Hybrid subconvexity bounds for twisted L-functions on GL(3)

Let q be a large prime, and χ the quadratic character modulo q. Let ϕ be a self-dual Hecke-Maass cusp form for SL(3, ℤ), and uj a Hecke-Maass cusp form for Г0(q) ⊆ SL(2, ℤ) with spectral parameter tj. We prove, for the first time, some hybrid subconvexity bounds for the twisted L-functions on GL(3), such as L(1/2,ϕ×uj×χ)≪ϕ,ε(q(1+|tj|))3/2−θ+ε,L(1/2+it,ϕ×χ)≪ϕ,ε(q(1+|t|))3/4−θ/2+ε,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L\left( {1/2,\phi \times {u_j} \times \chi } \right){ \ll _{\phi ,\varepsilon }}{\left( {q\left( {1 + \left| {{t_j}} \right|} \right)} \right)^{3/2 - \theta + \varepsilon }},L\left( {1/2 + it,\phi \times \chi } \right){ \ll _{\phi ,\varepsilon }}{\left( {q\left( {1 + \left| t \right|} \right)} \right)^{3/4 - \theta /2 + \varepsilon }},$$\end{document} for any ε > 0, where θ = 1/23 is admissible. The proofs depend on the first moment of a family of L-functions in short intervals. In order to bound this moment, we first use the approximate functional equations, the Kuznetsov formula, and the Voronoi formula to transform it to a complicated summation; and then we apply different methods to estimate it, which give us strong bounds in different aspects. We also use the stationary phase method and the large sieve inequalities.