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

TL;DR

This paper proposes a new approach to defining $p$-adic Hecke operators on modular curve cohomology, relating their action on weight one cusp forms to Stark units, and formulates a related conjecture.

Contribution

It introduces a novel $p$-adic framework for Hecke operators on weight one forms and states a conjecture linking these operators to Stark units, extending existing theories.

Findings

Formulation of a conjecture relating $p$-adic Hecke operators to Stark units.

Outline of a method to define $p$-adic Shimura classes and derived Hecke operators.

Connection to and extension of Harris-Venkatesh constructions.

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 $\Gamma_1(N)$ 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.