Comportement asympotique des hauteurs des points de Heegner
Journal de théorie des nombres de Bordeaux, Volume 21 (2009) no. 3, p. 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 : https://doi.org/10.5802/jtnb.700
Classification:  11G50,  11M41
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     author = {Ricotta, Guillaume and Templier, Nicolas},
     title = {Comportement asympotique des hauteurs des points de Heegner},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     publisher = {Universit\'e Bordeaux 1},
     volume = {21},
     number = {3},
     year = {2009},
     pages = {743-755},
     doi = {10.5802/jtnb.700},
     mrnumber = {2605545},
     zbl = {pre05774809},
     language = {fr},
     url = {http://www.numdam.org/item/JTNB_2009__21_3_743_0}
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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://www.numdam.org/item/JTNB_2009__21_3_743_0/

[1] Chandrasekharan, K.; Narasimhan, Raghavan Hecke’s functional equation and the average order of arithmetical functions, Acta Arith., Tome 6 (1960/1961), pp. 487-503 | MR 126423 | Zbl 0101.03703

[2] Chandrasekharan, K.; Narasimhan, Raghavan Functional equations with multiple gamma factors and the average order of arithmetical functions, Ann. of Math. (2), Tome 76 (1962), pp. 93-136 | MR 140491 | Zbl 0211.37901

[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 1111010 | Zbl 0745.11030

[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 1220457 | Zbl 0787.11020

[5] Duke, W.; Iwaniec, H. Estimates for coefficients of L-functions. III, Séminaire de Théorie des Nombres, Paris, 1989–90, Birkhäuser Boston, Boston, MA (Progr. Math.) Tome 102 (1992), pp. 113-120 | MR 1476732 | Zbl 0763.11024

[6] Duke, W.; Iwaniec, H. Estimates for coefficients of L-functions. IV, Amer. J. Math., Tome 116 (1994) no. 1, pp. 207-217 | MR 1262431 | Zbl 0820.11032

[7] Goldfeld, D.; Hoffstein, J.; D., Lieman An effective zero free region, Ann. of Math. (2), Tome 140 (1994) no. 2 | MR 1289494

[8] Gross, Benedict H.; Zagier, Don B. Heegner points and derivatives of L-series, Invent. Math., Tome 84 (1986) no. 2, pp. 225-320 | MR 833192 | Zbl 0608.14019

[9] Hafner, James Lee; Ivić, Aleksandar On sums of Fourier coefficients of cusp forms, Enseign. Math. (2), Tome 35 (1989) no. 3-4, pp. 375-382 | MR 1039952 | Zbl 0696.10020

[10] Iwaniec, Henryk On the order of vanishing of modular L-functions at the critical point, Sém. Théor. Nombres Bordeaux (2), Tome 2 (1990) no. 2, pp. 365-376 | Numdam | MR 1081731 | Zbl 0719.11029

[11] Iwaniec, Henryk; Kowalski, Emmanuel Analytic number theory, American Mathematical Society, Providence, RI, American Mathematical Society Colloquium Publications, Tome 53 (2004) | MR 2061214 | Zbl 1059.11001

[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 1111014 | Zbl 0735.11023

[13] Ricotta, Guillaume; Vidick, Thomas Hauteur asymptotique des points de Heegner, Canad. J. Math., Tome 60 (2008) no. 6, pp. 1406-1436 | MR 2462452 | Zbl 1195.11082 | Zbl pre05382118

[14] Shimura, Goro The special values of the zeta functions associated with cusp forms, Comm. Pure Appl. Math., Tome 29 (1976) no. 6, pp. 783-804 | MR 434962 | Zbl 0348.10015

[15] Silverman, Joseph H. The arithmetic of elliptic curves, Springer-Verlag, New York, Graduate Texts in Mathematics, Tome 106 (1992) (Corrected reprint of the 1986 original) | MR 1329092 | Zbl 0585.14026

[16] Taylor, Richard; Wiles, Andrew Ring-theoretic properties of certain Hecke algebras, Ann. of Math. (2), Tome 141 (1995) no. 3, pp. 553-572 | MR 1333036 | Zbl 0823.11030

[17] Wiles, Andrew Modular elliptic curves and Fermat’s last theorem, Ann. of Math. (2), Tome 141 (1995) no. 3, pp. 443-551 | MR 1333035 | Zbl 0823.11029