Subconvex bounds on $${{\mathrm{GL}}}_3$$ GL 3 via degeneration to frequency zero

Abstract

For a fixed cusp form π on GL 3 (Z) and a varying Dirichlet character χ of prime conductor q, we prove that the subconvex bound L π ⊗ χ, 1 2 q 3/4−δ holds for any δ < 1/36. This improves upon the earlier bounds δ < 1/1612 and δ < 1/308 obtained by Munshi using his GL 2 variant of the δ-method. The method developed here is more direct. We first express χ as the degenerate zero-frequency contribution of a carefully chosen summation formula à la Poisson. After an elementary "amplification" step exploiting the multiplicativity of χ , we then apply a sequence of standard manipulations (reciprocity, Voronoi, Cauchy-Schwarz and the Weil bound) to bound the contributions of the nonzero frequencies and of the dual side of that formula.