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

Francis Brown|arXiv (Cornell University)|Oct 22, 2017

Advanced Algebra and Geometry15 references3 citations

TL;DR

This paper constructs canonical real analytic cusp forms for SL₂(ℤ) associated to Hecke eigenforms, generalizing real analytic Eisenstein series. It establishes that mock modular forms of integral weight arise as algebro-geometric objects with Fourier coefficients proportional to $ n^{1-k}(a_n' + \rho a_n) $, where $ \rho $ is the normalized permanent of the period matrix, resolving the existence of weak harmonic lifts for level-one eigenforms.

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 $n eq 0$, 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.

Motivation & Objective

• To construct canonical real analytic cusp forms for SL₂(ℤ) associated to Hecke eigenforms of integral weight.
• To resolve the existence of weak harmonic lifts for every Hecke eigenform of integral weight and level one.
• To provide a conceptual explanation for the algebraicity of mock modular form coefficients in the CM case.
• To establish a link between the Fourier coefficients of mock modular forms and periods/quasi-periods of cusp forms via the single-valued involution.
• To demonstrate that the overlap between real analytic modular forms and Maass waveforms is fully captured by the constructed cusp forms.

Proposed method

• Constructs real analytic cusp forms $ \mathcal{H}(f)_{r,s} $ as canonical lifts of Hecke eigenforms using the real Frobenius (complex conjugation) on cohomology.
• Applies the single-valued involution $ \mathbf{s} $ on modular forms, lifting it to an involution on $ M_{n+2}^! $, to define canonical representatives.
• Uses the splitting $ M_{n+2}^! = D^{n+1}M_{-n}^! \oplus M_{n+2}^!/D^{n+1}M_{-n}^! $ to ensure uniqueness in the construction.
• Derives differential equations for $ \mathcal{H}(f)_{r,s} $ analogous to those for real analytic Eisenstein series, involving $ \partial $ and $ \overline{\partial} $ operators.
• Computes the period matrix and single-valued period matrix explicitly for $ \Delta \in S_{12} $, using Betti and de Rham cohomology.
• Derives the Fourier coefficients of mock modular forms as $ \alpha + \sum_n \frac{\sigma a_n' + \tau a_n}{n^{11}} q^n $, with $ \sigma, \tau $ from the single-valued involution.

Experimental results

Research questions

• RQ1 Can canonical real analytic cusp forms be constructed for every Hecke eigenform of integral weight and level one in SL₂(ℤ)?
• RQ2 How are the Fourier coefficients of mock modular forms related to periods and quasi-periods of cusp forms?
• RQ3 What is the role of the single-valued involution in constructing weak harmonic lifts and explaining the algebraicity of mock modular form coefficients?
• RQ4 To what extent do the constructed cusp forms span the space of eigenfunctions of the Laplacian in $ \mathcal{M}^{!} $?
• RQ5 Can the framework explain the algebraicity of mock modular form coefficients in the CM case through motivic periods?

Key findings

• The real analytic cusp forms $ \mathcal{H}(f)_{r,s} $ are constructed as canonical lifts of Hecke eigenforms, satisfying differential equations analogous to real analytic Eisenstein series.
• For $ \Delta \in S_{12} $, the single-valued involution yields $ \mathbf{s}(\Delta) = \sigma \Delta' + \tau \Delta $ with $ \sigma \approx -0.35207 $, $ \tau \approx -648.84093 $, and $ \rho = \tau / \sigma \approx 1842.8947269 $.
• The mock modular form $ M_{\Delta} $ has Fourier coefficients proportional to $ n^{-11}(a_n' + \rho a_n) $, matching Ono's formula to high numerical accuracy.
• The constant term in the mock modular form is $ \alpha = \frac{7! \cdot 13}{691} \cdot \frac{10!}{2^{11}} \sigma $, with the 691 denominator arising from the congruence $ \Delta \equiv \mathbb{G}_{12} \pmod{691} $.
• The Petersson norm of $ \Delta $ is computed as $ \approx 0.00000103536205 $, confirming positivity and consistency with known values.
• The determinant condition $ \det\begin{pmatrix} \eta^+ & \omega^+ \\ i\eta^- & i\omega^- \end{pmatrix} = 10! \cdot (2\pi i)^{11} $ is numerically verified, confirming the period matrix structure.