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

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 Lindelöf-on-average upper bound for the second moment of Dirichlet $L$-functions restricted to a coset, which is of independent interest.

48 pages

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