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Lower bound for the derivative in the point 1 of the \(L\)-function attached to an elliptic curve of Weil. (Minorant de la dérivée au point 1 de la fonction \(L\) attachée à une courbe elliptique de Weil.) (French) Zbl 0826.11028

Let \(E\) be an elliptic curve defined on \(\mathbb{Q}\), \(N\) its conductor and \(L(E, s)\) its zeta-function of Hasse-Weil. Suppose that \(E\) is a Weil curve and that \(L(E, 1)\neq 0\). Then, \(L' (E, 1)\neq 0\) and hence \(E\) is of rank 1. The author of this paper obtains that if \(E\) is an elliptic curve of Weil with prime conductor \(p\) satisfying \(L(E,1) =0\), then \(L' (E,1) =0\) or \(|L' (E, 1)|> (10^7 p^5 (1+ 1/p )^{3/2} H(E) )^{-1} \log p\) with \(H(E)= \sup(|c_4 (E) |^{1/2}, |c_6 (E) |^{1/2})\). The constants \(c_4 (E)\) and \(c_6 (E)\) are habitual invariants depending on \(E\).
Reviewer: M.Muro (Yanagido)

MSC:

11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
14H52 Elliptic curves
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