On Triple Product L-Functions and a Conjecture of Harris–Venkatesh

Abstract

Abstract Harris and Venkatesh made a conjecture relating the derived Hecke operators and the adjoint motivic cohomology in the setting of weight one modular forms. This conjecture was proved under some conditions in the dihedral case by Darmon–Harris–Rotger–Venkatesh. We use a new approach to prove more general cases of the conjecture (up to sign). Our approach relies on Waldspurger’s formula for the central value of Rankin L-series and Ichino’s formula for the triple product L-function.