Nonvanishing theorems for automorphic \(L\)-functions on \(\text{GL} (2)\)
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.
scientific article; zbMATH DE number 819353