Torsion and Tamagawa numbers
Annales de l'Institut Fourier, Volume 61 (2011) no. 5, p. 1995-2037

Let K be a number field, and let A/K be an abelian variety. Let c denote the product of the Tamagawa numbers of A/K, and let A(K) tors denote the finite torsion subgroup of A(K). The quotient c/|A(K) tors | is a factor appearing in the leading term of the L-function of A/K in the conjecture of Birch and Swinnerton-Dyer. We investigate in this article possible cancellations in this ratio. Precise results are obtained for elliptic curves over or quadratic extensions K/, and for abelian surfaces A/. The smallest possible ratio c/|E() tors | for elliptic curves over is 1/5, achieved only by the modular curve X 1 (11).

Soit K un corps de nombres, et soit A/K une variété abélienne. Dénotons par c le produit des nombres de Tamagawa de A/K, et par A(K) tors le sous-groupe fini des éléments de torsion de A(K). Le quotient c/|A(K) tors | apparaît dans la conjecture de Birch et Swinnerton-Dyer comme un facteur de la valeur du premier terme non-nul dans le développement limité en s=1 de la fonction L de A/K. Nous nous intéressons dans cet article aux diviseurs communs des entiers c et |A(K) tors |. Nous obtenons des résultats précis pour les courbes elliptiques sur ou sur une extension quadratique, et pour les surfaces abéliennes sur . La plus petite valeur de la fraction c/|E() tors | pour les courbes elliptiques sur est 1/5, obtenue seulement par la courbe modulaire X 1 (11)/.

DOI : https://doi.org/10.5802/aif.2664
Classification:  11G05,  11G10,  11G30,  11G35,  11G40,  14G05,  14G10
Keywords: Abelian variety over a global field, torsion subgroup, Tamagawa number, elliptic curve, abelian surface, dual abelian variety, Weil restriction
@article{AIF_2011__61_5_1995_0,
     author = {Lorenzini, Dino},
     title = {Torsion and Tamagawa numbers},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {61},
     number = {5},
     year = {2011},
     pages = {1995-2037},
     doi = {10.5802/aif.2664},
     mrnumber = {2961846},
     zbl = {1283.11088},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2011__61_5_1995_0}
}
Lorenzini, Dino. Torsion and Tamagawa numbers. Annales de l'Institut Fourier, Volume 61 (2011) no. 5, pp. 1995-2037. doi : 10.5802/aif.2664. http://www.numdam.org/item/AIF_2011__61_5_1995_0/

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