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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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