The Harris-Venkatesh conjecture for derived Hecke operators I: imaginary dihedral forms

TL;DR

This paper proves the Harris-Venkatesh conjecture for imaginary dihedral weight-one modular forms, establishing a precise link between derived Hecke operators and Stark units without assumptions on primality or ramification.

Contribution

It fully proves the conjecture in the imaginary dihedral case, extending prior results to the adelic setting and removing previous restrictions.

Findings

Proves Harris-Venkatesh conjecture for imaginary dihedral forms

Introduces Harris--Venkatesh period and two-variable optimal form

Removes assumptions on primality and ramification

Abstract

The Harris-Venkatesh conjecture posits a relationship between the action of derived Hecke operators on weight-one modular forms and Stark units. We prove the full Harris-Venkatesh conjecture for imaginary dihedral weight-one modular forms. This reproves results of Darmon-Harris-Rotger-Venkatesh, extends their work to the adelic setting, and removes all assumptions on primality and ramification from the imaginary dihedral case of the Harris-Venkatesh conjecture. This is accomplished by introducing two new key ingredients: the Harris--Venkatesh period on modular curves and the two-variable optimal form.