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A class of non-holomorphic modular forms I

Francis Brown

TL;DR

The paper develops a comprehensive framework for a class of real-analytic modular forms \\mathcal{M} with weights \\(r,s) and analyzes their rich algebraic and differential structure, including a nontrivial \mathfrak{sl}_2 action via \\partial and \\overline{\\partial}, a bigraded Laplacian, and a Petersson-type inner product. It introduces modular iterated primitives and the subspace \\mathcal{MI} (and \\mathcal{MI}^{E}) built from equivariant iterated integrals of Eisenstein series, linking to real-analytic Eisenstein series and to single-valued period phenomena. The work connects these modular primitives to modular graph functions from genus-one string perturbation theory and motives, and develops explicit constructions of equivariant double iterated integrals with orthogonality properties that separate Eisenstein data from cusp-form contributions. Collectively, the findings illuminate how real-analytic modular objects encode period data, SL$_2$-equivariant iterated integrals, and potential links to mixed motives, offering a structured approach to extending modular forms beyond the holomorphic setting. The framework also yields concrete realizations of real-analytic Eisenstein series and their primitives, and outlines pathways to meromorphic generalizations via \\mathcal{M}^{!} and iterated integrals.

Abstract

This introductory paper studies a class of real analytic functions on the upper half plane satisfying a certain modular transformation property. They are not eigenfunctions of the Laplacian and are quite distinct from Maass forms. These functions are modular equivariant versions of real and imaginary parts of iterated integrals of holomorphic modular forms, and are modular analogues of single-valued polylogarithms. The coefficients of these functions in a suitable power series expansion are periods. They are related both to mixed motives (iterated extensions of pure motives of classical modular forms), as well as the modular graph functions arising in genus one string perturbation theory. In an appendix, we use weakly holomorphic modular forms to write down modular primitives of cusp forms. Their coefficients involve the full period matrix (periods and quasi-periods) of cusp forms.

A class of non-holomorphic modular forms I

TL;DR

The paper develops a comprehensive framework for a class of real-analytic modular forms \\mathcal{M} with weights \\(r,s) and analyzes their rich algebraic and differential structure, including a nontrivial \mathfrak{sl}_2 action via \\partial and \\overline{\\partial}, a bigraded Laplacian, and a Petersson-type inner product. It introduces modular iterated primitives and the subspace \\mathcal{MI} (and \\mathcal{MI}^{E}) built from equivariant iterated integrals of Eisenstein series, linking to real-analytic Eisenstein series and to single-valued period phenomena. The work connects these modular primitives to modular graph functions from genus-one string perturbation theory and motives, and develops explicit constructions of equivariant double iterated integrals with orthogonality properties that separate Eisenstein data from cusp-form contributions. Collectively, the findings illuminate how real-analytic modular objects encode period data, SL-equivariant iterated integrals, and potential links to mixed motives, offering a structured approach to extending modular forms beyond the holomorphic setting. The framework also yields concrete realizations of real-analytic Eisenstein series and their primitives, and outlines pathways to meromorphic generalizations via \\mathcal{M}^{!} and iterated integrals.

Abstract

This introductory paper studies a class of real analytic functions on the upper half plane satisfying a certain modular transformation property. They are not eigenfunctions of the Laplacian and are quite distinct from Maass forms. These functions are modular equivariant versions of real and imaginary parts of iterated integrals of holomorphic modular forms, and are modular analogues of single-valued polylogarithms. The coefficients of these functions in a suitable power series expansion are periods. They are related both to mixed motives (iterated extensions of pure motives of classical modular forms), as well as the modular graph functions arising in genus one string perturbation theory. In an appendix, we use weakly holomorphic modular forms to write down modular primitives of cusp forms. Their coefficients involve the full period matrix (periods and quasi-periods) of cusp forms.

Paper Structure

This paper contains 53 sections, 35 theorems, 220 equations, 1 figure, 1 table.

Table of Contents

  1. Modular graph functions
  2. Properties
  3. Landscape
  4. Acknowledgements
  5. A class of functions $\mathcal{M}$
  6. First definitions
  7. $q$ - expansions and pole filtration
  8. Differential operators (Maass)
  9. Action of $\mathfrak{sl}_2$
  10. Almost holomorphic modular forms
  11. Bigraded Laplace operator
  12. Real analytic Petersson inner product
  13. Holomorphic projections Sturm.
  14. A picture of $\mathcal{M}$
  15. Primitives and obstructions
  16. ...and 38 more sections

Key Result

Lemma 2.2

Suppose that $f: \mathfrak{H} \rightarrow \mathbb C$ satisfies equation $(fbimod)$, and admits an expansion in the ring $\mathbb C [[q, \overline{q}]] [ \log q , \log \overline{q} ]$. Then $f \in \mathbb C[[q, \overline{q}]] [ \mathbb{L} ].$

Figures (1)

  • Figure 1: Two graphs

Theorems & Definitions (85)

  • Definition 1.1
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Example 2.3
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • ...and 75 more