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 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 -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 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 dans les progressions arithmétiques sont au cœur de la démonstration.
@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] Hecke’s functional equation and the average order of arithmetical functions, Acta Arith., Volume 6 (1960/1961), pp. 487-503 | MR | Zbl
[2] 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] Estimates for coefficients of -functions. I, Automorphic forms and analytic number theory (Montreal, PQ, 1989), Univ. Montréal, Montreal, QC, 1990, pp. 43-47 | MR | Zbl
[4] Estimates for coefficients of -functions. II, Proceedings of the Amalfi Conference on Analytic Number Theory (Maiori, 1989), Univ. Salerno, Salerno (1992), pp. 71-82 | MR | Zbl
[5] Estimates for coefficients of -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] Estimates for coefficients of -functions. IV, Amer. J. Math., Volume 116 (1994) no. 1, pp. 207-217 | MR | Zbl
[7] An effective zero free region, Ann. of Math. (2), Volume 140 (1994) no. 2 | MR
[8] Heegner points and derivatives of -series, Invent. Math., Volume 84 (1986) no. 2, pp. 225-320 | MR | Zbl
[9] On sums of Fourier coefficients of cusp forms, Enseign. Math. (2), Volume 35 (1989) no. 3-4, pp. 375-382 | MR | Zbl
[10] On the order of vanishing of modular -functions at the critical point, Sém. Théor. Nombres Bordeaux (2), Volume 2 (1990) no. 2, pp. 365-376 | EuDML | Numdam | MR | Zbl
[11] Analytic number theory, American Mathematical Society Colloquium Publications, 53, American Mathematical Society, Providence, RI, 2004 | MR | Zbl
[12] 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] Hauteur asymptotique des points de Heegner, Canad. J. Math., Volume 60 (2008) no. 6, pp. 1406-1436 | MR | Zbl
[14] 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] The arithmetic of elliptic curves, Graduate Texts in Mathematics, 106, Springer-Verlag, New York, 1992 (Corrected reprint of the 1986 original) | MR | Zbl
[16] Ring-theoretic properties of certain Hecke algebras, Ann. of Math. (2), Volume 141 (1995) no. 3, pp. 553-572 | MR | Zbl
[17] 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: