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Dino Lorenzini
Torsion and Tamagawa numbers
(Torsion et nombres de Tamagawa)
Annales de l'institut Fourier, 61 no. 5 (2011), p. 1995-2037, doi: 10.5802/aif.2664
Article: subscription required (your ip address: 54.163.67.67) | Reviews MR 2961846 | Zbl 1283.11088
Class. Math.: 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

Résumé - Abstract

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 $A(K)_{\textrm{tors}}$ denote the finite torsion subgroup of $A(K)$. The quotient $c/ |A(K)_{\textrm{tors}}|$ 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 $c/ |E(\mathbb{Q})_{\textrm{tors}}|$ for elliptic curves over $\mathbb{Q}$ is $1/5$, achieved only by the modular curve $X_1(11)$.

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