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

TL;DR

This paper establishes a new subconvex bound for L-functions of GL(3) cusp forms twisted by Dirichlet characters, using a novel approach that simplifies previous methods and achieves a better exponent.

Contribution

The paper introduces a more direct method to obtain subconvex bounds for GL(3) L-functions, improving upon previous results by reducing the exponent gap.

Findings

Achieved subconvex bound with δ < 1/36 for GL(3) L-functions

Developed a new technique expressing characters as zero-frequency contributions

Simplified the process compared to earlier methods like Munshi's

Abstract

For a fixed cusp form $\pi$ on $\operatorname{GL}_3(\mathbb{Z})$ and a varying Dirichlet character $\chi$ of prime conductor $q$, we prove that the subconvex bound \[ L(\pi \otimes \chi, \tfrac{1}{2}) \ll q^{3/4 - \delta} \] holds for any $\delta < 1/36$. This improves upon the earlier bounds $\delta < 1/1612$ and $\delta < 1/308$ obtained by Munshi using his $\operatorname{GL}_2$ 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.