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The \(p\)-adic Birch and Swinnerton-Dyer’s conjecture. (La conjecture de Birch et Swinnerton-Dyer \(p\)-adique.) (French) Zbl 1094.11025
Bourbaki seminar. Volume 2002/2003. Exposes 909–923. Paris: Société Mathématique de France (ISBN 2-85629-156-2/pbk). Astérisque 294, 251-319, Exp. No. 919 (2004).
Let \(E\) be an elliptic curve over \(\mathbb Q\) of rank \(r(E)\). For a prime number \(p\) factorize the Euler factor in \(p\) of the \(L\)-function \(L(E,s)\) as \((1-\alpha_1p^{-s})(1-\alpha_2p^{-s})\) and choose (whenever possible) \(\alpha\in\{\alpha_1,\alpha_2\}\) verifying \(v_p(\alpha)<1.\) Let \(L_{p,\alpha}(E,s)\) be the \(p\)-adic \(L\)-function of \(E\) associated to \(\alpha\).
The author explains the proof of the following important theorem of K. Kato [Astérisque 295, 117–290 (2004; Zbl 1142.11336)]: The order of the zero of \(L_{p,\alpha}(E,s)\) at \(s=1\) is \(\geq r(E)\) and \(\geq r(E)+1\) if \(\alpha=1\). If equality holds then the \(p\)-part of the Tate-Shafarevich group of \(E\) is finite an the \(p\)-adic regulator \(R_{p,\alpha}(E)\) is different from zero.
An extensive bibliography of 194 items closes this brilliant survey.
For the entire collection see [Zbl 1052.00010].

MSC:
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
11-02 Research exposition (monographs, survey articles) pertaining to number theory
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