Subconvex bounds on $${{\mathrm{GL}}}_3$$ 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 $$\begin{aligned} L\left( \pi \otimes \chi , \tfrac{1}{2}\right) \ll q^{3/4 - \delta } \end{aligned}$$ 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 a 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.