Let be a number field, and let be an abelian variety. Let denote the product of the Tamagawa numbers of , and let denote the finite torsion subgroup of . The quotient is a factor appearing in the leading term of the -function of 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 , and for abelian surfaces . The smallest possible ratio for elliptic curves over is , achieved only by the modular curve .
Soit un corps de nombres, et soit une variété abélienne. Dénotons par le produit des nombres de Tamagawa de , et par le sous-groupe fini des éléments de torsion de . Le quotient 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 de la fonction de . Nous nous intéressons dans cet article aux diviseurs communs des entiers et . 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 pour les courbes elliptiques sur est , obtenue seulement par la courbe modulaire .
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}, pages = {1995--2037}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {61}, number = {5}, year = {2011}, doi = {10.5802/aif.2664}, zbl = {1283.11088}, mrnumber = {2961846}, language = {en}, url = {http://www.numdam.org/articles/10.5802/aif.2664/} }
TY - JOUR AU - Lorenzini, Dino TI - Torsion and Tamagawa numbers JO - Annales de l'Institut Fourier PY - 2011 SP - 1995 EP - 2037 VL - 61 IS - 5 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.2664/ DO - 10.5802/aif.2664 LA - en ID - AIF_2011__61_5_1995_0 ER -
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/articles/10.5802/aif.2664/
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