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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