Automorphic Spectral Analysis Of A Self-Adjoint Operator Attached To A Triple-Product L-Function

Abstract

The spectral theory of unbounded self-adjoint operators applied to invariant Lapla- cians on arithmetic quotients gives information about analytic behavior of L-functions. Given three cuspforms f1, f2, f on SL2 and a strong subconvexity assumption on L(s, f1 × f2 × f ), we specify a natural Hilbert space of automorphic forms and a self-adjoint operator T such that the discrete spectrum (if any) of T is a subset of values s(s − 1) for L(s, f1 × f2 × f ) = 0. Self-adjointness of T implies real eigenvalues, which implies that any such s is on the critical line or in R.