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

Let q be a large prime, and chi the quadratic character modulo q. Let phi be a self-dual Hecke-Maass cusp form for SL(3, DOUBLE-STRUCK CAPITAL Z), and u(j) a Hecke-Maass cusp form for CYRILLIC CAPITAL LETTER GHE(0)(q) subset of SL(2, DOUBLE-STRUCK CAPITAL Z) with spectral parameter t(j). We prove, for the first time, some hybrid subconvexity bounds for the twisted L-functions on GL(3), such as L(1/2,phi xujx chi)MUCH LESS-THAN phi,epsilon(q(1+|tj|))3/2-theta+epsilon,L(1/2+it,phi x chi)MUCH LESS-THAN phi,epsilon(q(1+|t|))3/4-theta/2+epsilon,$$L\left( {1/2,\phi \times {u_j} \times \chi } \right){ \ll _{\phi ,\varepsilon }}{\left( {q\left( {1 + \left| {{t_j}} \right|} \right)} \right)<^>{3/2 - \theta + \varepsilon }},L\left( {1/2 + it,\phi \times \chi } \right){ \ll _{\phi ,\varepsilon }}{\left( {q\left( {1 + \left| t \right|} \right)} \right)<^>{3/4 - \theta /2 + \varepsilon }},$$\end{document} for any epsilon > 0, where theta = 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.