Table of Contents
Journal of Numbers
Volume 2014 (2014), Article ID 314173, 7 pages
http://dx.doi.org/10.1155/2014/314173
Research Article

On a Rankin-Selberg -Function over Different Fields

Department of Mathematics, St. Ambrose University, 518 W. Locust Street, Davenport, IA 52803, USA

Received 5 February 2014; Accepted 7 April 2014; Published 27 April 2014

Academic Editor: Emrah Kılıç

Copyright © 2014 Tim Gillespie. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Given two unitary automorphic cuspidal representations and defined on and , respectively, with and being Galois extensions of , we consider two generalized Rankin-Selberg -functions obtained by forcefully factoring   and   . We prove the absolute convergence of these -functions for . The main difficulty in our case is that the two extension fields may be completely unrelated, so we are forced to work either “downstairs” in some intermediate extension between and , or “upstairs” in some extension field containing the composite extension . We close by investigating some special cases when analytic continuation is possible and show that when the degrees of the extension fields and are relatively prime, the two different definitions give the same generating function.