FACTORIZATION OF AUTOMORPHIC L-FUNCTIONS AND THEIR ZERO STATISTICS

Abstract

In this paper we define a Rankin–Selberg L-function attached to two Galois invariant automorphic cuspidal representations of GL m(𝔸E) and GL m′(𝔸F) over cyclic Galois extensions E and F of prime degree. This differs from the classical case in that the two extension fields E and F could be completely unrelated to one another, and we exploit the existence of the automorphic induction functor over cyclic extensions (see [J. Arthur and L. Clozel, Simple Algebras, Base Change, and the Advanced Theory of the Trace Formula, Annals of Mathematics Studies, No. 120 (Princeton University Press, Princeton, NJ, 1989)]) to define the L-function. Using a result proved by C. S. Rajan, we prove a prime number theorem for this L-function, and proceed to calculate the n-level correlation function of high nontrivial zeros of a product L(s, π1)L(s, π2)…L(s, πk) where πi is a Galois invariant cuspidal representation of GL ni(𝔸Fi) with Fi a cyclic Galois extension of prime degree ℓi for i = 1,…,k, thus generalizing the results of Liu and Ye [Functoriality of automorphic L-functions through their zeros, Sci. China Ser. A51(1) (2008) 1–16].