Towards the p-adic derived Hecke algebra for weight one forms

2025/06/10 by Robin Zhang, Zhang, Robin
Mathematics · #11F41 #11F67 #11F75 #11F85 #11R27 (Secondary) #11R42 (Primary) 11G18 #Advanced Algebra and Geometry #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2506.09139

openalex publication_date 2025/06/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This note outlines an approach to defining p-adic Shimura classes and p-adic derived Hecke operators on the completed cohomology of modular curves from upcoming work by the author. After reviewing the modulo-p constructions of Harris and Venkatesh, we formulate a conjecture relating the action of p-adic derived Hecke operators on cusp forms of weight 1 and level Γ₁(N) to the p-adic logarithm of the Stark unit for the corresponding adjoint Deligne-Serre representation. This new p-adic conjecture can be viewed as complementary to the Harris-Venkatesh conjecture.

Citations

• The Harris-Venkatesh conjecture for derived Hecke operators II: a unified Stark conjecture
• The Harris-Venkatesh conjecture for derived Hecke operators III: local constants
• The Harris-Venkatesh conjecture for derived Hecke operators I: imaginary dihedral forms
• The derived Hecke algebra for dihedral weight one forms
• On triple product L-functions and a conjecture of Harris--Venkatesh
• Derived Hecke algebra for weight one forms

Related

• Derived Hecke algebra for weight one forms
• The Harris-Venkatesh conjecture for derived Hecke operators I: imaginary dihedral forms
• The Harris-Venkatesh conjecture for derived Hecke operators II: a unified Stark conjecture
• Hecke algebras and p-adic groups
• Traces on Hecke algebras and families of p-adic modular forms