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The Harris-Venkatesh conjecture for derived Hecke operators II: a unified Stark conjecture
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abstract
We prove a compatibility theorem between the Stark conjecture and the Harris-Venkatesh conjecture for imaginary dihedral modular forms of weight $1$. The key technical input is a general two-variable $\mathrm{PGL}_2$ Siegel-Weil formula that precisely gives the Laurent series coefficients of Eisenstein series as a dual theta lift of Eisenstein series. This two-variable Siegel-Weil formula is applied to Rankin-Selberg periods of imaginary dihedral optimal forms and a refinement of Stark's formula relating $L$-values with elliptic units, yielding compatibility between derived Hecke operators and adjoint $L$-values for Deligne-Serre representations. We formulate a general unified conjecture encompassing the Stark and Harris-Venkatesh conjectures and conceptually reinterpret the action of derived Hecke operators as modular Stark unit data at almost all primes.
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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.
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