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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