Hybrid Subconvexity Bounds For Twisted L-functions On GL(3)

The subconvexity problem for automorphic L-functions is one of the central problems in analytic number theory.In recent decades,by the hard work of many number theorists,the subconvexity problem for L-functions on GL(1)and GL(2)is almost solved.And in recent years,several breakthroughs are made to get the subconvexity bounds for GL(3)L-functions.However,so far all known subconvex-ity bounds on GL(3)are just for one aspect.In this paper we prove,for the first time,the hybrid subconvexity bounds for some twisted GL(3)L-functions.Pre-cisely,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 Γ0(q)(?)SL(2,Z)with spectral parameter tj.We prove the hybrid subconvexity bounds for the twisted L-functions(?)and(?)for any ε>0,where θ＝1/23 is admissible.To prove our main result,we will use two different methods to give the follow-ing two types of upper bounds for those L-functions:(?)(?)and(?)(?)for any ε>0.Then after some modification we can prove our main result.All these upper bounds will follow from an estimate for the first moment of a family of L-functions at the central point in short intervals.The major ingredient of this paper is a new treatment of the weight function on the right hand side of the Voronoi formula for SL(3,Z),which allows us to use the hybrid large sieve inequality and then shows our first type of upper bound for those L-functions.Noting that this result gives us a bound which is subconvexity in the q-aspect,and convexity in the t-aspect.And then we need a bound which is subconvexity in the t-aspect.Fortunately,we can obtain the second type of bounds above by modifying the previous work in this topic.And finally,we can prove our hybrid subconvexity bounds.