# Hecke Actions on Loops and Periods of Iterated Shimura Integrals

## TL;DR

This paper explores how classical Hecke correspondences act on conjugacy classes of the modular group and its profinite completion, inducing dual actions on rings of class functions related to iterated integrals of modular forms, with implications for mixed Hodge structures and Galois actions.

## Contribution

It introduces a new framework for Hecke actions on conjugacy classes and their induced dual actions on rings of class functions associated with iterated modular integrals, revealing non-commutative algebraic structures.

## Key findings

- Hecke correspondences act on conjugacy classes of SL_2(Z) and its profinite completion.
- Induces a dual action on rings of class functions with mixed Hodge structures.
- Hecke operators preserve Hodge structures and commute with Galois actions, but generate a non-commutative algebra.

## Abstract

In this paper we show that the classical Hecke correspondences T_N, N>0, act on the free abelian groups generated by the conjugacy classes of the modular group SL_2(Z) and the conjugacy classes of its profinite completion. We show that this action induces a dual action on the ring of class functions of a certain relative unipotent completion of the modular group. This ring contains all iterated integrals of modular forms that are constant on conjugacy classes. It possesses a natural mixed Hodge structure and, after tensoring with Q_ell$, a natural action of the absolute Galois group. Each Hecke operator preserves this mixed Hodge structure and commutes with the action of the absolute Galois group. Unlike in the classical case, the algebra generated by these Hecke operators is not commutative. The appendix by Pham Tiep is not included. It can be found at arXiv:2303.02807.