# Derived Hecke algebra for weight one forms

## TL;DR

This paper investigates the action of the derived Hecke algebra on weight one modular forms, proposing a conjecture linking it to Stark units and verifying it through extensive numerical computations.

## Contribution

It introduces a conjecture connecting derived Hecke algebra actions on weight one forms to Stark units and provides numerical evidence supporting this conjecture.

## Key findings

- Numerical verification of the conjecture for levels 23 and 31.
- Confirmation of the conjecture for many derived Hecke operators at primes less than 200.
- Utilization of Merel's pairing evaluation between Shimura and cuspidal subgroups.

## Abstract

We study the action of the derived Hecke algebra on the space of weight one forms. By analogy with the topological case, we formulate a conjecture relating this to a certain Stark unit. We verify the truth of the conjecture numerically, for the weight one forms of level $23$ and $31$, and many derived Hecke operators at primes less than $200$. Our computation depends in an essential way on Merel's evaluation of the pairing between the Shimura and cuspidal subgroups of $J_0(q)$.