Torsion and Tamagawa Numbers
نویسنده
چکیده
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/Q. The smallest possible ratio c/|E(Q)tors| for elliptic curves over Q is 1/5, achieved only by the modular curve X1(11).
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