The derived Hecke algebra for dihedral weight one forms

TL;DR

This paper investigates the action of the derived Hecke algebra on dihedral weight one forms, establishing a connection to Stark units and symmetric square L-functions through advanced algebraic and automorphic techniques.

Contribution

It introduces a new relationship between the derived Hecke algebra action and Stark units for dihedral weight one forms, proving a conjecture by the second and fourth authors.

Findings

Established a link between derived Hecke algebra action and Stark units.

Proved a conjecture relating Hecke algebra to special L-values.

Utilized theta correspondence and Eisenstein elements in the proof.

Abstract

We study the action of the derived Hecke algebra in the setting of dihedral weight one forms, and prove a conjecture of the second- and fourth- named authors relating this action to certain Stark units associated to the symmetric square L-function. The proof exploits the theta correspondence between various Hecke modules as well as ideas of Merel and Lecouturier on higher Eisenstein elements.