On \(L\)-functions with poles satisfying Maaß's functional equation
The main result of this paper is a converse theorem for Maaß forms, in the spirit of \textit{E. Hecke} [Math. Ann. 112, 664--699 (1936; Zbl 0014.01601)], \textit{H. Maaß} [Math. Ann. 121, 141--183 (1949; Zbl 0033.11702)] stating that any Dirichlet series which has meromorphic continuation to \(\mathbb C\) with a certain functional equation and appropriate analytic properties is, in fact, the Dirichlet series attached to a Maaß form. The emphasis of the present work (as well as the author's previous work [\textit{R. Raghunathan} [ Stud. Math., Tata Inst. Fundam. Res. 15, 127--142 (2001; Zbl 1195.11065)] dealing with holomorphic forms) is to relax the conditions which must be placed on the number and order of the poles of the meromorphically continued function. For example, it is typical to assume that the \(L\) function has no more than four poles, each simple and at a specified location. The present result requires only that the number of poles is finite and that each is of finite order. The argument can be described as a ``bootstrap'' method: it shows that the stronger conditions normally required actually follow from this weaker condition, together with the growth properties and functional equation. This is accomplished by computing an asymptotic expansion for the ``period function'' of \textit{J. Lewis} and \textit{D. Zagier} [Ann. Math. (2) 153, No. 1, 191--258 (2001; Zbl 1061.11021)] in two different ways and comparing the results. Unfortunately, the author states that the method works ``only in the context of modular or Maaß forms associated to the full modular group,'' noting that certain Dirichlet series constructed in \textit{M. I. Knopp} [Invent. Math. 117, No. 3, 361--372 (1994; Zbl 0816.11032)] would provide a counterexample to the hypothetical extension to higher level. The final section contains an application to comparing the zero sets of \(L\) functions attached to distinct automorphic forms.
scientific article; zbMATH DE number 5723350
scientific article