# Torsion and Tamagawa numbers

```@article{Lorenzini2011TorsionAT,
title={Torsion and Tamagawa numbers},
author={Dino J. Lorenzini},
journal={Annales de l'Institut Fourier},
year={2011},
volume={61},
pages={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 Q or quadratic extensions K/Q, and for abelian surfaces A…

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