A class of non-holomorphic modular forms III: real analytic cusp forms for $$\mathrm {SL}_2(\mathbb {Z})$$ SL 2 ( Z )

Abstract

We define canonical real analytic versions of modular forms of integral weight for the full modular group, generalising real analytic Eisenstein series. They are harmonic Maass waveforms with poles at the cusp, whose Fourier coefficients involve periods and quasi-periods of cusp forms, which are conjecturally transcendental. In particular, we settle the question of finding explicit 'weak harmonic lifts' for every eigenform of integral weight k and level one. We show that mock modular forms of integral weight are algebro-geometric and have Fourier coefficients proportional to n 1−k (a n + ρa n ) for n = 0, where ρ is the normalised permanent of the period matrix of the corresponding motive, and a n , a n are the Fourier coefficients of a Hecke eigenform and a weakly holomorphic Hecke eigenform, respectively. More generally, this framework provides a conceptual explanation for the algebraicity of the coefficients of mock modular forms in the CM case.

Generating series of all modular graph forms from iterated Eisenstein integrals

“…construction [46,49,75] of non-holomorphic modular forms, the series Y τ η is engineered to simplify the holomorphic derivative (2.33) at the expense of the more lengthy expression (2.37) for the antiholomorphic one.…”

A Class of Nonholomorphic Modular Forms Ii: Equivariant Iterated Eisenstein integrals

“…For this, one must allow poles at the cusps, for example. The length one part of this class is the theory of weak harmonic Maass forms and mock modular forms of level one [7], but involves new functions thereafter.…”

“…Writing sv in terms of (b sv , φ sv ) above, we deduce that The idea of this proof applies in a much more general setting. It may be possible to circumvent the final step (7), which appeals to deep results about the category of mixed Tate motives over Z by a direct argument following the procedure outlined in Remark 7.3. However, it is likely that this would forfeit some of the constraints described in Section 7.1, which would become conjectures given the current state of knowledge.…”

The SAGEX review on scattering amplitudes Chapter 10: Selected topics on modular covariance of type IIB string amplitudes and their N=4 supersymmetric Yang–Mills duals

“…This mixing with holomorphic cusp forms becomes more manifest once we choose to represent the generalised Eisenstein series as particular combinations of iterated integrals of holomorphic Eisenstein series [98,[142][143][144][145]. The Eichler-Shimura theorem [146,147] and the work of Brown [115,117,118,148] on iterated integrals of general holomorphic modular forms makes it very plausible that holomorphic cusp forms make an appearance.…”