Twisted moments of L-functions and spectral reciprocity

Abstract

We establish a reciprocity formula that expresses the fourth moment of automorphic $L$-functions of level $q$ twisted by the $\ell$th Hecke eigenvalue as the fourth moment of automorphic $L$-functions of level $\ell$ twisted by the $q$th Hecke eigenvalue. Direct corollaries include subconvexity bounds for $L$-functions in the level aspect and a short proof of an upper bound for the fifth moment of automorphic $L$-functions.