Modularity of the Rankin-Selberg L-Series, and Multiplicity One for SL(2)

Abstract

Let f, g be primitive cusp forms, holomorphic or otherwise, on the upper half-plane H of levels N,M respectively, with (unitarily normalized) L-functions L(s, f) = [equation] and L(s, g) = [equation]. When p does not divide N (resp. M), the inverse roots αp, βp (resp. α′p, β′p ) are nonzero with sum ap (resp. bp). For every p prime to NM, set Lp(s, f × g) = [(1 − αpα′pp−s)(1 − αpβ′pp−s)(1 − βpα′pp−s)(1 − βpβ′pp−s)]^−1. Let L∗(s, f × g) denote the (incomplete Euler) product of Lp(s, f × g) over all p not dividing NM. This is closely related to the convolution L-series [sum over n≥1] a[sub]n b[sub] n n^−s, whose miraculous properties were first studied by Rankin and Selberg.