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

For a fixed cusp form $π$ on $\operatorname{GL}_3(\mathbb{Z})$ and a varying Dirichlet character $χ$ 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 $δ< 1/36$. This improves upon the earlier bounds $δ< 1/1612$ and $δ< 1/308$ obtained by Munshi using his $\operatorname{GL}_2$ 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.

No reviews yet.

No comments yet.