Strong Hybrid Subconvexity for Twisted Selfdual $\mathrm{GL}_3$ $L$-Functions

TL;DR

This paper establishes strong hybrid subconvex bounds for selfdual GL_3 L-functions twisted by Dirichlet characters and for certain GL_3×GL_2 Rankin-Selberg L-values, advancing the understanding of their size in various aspects.

Contribution

It introduces a novel spectral reciprocity approach and a Lindelöf-on-average bound to achieve near-optimal subconvex bounds for these L-functions.

Findings

Proved hybrid subconvex bounds in q and t aspects for GL_3 L-functions.

Established analogous bounds for GL_3×GL_2 Rankin-Selberg L-values.

Developed a spectral reciprocity formula linking different L-function moments.

Abstract

We prove strong hybrid subconvex bounds simultaneously in the $q$ and $t$ aspects for $L$-functions of selfdual $\mathrm{GL}_3$ cusp forms twisted by primitive Dirichlet characters. We additionally prove analogous hybrid subconvex bounds for central values of certain $\mathrm{GL}_3 \times \mathrm{GL}_2$ Rankin-Selberg $L$-functions. The subconvex bounds that we obtain are strong in the sense that, modulo current knowledge on estimates for the second moment of $\mathrm{GL}_3$ $L$-functions, they are the natural limit of the first moment method pioneered by Li and by Blomer. The method of proof relies on an explicit $\mathrm{GL}_3 \times \mathrm{GL}_2 \leftrightsquigarrow \mathrm{GL}_4 \times \mathrm{GL}_1$ spectral reciprocity formula, which relates a $\mathrm{GL}_2$ moment of $\mathrm{GL}_3 \times \mathrm{GL}_2$ Rankin-Selberg $L$-functions to a $\mathrm{GL}_1$ moment of $\mathrm{GL}_4 \times \mathrm{GL}_1$ Rankin-Selberg $L$-functions. A key additional input is a Lindel\"of-on-average upper bound for the second moment of Dirichlet $L$-functions restricted to a coset, which is of independent interest.