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On \(p\)-adic analogues of the conjectures of Birch and Swinnerton-Dyer. (English) Zbl 0699.14028
In this paper \(p\)-adic analogues of the Birch and Swinnerton-Dyer conjecture are proposed for the case of Weil elliptic curves. The basic new phenomenon, studied by the authors, is the possibility of appearance of a “redundant” zero in the center of the critical strip for the \(p\)-adic \(L\)-function in comparison with its Archimedean analogue. This phenomenon corresponds to the fact, that the “extended Mordell-Weil group” together with a \(p\)-adic height on it arises, in a natural way, on an elliptic curve with multiplicative reduction in \(p\). In so doing, the rank of the extended group is one plus the rank of the group of rational points. The comparison of the classical Swinnerton-Dyer conjecture with its \(p\)-adic analogue leads to the hypothesis that the ratio \(L'_ p(E,1)/\)(“algebraic part” of \(L(E,1))\) is expressed by the \(p\)-adic multiplicative period of the curve. An analogous ratio can be formed by replacing the \(L\)-function of the curve by the \(L\)-function of a parabolic newform of an arbitrary weight. The authors formulate and check numerically the hypothesis that this value (the algebro-geometric sense of which is not clear) does not change in case of twisting the form with the help of such a Dirichlet character \(\psi\) that \(\psi (p)=1\).

14G20 Local ground fields in algebraic geometry
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
11G07 Elliptic curves over local fields
14H25 Arithmetic ground fields for curves
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
14G05 Rational points
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