Comportement asympotique des hauteurs des points de Heegner
Journal de théorie des nombres de Bordeaux, Volume 21 (2009) no. 3, pp. 743-755.

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.

Le terme principal de la moyenne, sur les discriminants quadratiques satisfaisant la condition de Heegner, de la hauteur de Néron-Tate des points de Heegner d’une courbe elliptique rationnelle E a été déterminé dans [13]. Les auteurs ont également conjecturé l’expression du terme suivant. Dans cet article, il est démontré que cette expression est correcte et une asymptotique précise, qui sauve une puissance dans le terme d’erreur, est obtenue. Les annulations des coefficients de Fourier de formes sur GL 2 dans les progressions arithmétiques sont au cœur de la démonstration.

DOI: 10.5802/jtnb.700
Classification: 11G50, 11M41
Ricotta, Guillaume 1; Templier, Nicolas 2

1 Université Bordeaux 1 Institut de Mathématiques de Bordeaux Laboratoire A2X 351, cours de la libération 33405 Talence Cedex, France
2 Université Montpellier 2 Institut de Mathématiques et de Modélisation de Montpellier Case courrier 051 Place Eugène Bataillon 34095 Montpellier Cedex, France
@article{JTNB_2009__21_3_743_0,
     author = {Ricotta, Guillaume and Templier, Nicolas},
     title = {Comportement asympotique des hauteurs des points de {Heegner}},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {743--755},
     publisher = {Universit\'e Bordeaux 1},
     volume = {21},
     number = {3},
     year = {2009},
     doi = {10.5802/jtnb.700},
     mrnumber = {2605545},
     language = {fr},
     url = {http://archive.numdam.org/articles/10.5802/jtnb.700/}
}
TY  - JOUR
AU  - Ricotta, Guillaume
AU  - Templier, Nicolas
TI  - Comportement asympotique des hauteurs des points de Heegner
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2009
SP  - 743
EP  - 755
VL  - 21
IS  - 3
PB  - Université Bordeaux 1
UR  - http://archive.numdam.org/articles/10.5802/jtnb.700/
DO  - 10.5802/jtnb.700
LA  - fr
ID  - JTNB_2009__21_3_743_0
ER  - 
%0 Journal Article
%A Ricotta, Guillaume
%A Templier, Nicolas
%T Comportement asympotique des hauteurs des points de Heegner
%J Journal de théorie des nombres de Bordeaux
%D 2009
%P 743-755
%V 21
%N 3
%I Université Bordeaux 1
%U http://archive.numdam.org/articles/10.5802/jtnb.700/
%R 10.5802/jtnb.700
%G fr
%F JTNB_2009__21_3_743_0
Ricotta, Guillaume; Templier, Nicolas. Comportement asympotique des hauteurs des points de Heegner. Journal de théorie des nombres de Bordeaux, Volume 21 (2009) no. 3, pp. 743-755. doi : 10.5802/jtnb.700. http://archive.numdam.org/articles/10.5802/jtnb.700/

[1] Chandrasekharan, K.; Narasimhan, Raghavan Hecke’s functional equation and the average order of arithmetical functions, Acta Arith., Volume 6 (1960/1961), pp. 487-503 | MR | Zbl

[2] Chandrasekharan, K.; Narasimhan, Raghavan Functional equations with multiple gamma factors and the average order of arithmetical functions, Ann. of Math. (2), Volume 76 (1962), pp. 93-136 | MR | Zbl

[3] Duke, W.; Iwaniec, H. Estimates for coefficients of L-functions. I, Automorphic forms and analytic number theory (Montreal, PQ, 1989), Univ. Montréal, Montreal, QC, 1990, pp. 43-47 | MR | Zbl

[4] Duke, W.; Iwaniec, H. Estimates for coefficients of L-functions. II, Proceedings of the Amalfi Conference on Analytic Number Theory (Maiori, 1989), Univ. Salerno, Salerno (1992), pp. 71-82 | MR | Zbl

[5] Duke, W.; Iwaniec, H. Estimates for coefficients of L-functions. III, Séminaire de Théorie des Nombres, Paris, 1989–90 (Progr. Math.), Volume 102, Birkhäuser Boston, Boston, MA, 1992, pp. 113-120 | MR | Zbl

[6] Duke, W.; Iwaniec, H. Estimates for coefficients of L-functions. IV, Amer. J. Math., Volume 116 (1994) no. 1, pp. 207-217 | MR | Zbl

[7] Goldfeld, D.; Hoffstein, J.; D., Lieman An effective zero free region, Ann. of Math. (2), Volume 140 (1994) no. 2 | MR

[8] Gross, Benedict H.; Zagier, Don B. Heegner points and derivatives of L-series, Invent. Math., Volume 84 (1986) no. 2, pp. 225-320 | MR | Zbl

[9] Hafner, James Lee; Ivić, Aleksandar On sums of Fourier coefficients of cusp forms, Enseign. Math. (2), Volume 35 (1989) no. 3-4, pp. 375-382 | MR | Zbl

[10] Iwaniec, Henryk On the order of vanishing of modular L-functions at the critical point, Sém. Théor. Nombres Bordeaux (2), Volume 2 (1990) no. 2, pp. 365-376 | EuDML | Numdam | MR | Zbl

[11] Iwaniec, Henryk; Kowalski, Emmanuel Analytic number theory, American Mathematical Society Colloquium Publications, 53, American Mathematical Society, Providence, RI, 2004 | MR | Zbl

[12] Rankin, R. A. Sums of cusp form coefficients, Automorphic forms and analytic number theory (Montreal, PQ, 1989), Univ. Montréal, Montreal, QC, 1990, pp. 115-121 | MR | Zbl

[13] Ricotta, Guillaume; Vidick, Thomas Hauteur asymptotique des points de Heegner, Canad. J. Math., Volume 60 (2008) no. 6, pp. 1406-1436 | MR | Zbl

[14] Shimura, Goro The special values of the zeta functions associated with cusp forms, Comm. Pure Appl. Math., Volume 29 (1976) no. 6, pp. 783-804 | MR | Zbl

[15] Silverman, Joseph H. The arithmetic of elliptic curves, Graduate Texts in Mathematics, 106, Springer-Verlag, New York, 1992 (Corrected reprint of the 1986 original) | MR | Zbl

[16] Taylor, Richard; Wiles, Andrew Ring-theoretic properties of certain Hecke algebras, Ann. of Math. (2), Volume 141 (1995) no. 3, pp. 553-572 | MR | Zbl

[17] Wiles, Andrew Modular elliptic curves and Fermat’s last theorem, Ann. of Math. (2), Volume 141 (1995) no. 3, pp. 443-551 | MR | Zbl

Cited by Sources: