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^{prime } + rho a_n) for nne 0, where rho is the normalised permanent of the period matrix of the corresponding motive, and a_n, a_n^{prime } 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.
Highlights
Let H denote the upper-half plane with the usual left action by = SL2(Z)
This paper is the third in a series [2,3] studying subspaces of the vector space M! of real analytic functions f : H → C which are modular of weights (r, s) for r, s ∈ Z, i.e
There exists a unique family of real analytic modular functions
Summary
Of modular iterated integrals, generated from weakly holomorphic modular forms by repeatedly taking primitives with respect to ∂ and ∂ In this instalment, we describe the subspace. Of modular iterated integrals of length one These correspond to a modular incarnation of the abelian quotient of the relative completion of the fundamental group [4,15] of the moduli stack of elliptic curves M1,1. They span the first level in an infinite tower of non-abelian or ‘mixed’ modular functions whose general definition was given in
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