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

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 asubn bsub n n^−s, whose miraculous properties were first studied by Rankin and Selberg.