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Théorie d’Iwasawa p-adique locale et globale. (Local and global p-adic Iwasawa theory). (French) Zbl 0715.11030
The author proves the following theorem. Let E be an elliptic curve defined over \({\mathbb{Q}}\) and let \(p\geq 5\) be a prime number for which E has good supersingular reduction. If E(\({\mathbb{Q}})\) and the p-primary component of the Tate-Shafarevich group are finite, then \(E({\mathbb{Q}}_{\infty})\) is a finitely generated \({\mathbb{Z}}\)-module, where \({\mathbb{Q}}_{\infty}/{\mathbb{Q}}\) is the unique \({\mathbb{Z}}_ p\)-extension of \({\mathbb{Q}}\). This result was proved in the case of ordinary reduction by B. Mazur [Invent. Math. 18, 183-266 (1972; Zbl 0245.14015)]. The present proof relies on the nonvanishing of a certain p-adic L-function, constructed using the Selmer groups of E over the intermediate fields in the \({\mathbb{Z}}_ p\)-extension. One of the ingredients is a nice use of the ring of norms, constructed by J.-M. Fontaine and J.-P. Wintenberger [C. R. Acad. Sci., Paris, Ser. A 288, 367-370, 441-444 (1979; Zbl 0403.12018, Zbl 0475.12020)], which permits a generalization of the usual sequence of points compatible with respect to the trace in a \({\mathbb{Z}}_ p\)- extension. In a final section, the author discusses p-adic heights attached to supersingular elliptic curves.
Reviewer: L.Washington

MSC:
11G05 Elliptic curves over global fields
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
11R23 Iwasawa theory
14H52 Elliptic curves
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