The paper defines p-adic Shimura classes and derived Hecke operators and conjectures that their action on weight-one forms equals the p-adic logarithm of a Stark unit.
The Harris-Venkatesh conjecture for derived Hecke operators I: imaginary dihedral forms
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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.
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Towards the $p$-adic derived Hecke algebra for weight one forms
The paper defines p-adic Shimura classes and derived Hecke operators and conjectures that their action on weight-one forms equals the p-adic logarithm of a Stark unit.