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 [7–9]) 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
References
[1]
R. Rankin, “Contributions to the theory of Ramanujan's function τ(n) and similar arithmetical functions, I and II,” Proceedings of the Cambridge Philosophical Society, vol. 35, pp. 351–372, 1939.
[2]
A. Selberg, “Bermerkungen uber eine Dirichletsche Reihe, die mit der Modulformen nahe verbunden ist,” Archiv for Mathematik og Naturvidenskab, vol. 43, pp. 47–50, 1940.
[3]
A. Selberg, “On the estimation of Fourier coefficients of modular forms,” in Proceedings of Symposia in Pure Mathematics, vol. 8, pp. 1–15, American Mathematical Society, 1965.
[4]
H. Jacquet, I. I. Piatetski-Shapiro, and J. Shalika, “Rankin-Selberg convolutions,” The American Journal of Mathematics, vol. 105, pp. 367–464, 1983.
[5]
H. Jacquet and J. A. Shalika, “On Euler products and the classification of automorphic representations I, II,” The American Journal of Mathematics, vol. 103, pp. 499–558, 777–815, 1981.
[6]
J. Liu and Y. Ye, “Selberg's orthogonality conjecture for automorphic L-functions,” The American Journal of Mathematics, vol. 127, no. 4, pp. 837–849, 2005.
[7]
J. Liu and Y. Ye, “Zeros of automorphic L-functions and noncyclic base change,” in Number Theory: Tradition and Modernization, pp. 119–152, Springer, New York, NY, USA, 2005.
[8]
J. Liu and Y. Ye, “Correlation of zeros of automorphic L-functions,” Science in China, Series A: Mathematics, vol. 51, no. 7, pp. 1147–1166, 2008.
[9]
J. Liu and Y. Ye, “Functoriality of automorphic L-functions through their zeros,” Science in China, Series A: Mathematics, vol. 52, no. 1, pp. 1–16, 2009.
[10]
F. Shahidi, “On certain L-functions,” The American Journal of Mathematics, vol. 103, pp. 297–355, 1981.
[11]
C. Moeglin and J. L. Waldspurger, “Le spectre résiduel de ,” Annales Scientifiques de l'école Normale Supérieure, vol. 22, no. 4, pp. 605–674, 1989.
[12]
H. Jacquet, “Principal L-functions of the linear group,” Proceedings of Symposia in Pure Mathematics, vol. 33, part 2, pp. 63–86, 1979.
[13]
J. Arthur L, Clozel, Simple Algebras, Base Change, and the Advanced Theory of the Trace Formula, vol. 120 of Annals of Mathematics Studies, Princeton University Press, 1989.
[14]
Z. Rudnick and P. Sarnak, “Zeros of principal L-functions and random matrix theory,” Duke Mathematical Journal, vol. 88, pp. 269–322, 1996.
[15]
T. Gillespie and G. Ji, “Prime number theorems for Rankin-Selberg L-functions over number fields,” Science China Mathematics, vol. 54, no. 1, pp. 35–46, 2011.
[16]
T. Gillespie, “Factorization of automorphic L-functions and their zero statistics,” International Journal of Number Theory, vol. 9, no. 6, p. 1367, 2013.
[17]
J. W. Cogdell and I. I. Piatetski-Shapiro, “Converse theorems for ,” Publications Mathématiques de L'Institut des Hautes Scientifiques, vol. 79, no. 1, pp. 157–214, 1994.
[18]
H. H. Kim, “An example of non-normal quintic automorphic induction and modularity of symmetric powers of cusp forms of icosahedral type,” Inventiones Mathematicae, vol. 156, no. 3, pp. 495–502, 2004.
[19]
J. Rogawski, “Functoriality and the artin conjecture,” Proceedings of Symposia in Pure Mathematics, vol. 61, pp. 331–353, 1997.
[20]
D. Ramakrishnan, “Modularity of the Rankin-Selberg L-series, and multiplicity one for SL(2),” Annals of Mathematics, vol. 152, no. 1, pp. 45–111, 2000.
