%0 Journal Article
%T On a Rankin-Selberg -Function over Different Fields
%A Tim Gillespie
%J Journal of Numbers
%D 2014
%R 10.1155/2014/314173
%X 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. 1. Introduction The generating function attached to pairs of automorphic representations has its origins in the papers of Rankin [1] and Selberg [2, 3]. As a consequence of the knowledge of the location and multiplicities of the poles of these -functions, they obtained nonvanishing results on the edge of the critical strip for Hecke¡¯s -functions and asymptotic estimates for the growth of Fourier coefficients of modular forms. Moreover, considering the adelic setting in [4], Rankin-Selberg convolutions were applied to obtain more precise information regarding constant terms of Eisenstein series, and in [5] the analytic properties of Rankin-Selberg -functions were used to obtain multiplicity-one results. More recently, using a version of Selberg orthogonality [6], Liu and Ye computed the -level correlation function of high nontrivial zeros of automorphic -functions (see [7¨C9]) to give insight into the distribution of prime numbers. In the classical setting as in [4, 10, 11] we are given a Galois extension of and two automorphic cuspidal representations with unitary central characters on and £¿£¿on . Suppose for any finite place of we have the associated conjugacy classes given by the Langlands correspondence in determined by , and in associated with . Then the finite part of the Rankin-Selberg -function is given by the product of local factors where and denotes the cardinality of the residue field at a finite place of . We will always write , where lies over the prime in and denotes the modular degree (which will be the same for any lying over since is Galois) where denotes the ring of integers in with unique prime ideal . If a finite place of lies over the prime in then we will always denote£¿£¿ . The Euler product in (3) is known to converge absolutely for (see [4]), and this result is crucial in
%U http://www.hindawi.com/journals/jn/2014/314173/