A Burgess-like subconvex bound for twisted L-functions

Abstract

Let g be a cuspidal newform (holomorphic or Maass) of arbitrary level and nebentypus, χ a primitive character of conductor q , and s a point on the critical line ℜ s = ½. It is proved that , where ε > 0 is arbitrary and θ = is the current known approximation towards the Ramanujan–Petersson conjecture (which would allow θ = 0); moreover, the dependence on s and all the parameters of g is polynomial. This result is an analog of Burgess' classical subconvex bound for Dirichlet L -functions. In Appendix 2 the above result is combined with a theorem of Waldspurger and the adelic calculations of Baruch–Mao to yield an improved uniform upper bound for the Fourier coefficients of holomorphic half-integral weight cusp forms.