The Harris-Venkatesh conjecture for derived Hecke operators II: a unified Stark conjecture

2023/05/15 by Zhang, Robin · 1 citation
#11F67 #11F70 #11F75 #11R42 (Primary) 11F41 #22E55 (Secondary) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2305.08956

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 PGL₂ 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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