Iwasawa-theory of abelian varieties at primes of non-ordinary reduction
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 \textit{B. Mazur} [Invent. Math. 18, 183-266 (1972; Zbl 0245.14015)] and the formulas for \(L(1)\) were established by \textit{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 \textit{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.
\(p\)-adic \(L\)-function of an abelian variety
scientific article; zbMATH DE number 784072