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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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