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Iwasawa-theory of abelian varieties at primes of non-ordinary reduction. (English) Zbl 0847.14026
The well known Birch and Swinnerton-Dyer conjectures for abelian varieties can be reformulated for $$p$$-adic $$L$$-functions. There are two approaches to a definition of $$p$$-adic $$L$$-function of an abelian variety $$A$$, the analytical one and the algebraic one. The last one working with the Selmer groups of infinite cyclotomic extensions is more directly related with the Birch-Swinnerton-Dyer formulas for value at $$s = 1$$. The algebraic definition was given by B. Mazur [Invent. Math. 18, 183-266 (1972; Zbl 0245.14015)] and the formulas for $$L(1)$$ were established by P. Schneider [Invent. Math. 71, 251-293 (1983; Zbl 0511.14010)]. The severe restriction used in these papers was that the abelian variety must have an ordinary reduction at the primes $$p$$. This restriction was overcome by B. Perrin-Riou in Invent. Math. 99, No. 2, 247-292 (1990; Zbl 0715.11030) for supersingular elliptic curves.
The author gives here an extension to the case of general abelian varieties $$A$$. It is defined some module $$L_0$$ over Iwasawa algebra of $$A$$ and the $$L$$-function is a characteristic function related to $$L_0$$. The part of the Birch-Swinnerton-Dyer formulas is checked.

##### MSC:
 14K05 Algebraic theory of abelian varieties 11R23 Iwasawa theory 14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) 11S40 Zeta functions and $$L$$-functions 11G40 $$L$$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture 11G25 Varieties over finite and local fields 11G10 Abelian varieties of dimension $$> 1$$
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