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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 $$\psi (p)=1$$.

MSC:
 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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