Rankin–Selberg L-functions and “beyond endoscopy”

Abstract

Let f and g be two holomorphic cuspidal Hecke eigenforms on the full modular group $$ \text {SL}_{2}({\mathbb {Z}}). $$ We show that the Rankin–Selberg L-function $$L(s, f \times g)$$ has no pole at $$s=1$$ unless $$ f=g $$, in which case it has a pole with residue $$ \frac{3}{\pi }\frac{(4\pi )^{k}}{\Gamma (k)} \Vert f \Vert ^2 $$, where $$ \Vert f\Vert $$ is the Petersson norm of f. Our proof uses the Petersson trace formula and avoids the Rankin–Selberg method.