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Nonvanishing of \(L\)-functions for \(\mathrm{GL}(2)\). (English) Zbl 0677.10020

Let \(F\) be a number field. To each irreducible cuspidal representation \(\pi\) of \(\mathrm{GL}(2)\) over \(F\) one may associate the \(L\)-function \(L(s,\pi)\). It is shown that for any given \(s\in {\mathbb C}\) there are infinitely many primitive ray class characters \(\chi\) of \(F\) such that \(L(s,\pi \otimes \chi)\neq 0\). These characters \(\chi\) may be taken unramified outside a finite set of places of \(F\). The corresponding statement is also proved for ‘automorphic representations of \(\mathrm{GL}(1)'\), i.e. continuous characters of \(F^*_ A/F^*.\)
The proof uses a wide arsenal of number theoretical methods. An important step is a proposition providing for any \(\epsilon >0\) infinitely many integral ideals \({\mathfrak q}\) of \(F\) such that the ray class group of \(F\) modulo \({\mathfrak q}\) has more than \(N({\mathfrak q})^{1-\varepsilon}\) elements, and such that the norm \(N({\mathfrak q})\) over the rationals is the product of different rational primes outside a given finite set of primes.
Consequences of the theorem are discussed in the introduction. They concern the algebraicity of \(L(s,\pi)\) at special points, and even intersection homology of the Baily-Borel compactification of a Hilbert modular variety.
Reviewer: R.W.Bruggeman

MSC:

11F70 Representation-theoretic methods; automorphic representations over local and global fields
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11R23 Iwasawa theory
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References:

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