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Poincaré series for modular graph forms at depth two. Part II. Iterated integrals of cusp forms

Dorigoni, Daniele and Kleinschmidt, Axel and Schlotterer, Oliver (2022) 'Poincaré series for modular graph forms at depth two. Part II. Iterated integrals of cusp forms.', Journal of High Energy Physics, 2022 (1). p. 134.


We continue the analysis of modular invariant functions, subject to inhomogeneous Laplace eigenvalue equations, that were determined in terms of Poincaré 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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Publisher statement:This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Date accepted:25 December 2021
Date deposited:11 May 2022
Date of first online publication:25 January 2022
Date first made open access:11 May 2022

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