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Multiple Modular Values and the relative completion of the fundamental group of $M_{1,1}$

Francis Brown

TL;DR

The paper develops a comprehensive framework for understanding how iterated extensions of motives attached to modular forms may be realized through the relative completion of the fundamental group of modular curves, focusing on the genus one case $\\mathcal{M}_{1,1}$ and the full modular group. It builds an analytic theory of totally holomorphic periods via iterated Shimura/Eichler integrals and encodes these periods in a canonical holomorphic cocycle, then anchors the structure in a Tannakian/Hodge-theoretic setting by describing the automorphism group acting on the relative completion. The work constructs explicit zeta-like and modular elements, proves freeness results for the acting Lie algebra, and derives transference principles that link periods of Eisenstein and cusp data to special values of L-functions. By combining Rankin–Selberg type arguments with Haberlund-type formulas, it connects double Eisenstein integrals to $L$-values beyond the critical strip, offering a path toward Beilinson-type conjectures for Rankin–Selberg convolutions and promising extensions to broader congruence subgroups. The results illuminate how nontrivial Galois actions on relative completions of modular curves can encode rich arithmetic information with potential applications in perturbative quantum field theory and modular graph functions.

Abstract

Multiple modular values are a common generalisation of multiple zeta values and periods of modular forms, and are periods of a hypothetical Tannakian category of mixed modular motives. They are given by regularised iterated integrals on the upper half plane generalising the iterated Shimura integrals of Manin. In this paper, some first properties of the underlying theory are established in the case of the full modular group: in particular, the relationship with special values of L-functions of modular forms at all positive integers; and the action of the conjectural motivic Galois group via a certain group of automorphisms.

Multiple Modular Values and the relative completion of the fundamental group of $M_{1,1}$

TL;DR

The paper develops a comprehensive framework for understanding how iterated extensions of motives attached to modular forms may be realized through the relative completion of the fundamental group of modular curves, focusing on the genus one case and the full modular group. It builds an analytic theory of totally holomorphic periods via iterated Shimura/Eichler integrals and encodes these periods in a canonical holomorphic cocycle, then anchors the structure in a Tannakian/Hodge-theoretic setting by describing the automorphism group acting on the relative completion. The work constructs explicit zeta-like and modular elements, proves freeness results for the acting Lie algebra, and derives transference principles that link periods of Eisenstein and cusp data to special values of L-functions. By combining Rankin–Selberg type arguments with Haberlund-type formulas, it connects double Eisenstein integrals to -values beyond the critical strip, offering a path toward Beilinson-type conjectures for Rankin–Selberg convolutions and promising extensions to broader congruence subgroups. The results illuminate how nontrivial Galois actions on relative completions of modular curves can encode rich arithmetic information with potential applications in perturbative quantum field theory and modular graph functions.

Abstract

Multiple modular values are a common generalisation of multiple zeta values and periods of modular forms, and are periods of a hypothetical Tannakian category of mixed modular motives. They are given by regularised iterated integrals on the upper half plane generalising the iterated Shimura integrals of Manin. In this paper, some first properties of the underlying theory are established in the case of the full modular group: in particular, the relationship with special values of L-functions of modular forms at all positive integers; and the action of the conjectural motivic Galois group via a certain group of automorphisms.

Paper Structure

This paper contains 186 sections, 92 theorems, 719 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Genus one
  4. A group of automorphisms
  5. Extensions
  6. Applications
  7. Outlook
  8. Plan
  9. Acknowledgement
  10. Basic notation and reminders
  11. Modular forms
  12. Tensor algebras
  13. Group cohomology
  14. Cup products
  15. Relative cohomology
  16. ...and 171 more sections

Key Result

Theorem 1.1

The images of these elements in $(\mathrm{Lie}\, \mathbb{A}^{dR}\otimes \overline{\mathbb Q})^{ab}$ are non-zero. Any choice of lift of these elements to $\mathrm{Lie}\, \mathbb{A}^{dR}\otimes \overline{\mathbb Q}$. acts non-trivially upon $\mathcal{G} ^{dR}_{1,1} \times \overline{\mathbb Q}$.

Figures (2)

  • Figure 1:
  • Figure 2:

Theorems & Definitions (241)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Remark 3.1
  • Proposition 3.2
  • proof
  • Remark 4.1
  • Remark 4.2
  • Definition 4.3
  • ...and 231 more