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On $$L$$-functions of elliptic curves and cyclotomic towers. (English) Zbl 0565.14006
Let $$f$$ be a normalized new form of weight 2, character $$\psi$$, and level $$N$$. Let $$P$$ be a finite set of primes not dividing $$N$$, and let $$X$$ be the set of all primitive Dirichlet characters which are unramified outside $$P$$ and infinity. For $$\chi\in X$$, let $$L(s,f,\chi)$$ be the $$L$$-function attached to $$f$$ and $$\chi$$. The main result in this paper is the following theorem:
For all but finitely many $$\chi\in X$$, $$L(1,f,\chi)\neq 0$$.
Two consequences of this result are given. One is a conjecture of B. Mazur and P. Swinnerton-Dyer [Invent. Math. 25, 1–61 (1974; Zbl 0281.14016)] to the effect that the $$p$$-adic $$L$$-function attached to a Weil curve over an abelian number field is not identically zero.
The other is as follows: Let $$E$$ be an elliptic curve defined over $$\mathbb Q$$ with complex multiplication by the ring of integers of an imaginary quadratic number field, let $$P$$ be a finite set of primes where $$E$$ has good reduction. Let $$L$$ be the maximal abelian extension of $$\mathbb Q$$ unramified outside $$P$$ and infinity, and let $$E(L)$$ be the group of $$L$$-rational points on $$E$$. Then $$E(L)$$ is finitely generated.
This generalizes a result of K. Rubin and A. Wiles [Number theory related to Fermat’s last theorem, Proc. Conf., Prog. Math. 26, 237–254 (1982; Zbl 0519.14017)]. If the conjectures of Taniyama-Weil and Birch-Swinnerton-Dyer hold, then the restriction to the case of complex multiplication can be removed.

##### MSC:
 11G40 $$L$$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture 11R23 Iwasawa theory 14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
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