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On a conjecture of Agashe. (English) Zbl 07397999
Summary: Let $$E/\mathbb{Q}$$ be an optimal elliptic curve, $$-D$$ be a negative fundamental discriminant coprime to the conductor $$N$$ of $$E/\mathbb{Q}$$ and let $$E^{-D}/\mathbb{Q}$$ be the twist of $$E/\mathbb{Q}$$ by $$-D$$. A conjecture of Agashe predicts that if $$E^{-D}/\mathbb{Q}$$ has analytic rank 0, then the square of the order of the torsion subgroup of $$E^{-D}/\mathbb{Q}$$ divides the product of the order of the Shafarevich-Tate group of $$E^{-D}/\mathbb{Q}$$ and the orders of the arithmetic component groups of $$E^{-D}/\mathbb{Q}$$, up to a power of 2. This conjecture can be viewed as evidence for the second part of the Birch and Swinnerton-Dyer conjecture for elliptic curves of analytic rank zero. We provide a proof of a slightly more general statement without using the optimality hypothesis.
##### MSC:
 11G05 Elliptic curves over global fields 11G07 Elliptic curves over local fields 11G40 $$L$$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
ecdata; LMFDB
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