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Poincar\'e series for modular graph forms at depth two. II. Iterated integrals of cusp forms

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arxiv 2109.05018 v3 pith:UGVLGXKJ submitted 2021-09-10 hep-th math.NT

Daniele Dorigoni , Axel Kleinschmidt , Oliver Schlotterer This is my paper · ORCID
classification hep-thmath.NT
keywords modularseriesformsintegralsiteratedcuspdeptheisenstein
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We continue the analysis of modular invariant functions, subject to inhomogeneous Laplace eigenvalue equations, that were determined in terms of Poincar\'e series in a companion paper. The source term of the Laplace equation is a product of (derivatives of) two non-holomorphic Eisenstein series whence the modular invariants are assigned depth two. These modular invariant functions can sometimes be expressed in terms of single-valued iterated integrals of holomorphic Eisenstein series as they appear in generating series of modular graph forms. We show that the set of iterated integrals of Eisenstein series has to be extended to include also iterated integrals of holomorphic cusp forms to find expressions for all modular invariant functions of depth two. The coefficients of these cusp forms are identified as ratios of their L-values inside and outside the critical strip.

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Cited by 3 Pith papers

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  1. Resurgent Lambert series with characters

    math.NT 2025-07 conditional novelty 7.0 of 10

    A complete transseries expansion is derived for doubly Dirichlet-twisted Lambert series near q=1, with applications to quantum modularity and topological string spectral traces.

  2. From Modular Graph Forms to Iterated Integrals

    hep-th 2025-02 conditional novelty 7.0 of 10

    A tree-based algorithm converts modular graph forms into equivariant iterated Eisenstein integrals, is implemented for topologies up to four vertices, and is used to extract the alpha'^8 zeta3 zeta5 term of the four-g...

  3. Equivariant primitives of Eisenstein series for congruence subgroups

    math.NT 2025-02 accept novelty 6.0 of 10

    Equivariant primitives of Eisenstein series for principal congruence subgroups are shown to equal the corresponding non-holomorphic Eisenstein series, including new weight-two cases expressed via single-valued logarithms.

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