Journal of the Ramanujan Mathematical Society

Volume 37, Issue 2, June 2022  pp. 139–145.

Rankin-Selberg L-functions and “Beyond Endoscopy” II

Authors:  Satadal Ganguly and Ramdin Mawia
Author institution:Theoretical Statistics and Mathematics Unit, Indian Statistical Institute, 203 Barrackpore Trunk Road, Kolkata 700108, India

Summary:  Suppose f and g are Hecke cusp forms on SL{2}(Z), where f is holomorphic and g is either a holomorphic form or a Maaß form. We assume both f and g are eigenfunctions of all the Hecke operators. Using Langlands’ “Beyond Endoscopy” approach, we prove that the Rankin-Selberg convolution L(s, f × g) admits holomorphic extension to the region ℜs > 1/2 unless g = f, in which case the L-function has a pole at s = 1 with residue 3/π (4π)k/Γ(k) ||f||{2}, where ||f|| is the Petersson norm of f and k is the weight of f.