec, H. et Kowalski, E., Analytic number theory. AmericanMathematical Society Colloquium Publications 53. AmericanMathematical Society, Providence RI, 2004.Google Scholar
[Ma] Manin, J., Parabolic points and zeta functions of modular curves. Izv. Akad. Nauk SSSR Ser. Mat. 36(1972), 19–66 (Russian).Google Scholar
[Mu] Murty, M. R., Bounds for congruence primes. Dans: Automorphic Forms, Automorphic Representations, and Arithmetic. Proc. Sympos. Pure Math. 66, AmericanMathematical Society, Providence RI, 1999, pp. 177–192.Google Scholar
[PePo] Perelli, A. et Pomykala, J., Averages of twisted elliptic L-functions. Acta Arith. 80(1997), no. 2, 149–163.Google Scholar
[RoSi] Rosen, M. et Silverman, J. H., On the independence of Heegner points associated to distinct quadratic imaginary fields. J. Number Theory 127(2007), no. 1, 10–36.Google Scholar
[TaWi] Taylor, R. et Wiles, A., Ring-theoretic properties of certain Hecke algebras. Ann. of Math. 141(1995), no. 2, 553–572.Google Scholar
[Sh] Shimura, G., On the holomorphy of a certain Dirichlet series. Proc. London Math. Soc. 31(1975), no. 1, 79–98.Google Scholar
[Si] Silverman, J. H., The arithmetic of elliptic curves. Graduate Texts inMathematics 106, Springer-Verlag, New York, 1986.Google Scholar
[Wa] Watkins, M., Computing the modular degree of an elliptic curve. Experiment. Math. 11(2002) no. 4, 487–502.Google Scholar
[Wa2] Watkins, M., Some remarks on Heegner point computations. http://arxiv.org/abs/math.NT/0506325. Google Scholar
[Wi] Wiles, A., Modular elliptic curves and Fermat's last theorem. Ann. of Math. 141(1995), no. 3, 443–551.Google Scholar