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Guillaume Ricotta; Nicolas Templier
Comportement asympotique des hauteurs des points de Heegner
Journal de théorie des nombres de Bordeaux, 21 no. 3 (2009), p. 743-755, doi: 10.5802/jtnb.700
Article PDF | Reviews MR 2605545 | Zbl pre05774809
Class. Math.: 11G50, 11M41

Résumé - Abstract

Asymptotic behaviour for the averaged height of Heegner points

The leading order term for the average, over quadratic discriminants satisfying the so-called Heegner condition, of the Néron-Tate height of Heegner points on a rational elliptic curve $E$ has been determined in [13]. In addition, the second order term has been conjectured. In this paper, we prove that this conjectured second order term is the right one; this yields a power saving in the remainder term. Cancellations of Fourier coefficients of $GL_2$-cusp forms in arithmetic progressions lie in the core of the proof.

Bibliography

[1] K. Chandrasekharan & Raghavan Narasimhan, Hecke’s functional equation and the average order of arithmetical functions, Acta Arith. 6 (1960/1961), p. 487-503 Article |  MR 126423 |  Zbl 0101.03703
[2] K. Chandrasekharan & Raghavan Narasimhan, Functional equations with multiple gamma factors and the average order of arithmetical functions, Ann. of Math. (2) 76 (1962), p. 93-136  MR 140491 |  Zbl 0211.37901
[3] W. Duke & H. Iwaniec, Estimates for coefficients of $L$-functions. I, Automorphic forms and analytic number theory (Montreal, PQ, 1989), Univ. Montréal, 1990, p. 43–47  MR 1111010 |  Zbl 0745.11030
[4] W. Duke & H. Iwaniec, Estimates for coefficients of $L$-functions. II, in Proceedings of the Amalfi Conference on Analytic Number Theory (Maiori, 1989), Univ. Salerno, 1992, p. 71-82  MR 1220457 |  Zbl 0787.11020
[5] W. Duke & H. Iwaniec, Estimates for coefficients of $L$-functions. III, Séminaire de Théorie des Nombres, Paris, 1989–90, Progr. Math. 102, Birkhäuser Boston, 1992, p. 113–120  MR 1476732 |  Zbl 0763.11024
[6] W. Duke & H. Iwaniec, Estimates for coefficients of $L$-functions. IV, Amer. J. Math. 116 (1994), p. 207-217  MR 1262431 |  Zbl 0820.11032
[7] D. Goldfeld, J. Hoffstein & Lieman D., An effective zero free region, Ann. of Math. (2) 140 (1994)  MR 1289494
[8] Benedict H. Gross & Don B. Zagier, Heegner points and derivatives of $L$-series, Invent. Math. 84 (1986), p. 225-320  MR 833192 |  Zbl 0608.14019
[9] James Lee Hafner & Aleksandar Ivić, On sums of Fourier coefficients of cusp forms, Enseign. Math. (2) 35 (1989), p. 375-382  MR 1039952 |  Zbl 0696.10020
[10] Henryk Iwaniec, On the order of vanishing of modular $L$-functions at the critical point, Sém. Théor. Nombres Bordeaux (2) 2 (1990), p. 365-376 Cedram |  MR 1081731 |  Zbl 0719.11029
[11] Henryk Iwaniec & Emmanuel Kowalski, Analytic number theory, American Mathematical Society Colloquium Publications 53, American Mathematical Society, Providence, RI, 2004  MR 2061214 |  Zbl 1059.11001
[12] R. A. Rankin, Sums of cusp form coefficients, Automorphic forms and analytic number theory (Montreal, PQ, 1989), Univ. Montréal, 1990, p. 115–121  MR 1111014 |  Zbl 0735.11023
[13] Guillaume Ricotta & Thomas Vidick, Hauteur asymptotique des points de Heegner, Canad. J. Math. 60 (2008), p. 1406-1436  MR 2462452 |  Zbl pre05382118
[14] Goro Shimura, The special values of the zeta functions associated with cusp forms, Comm. Pure Appl. Math. 29 (1976), p. 783-804  MR 434962 |  Zbl 0348.10015
[15] Joseph H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics 106, Springer-Verlag, New York, 1992, Corrected reprint of the 1986 original  MR 1329092 |  Zbl 0585.14026
[16] Richard Taylor & Andrew Wiles, Ring-theoretic properties of certain Hecke algebras, Ann. of Math. (2) 141 (1995), p. 553-572  MR 1333036 |  Zbl 0823.11030
[17] Andrew Wiles, Modular elliptic curves and Fermat’s last theorem, Ann. of Math. (2) 141 (1995), p. 443-551  MR 1333035 |  Zbl 0823.11029