Torsion and Tamagawa numbers
Let be a number field, and let be an abelian variety. Let denote the product of the Tamagawa numbers of , and let denote the finite torsion subgroup of . The quotient is a factor appearing in the leading term of the -function of 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 or quadratic extensions , and for abelian surfaces . The smallest possible ratio...