Nonvanishing theorems for automorphic \(L\)-functions on \(\text{GL} (2)\). (English) Zbl 0847.11026

Let \(\pi\) be a cuspidal automorphic representation of \(\text{GL}(2, \mathbb{A})\) and let \(L(s, \chi\otimes \pi)\) denote the standard \(L\)-function of \(\pi\) twisted by a character \(\chi\). The following is proved. If \(\pi\) is not equivalent to its dual, there are infinitely many quadratic characters \(\chi\) such that \(L({1\over 2}, \chi\otimes \pi)\neq 0\). Here, the local components of \(\chi\) may be prescribed at a finite number of places. If \(\pi\) is equivalent to its dual, either the same is true or there are infinitely many \(\chi\) such that \(L(s, \chi\otimes \pi)\) has a simple zero at \(s= {1\over 2}\). The proof uses a Dirichlet series with twisted \(L\)-functions as coefficients and which is obtained from a Rankin-Selberg integral involving Eisenstein series on the double cover of \(\text{GL}(2, \mathbb{A})\). A brief history of nonvanishing results for these \(L\)-functions is given in the introduction of the paper.


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