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L-functions and division towers. (English) Zbl 0656.14013

In a previous paper [Invent. Math. 75, 409-423 (1984; Zbl 0565.14006] on the vanishing of \(L(1,s,\chi)\) for almost all primitive Dirichlet characters \(\chi\) unramified outside a fixed finite set P of primes and infinity (f being a new cusp form of weight 2), the author had to make a restriction on the given set P. This restriction is here removed and a generalization is made to the case of \(L(k/2,f,\chi)\), where f is of even weight k. Both improvements could be made, the author tells us, after indications given to him by M.-F. Vignéras. A second theorem concerns the Hecke L-function \(L(s,\tau)\) for an imaginary quadratic field K. One fixes the automorphic representation \(\pi_ 1\) of \(GL(2)/K\) determined by a new form of any weight, character and level and a Hecke character \(\phi\) of K of a particular kind and one lets \(\tau\) run over the twists \(\pi_ 1\otimes \phi \otimes \chi\), where \(\chi\) is a Hecke character of finite order, unramified outside a given finite set P of ideals of K (satisfying some condition). Then for almost all such \(\tau\), \(L(1/2,\tau)\neq 0.\)
An application is mentioned to the uniform boundedness of the rank of the group \(E(K_*)\), where E is an elliptic curve over \({\mathbb{Q}}\) and \(K_*\) runs through the finite extensions of K contained in a fixed extension \(K_{\infty}\), both E and \(K_{\infty}\) being subjected to conditions with respect to the given K and P.
Reviewer: J.H.de Boer

MSC:

14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)

Citations:

Zbl 0565.14006
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References:

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