Torsion and Tamagawa numbers

Lorenzini, Dino

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

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 $A{\left(K\right)}_{\mathrm{tors}}$ le sous-groupe fini des éléments de torsion de $A\left(K\right)$. Le quotient ${c/|A\left(K\right)}_{\mathrm{tors}}|$ 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 ${|A\left(K\right)}_{\mathrm{tors}}|$. Nous obtenons des résultats précis pour les courbes elliptiques sur $\mathbb{Q}$ ou sur une extension quadratique, et pour les surfaces abéliennes sur $\mathbb{Q}$. La plus petite valeur de la fraction ${c/|E\left(\mathbb{Q}\right)}_{\mathrm{tors}}|$ pour les courbes elliptiques sur $\mathbb{Q}$ est $1/5$, obtenue seulement par la courbe modulaire $X_1\left(11\right)/\mathbb{Q}$.

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 $A{\left(K\right)}_{\mathrm{tors}}$ denote the finite torsion subgroup of $A\left(K\right)$. The quotient ${c/|A\left(K\right)}_{\mathrm{tors}}|$ 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 $\mathbb{Q}$ or quadratic extensions $K/\mathbb{Q}$, and for abelian surfaces $A/\mathbb{Q}$. The smallest possible ratio ${c/|E\left(\mathbb{Q}\right)}_{\mathrm{tors}}|$ for elliptic curves over $\mathbb{Q}$ is $1/5$, achieved only by the modular curve $X_1\left(11\right)$.

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

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