Subconvex bounds on GL(3) via degeneration to frequency zero

Abstract

For a fixed cusp form $\pi$ on ${\mathrm{GL}}_3(\mathbb{Z})$ and a varying Dirichlet character $\chi$ of prime conductor $q$, we prove that the subconvex bound \[ L(π\otimes χ, \tfrac{1}{2}) \ll q^{3/4 - δ} \] holds for any $\delta <1/36$. This improves upon the earlier bounds $\delta <1/1612$ and $\delta <1/308$ obtained by Munshi using his ${\mathrm{GL}}_2$ variant of the $\delta$-method. The method developed here is more direct. We first express $\chi$ as the degenerate zero-frequency contribution of a carefully chosen summation formula à la Poisson. After an elementary "amplification" step exploiting the multiplicativity of $\chi$, 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.

17 pages; to appear in Math. Annalen; minor corrections