Torsion and Tamagawa numbers

Lorenzini, Dino

doi:10.5802/aif.2664

Torsion and Tamagawa numbers
[Torsion et nombres de Tamagawa]

Lorenzini, Dino ¹

¹ University of Georgia Department of mathematics Athens, GA 30602 (USA)

Annales de l'Institut Fourier, Tome 61 (2011) no. 5, pp. 1995-2037.

Résumé
Abstract

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 $ℚ$ ou sur une extension quadratique, et pour les surfaces abéliennes sur $ℚ$ . La plus petite valeur de la fraction ${c / | E (ℚ)}_{tors} |$ pour les courbes elliptiques sur $ℚ$ est $1 / 5$ , obtenue seulement par la courbe modulaire $X_{1} (11) / ℚ$ .

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 $ℚ$ or quadratic extensions $K / ℚ$ , and for abelian surfaces $A / ℚ$ . The smallest possible ratio ${c / | E (ℚ)}_{tors} |$ for elliptic curves over $ℚ$ is $1 / 5$ , achieved only by the modular curve $X_{1} (11)$ .

MR Zbl

DOI : 10.5802/aif.2664

Classification : 11G05, 11G10, 11G30, 11G35, 11G40, 14G05, 14G10
Mots-clés : Abelian variety over a global field, torsion subgroup, Tamagawa number, elliptic curve, abelian surface, dual abelian variety, Weil restriction

Affiliations des auteurs :

Lorenzini, Dino ¹

¹ University of Georgia Department of mathematics Athens, GA 30602 (USA)

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Lorenzini, Dino. Torsion and Tamagawa numbers. Annales de l'Institut Fourier, Tome 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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