A note on \(\text{GL}_2\) converse theorems
In this note the authors obtain a kind of hybrid version of \textit{A. Weil}'s [Math. Ann. 168, 149-156 (1967; Zbl 0158.08601)] and \textit{J. B. Conrey} and \textit{D. W. Farmer}'s [Int. Math. Res. Not. 1995, 445-463 (1995; Zbl 0849.11042)] converse theorems. Weil proved that modular forms \(f\in M_k(\Gamma_0(q))\) are characterized by the functional equation for twists of associated Dirichlet series \(L_f (s)\). For \(q\in\{5, 6,\dots, 12, 14, 15, 16, 17, 23\}\) Conrey and Farmer showed that the assumption on twists can be replaced by the assumption of \(L_f (s)\) having an Euler product of the appropriate form. The authors prove a converse theorem for all \(q\geq 1\) under the assumption of the Euler product and, moreover, of the functional equation for the twists by all primitive characters to a \textit{single} suitable prime modulus \(r\). They also give an algorithm for finding \(r\) and obtain an upper bound for \(r\).
scientific article; zbMATH DE number 1754828