The Harris-Venkatesh conjecture for derived Hecke operators III: local constants

Abstract

The first two papers in this series prove the Harris-Venkatesh conjecture and its refinement with the Stark conjecture for imaginary dihedral modular forms of weight $1$. This paper explicitly describes the constants appearing in the Harris-Venkatesh (plus Stark) conjecture for dihedral modular forms by evaluating $\mathrm{GL}(2)\times \mathrm{GL}(2)$ Rankin--Selberg periods and zeta integrals on newforms and optimal forms. One consequence is a formula for the ratio between Petersson norms and adjoint $L$-values. Our calculations also extend to exotic modular forms whose level is odd or whose Deligne-Serre representation is $2$-ordinary.

46 pages, sequel to arXiv:2301.00570 and arXiv:2305.08956; minor revisions and added Petersson norm calculations originally in arXiv:2305.08956