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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
Software:
ecdata; LMFDB
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