Hecke Actions on Loops and Periods of Iterated Shimura Integrals

2023/03/01 by Richard Hain, Hain, Richard · 1 citation
Mathematics · #Advanced Algebra and Geometry #Advanced Mathematical Identities #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT) #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2303.00143

openalex publication_date 2023/03/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 SL₂(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.

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