Ramanujan conjecture for GL(2) over number fields

Establish that every cuspidal automorphic form on GL(2) over an arbitrary number field k satisfies the 0‑Ramanujan hypothesis, namely that for each place v and every local Whittaker function W_v in the Whittaker model of the corresponding local representation, the integral ∫_{k_v^×} W_v(a(t,1)) |t|_v^s dt converges absolutely whenever Re(s) > 0.

Background

The paper introduces a γ-Ramanujan hypothesis tailored to the analytic estimates needed for their twisted Shintani zeta function analysis; it requires absolute convergence of certain local Whittaker integrals for Re(s) greater than a parameter γ. This framework quantifies how far current bounds are from the Ramanujan conjecture and directly controls the domains of analytic continuation obtained in the main theorems.

Within this setup, the authors summarize the state of knowledge: Kim–Shahidi proved that every cusp form on GL(2) is 1/9‑Ramanujan, and Blomer–Brumley established the bound 7/64 towards Ramanujan, which the authors use to obtain their explicit domains. They note that if the full Ramanujan conjecture held (i.e., 0‑Ramanujan), their bounds would improve accordingly, highlighting the conjecture’s central role in tightening analytic continuation regions.