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arXiv · 2301.00612

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

Robin Zhang
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Abstract

The first two papers in this series prove the Harris-Venkatesh conjecture on derived Hecke operators 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 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. These calculations also extend to exotic modular forms of odd level and to exotic modular forms with $2$-ordinary Deligne-Serre representation.

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Robin Zhang. 2023-01-02. The Harris-Venkatesh conjecture for derived Hecke operators III: local constants. https://arxiv.org/abs/2301.00612

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