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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