P-adic L-functions and P-adie Periods of Modular Forms*
نویسنده
چکیده
Let E be an elliptic curve which is defined over Q and has stable reduction modulo a given prime p. Assuming that E is modular, one can associate to E a p-adic L-function Lp(E, s). (See [-Mz-SwD, A-V, Vi, Mz-T-T] for its construction in various cases.) This function is defined by a certain interpolation property and is analytic for seZp. In this paper, we will assume that E has split multiplicative reduction at p. Under this assumption the interpolation property implies that Lp(E, 1)=0. We will prove a formula for Ep(E, 1) which was discovered experimentally by Mazur, Tate, and Teitelbaum [Mz-T-T]. By Tate's p-adic urfiformization theory, there is a p-adic integer q~:epZp (which we refer to as the Tate period for E) and a p-adic analytic isomorphism
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