A Pólya-Hilbert operator for automorphic \(L\)-functions (Q1602598)

The author proves that to any automorphic \(L\)-function there exists a natural so-called Pólya-Hilbert operator whose spectrum coincides with the set of zeros of the \(L\)-function on the critical line. This result generalizes a part of \textit{A. Connes}' recent paper [Sel. Math., New Ser. 5, 29-106 (1999; Zbl 0945.11015)] to the case of GL\(_n\). The investigations of the present paper lead to a new proof of the meromorphicity of automorphic \(L\)-functions, which gives new insights into the analytic properties of these functions.