On triple product L-functions and a conjecture of Harris--Venkatesh

TL;DR

This paper proves a more general case of Harris and Venkatesh's conjecture relating Hecke operators and motivic cohomology for weight one modular forms, using Waldspurger's and Ichino's formulas.

Contribution

It introduces a new approach to prove the conjecture in broader cases, extending previous dihedral case results.

Findings

Confirmed the conjecture up to sign in more general cases

Utilized Waldspurger's formula for central L-values

Applied Ichino's formula for triple product L-functions

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.