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The Birch and Swinnerton-Dyer formula for elliptic curves of analytic rank one. (English) Zbl 1401.11103

Authors’ abstract: “Let \(E/\mathbb{Q}\) be a semistable elliptic curve such that \(ord_{s=1}L(E, s) = 1\). We prove the \(p\)-part of the Birch and Swinnerton-Dyer formula for \(E/\mathbb{Q}\) for each prime \(p\geq 5\) of good reduction such that \(E[p]\) is irreducible: \[ \mathrm{ord}_p\biggl(\frac{L'(E, 1)} {\Omega_E\cdot\mathrm{Reg}(E/\mathbb{Q})}\biggr)= \mathrm{ord}_p\biggl(\#Ш(E/Q) \prod_{l\leq \infty}c_{l}(E/\mathbb{Q})\biggr). \] This formula also holds for \(p = 3\) provided \(a_p(E) =0\) if \(E\) has supersingular reduction at \(p\).”
Particular cases of this theorem have been obtained by W. Zhang [Camb. J. Math. 2, No. 2, 191–253 (2014; Zbl 1390.11091)] and by A. Berti et al. [in: Elliptic curves, modular forms and Iwasawa theory. In honour of John H. Coates’ 70th birthday, Cambridge, UK, March 2015. Proceedings of the conference and the workshop. Cham: Springer. 1–31 (2016; Zbl 1411.11063)] but the paper under review gives the proof for the general case. The key method of the proof is to give the lower and upper bounds of Tate-Shafarevich group \(\#Ш(E/Q)[p^{\infty}]\) by combining Iwasawa theory, the \(p\)-adic Waldspurger formula and the work of Gross-Zagier and Kolyvagin among other results. The contents are well written and the authors give a very clear perspective for the theory of the \(p\)-part of the Birch and Swinnerton-Dyer formula. Indeed, F. Castella [Camb. J. Math. 6, No. 1, 1–23 (2018; Zbl 1400.11139)] generalizes this result to the case when the prime \(p\) divides the conductor of \(E\) by combining their methods with the Hida theory.

MSC:

11G05 Elliptic curves over global fields
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
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