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

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 $$L(\pi \otimes \chi ,\frac{1}{2})\ll q^{3\mathrm{/}4-\delta }$$ holds for any $\delta <1\mathrm{/}36$. This improves upon the earlier bounds $\delta < 1/1612$and$\delta < 1/308$obtainedbyMunshiusinghis$\operatorname{GL}_2$variantofthe$\delta$-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.