Friedberg, Solomon; Hoffstein, Jeffrey Nonvanishing theorems for automorphic \(L\)-functions on \(\text{GL} (2)\). (English) Zbl 0847.11026 Ann. Math. (2) 142, No. 2, 385-423 (1995). 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. Reviewer: J.G.M.Mars (Utrecht) Cited in 4 ReviewsCited in 51 Documents MSC: 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 Keywords:nonvanishing theorems; cuspidal automorphic representation; quadratic characters; twisted \(L\)-functions; Rankin-Selberg integral × Cite Format Result Cite Review PDF Full Text: DOI