[Paper Review] A polyharmonic Maass form of depth 3/2 for SL_2(Z)

Scott Ahlgren, Nickolas Andersen+1 more|arXiv (Cornell University)|Jul 19, 2017

Advanced Mathematical Identities15 references3 citations

TL;DR

This paper constructs a polyharmonic Maass form of weight $1/2$ and depth $3/2$ for $\mathrm{SL}_2(\mathbb{Z})$, providing a direct realization of a form previously known only via regularization or Poincaré series. It introduces an extended inner product to interpret the intractable coefficients of square-index Fourier modes, linking them to traces of modular functions and generalized Hurwitz class numbers.

ABSTRACT

Duke, Imamoglu, and Toth constructed a polyharmonic Maass form of level 4 whose Fourier coefficients encode real quadratic class numbers. A more general construction of such forms was subsequently given by Bruinier, Funke, and Imamoglu. Here we give a direct construction of such a form for the full modular group and study the properties of its coefficients. We give interpretations of the coefficients of the holomorphic parts of each of these polyharmonic Maass forms as inner products of certain weakly holomorphic modular forms and harmonic Maass forms. The coefficients of square index are particularly intractable; in order to address these, we develop various extensions of the usual normalized Peterson inner product using a strategy of Bringmann, Ehlen and Diamantis.

Motivation & Objective

• To construct a polyharmonic Maass form of weight $1/2$ and depth $3/2$ for the full modular group $\mathrm{SL}_2(\mathbb{Z})$, providing a direct realization of such forms beyond level structures.
• To resolve the intractability of Fourier coefficients at square indices in polyharmonic Maass forms, which arise from poles in Poincaré series constructions.
• To extend the regularized inner product using a strategy inspired by Bringmann, Ehlen, and Diamantis to define a new inner product $\langle \cdot, \cdot \rangle_1$ for weakly holomorphic and harmonic Maass forms.
• To interpret the coefficients of the holomorphic part of the form as inner products involving weakly holomorphic modular forms and harmonic Maass forms.
• To establish a precise formula linking the coefficients of the square-index terms to traces of modular functions and generalized Hurwitz class numbers $h^*(d)$.

Proposed method

• Constructs a polyharmonic Maass form $\bm{Z}(\tau)$ of weight $1/2$ and depth $3/2$ on $\Gamma_0(4)$, later extended to $\mathrm{SL}_2(\mathbb{Z})$, via a regularized theta lift from the constant function $1$.
• Introduces a new inner product $\langle h_d, F \rangle_1$ defined via analytic continuation of an integral with exponential damping $e^{-wy}$, valid at $w=0$.
• Uses the operator $\xi_k = 2iy^k \overline{\partial / \partial \overline{\tau}}$ to define polyharmonicity, with $\xi_{1/2} \bm{Z} = -2 \widehat{\bm{Z}}_-$, linking to known harmonic Maass forms.
• Employs the generalized Hurwitz function $h^*(d)$, defined via regulators $R(d)$ and class numbers, to express Fourier coefficients uniformly across $d > 0$.
• Applies the strategy of Bringmann, Ehlen, and Diamantis to handle exponentially growing terms in the case of square discriminants, enabling the extension of the inner product.
• Derives an explicit formula for $\sqrt{24} \langle h_d, F \rangle_1 = -\operatorname{Tr}_d(f) + \chi_{12}(\sqrt{d})\left(\operatorname{Tr}_1(f) - 12 \frac{h^*(d)}{\sqrt{d}} - i\right)$, resolving the square-index coefficient problem.

Experimental results

Research questions

• RQ1 How can a polyharmonic Maass form of depth $3/2$ and weight $1/2$ be explicitly constructed for the full modular group $\mathrm{SL}_2(\mathbb{Z})$?
• RQ2 What is the interpretation of the Fourier coefficients of square index in such forms, which are not captured by standard regularized inner products?
• RQ3 Can the regularized inner product be extended to handle cases where the Fourier coefficients grow exponentially, such as when $d$ is a perfect square?
• RQ4 How are the coefficients of the holomorphic part of the form related to traces of modular functions and generalized Hurwitz class numbers?
• RQ5 What is the precise relationship between the coefficients of the square-index terms and the modular function $f$ on $\Gamma_0(6)$?

Key findings

• The paper constructs a polyharmonic Maass form $\bm{Z}(\tau)$ of weight $1/2$ and depth $3/2$ for $\mathrm{SL}_2(\mathbb{Z})$, with Fourier expansion involving $h^*(d)/\sqrt{d}$ for $d > 0$, $\beta_{1/2}(4\pi|d|y)q^d$ for $d < 0$, and a non-holomorphic term $\alpha(4n^2y)q^{n^2}$.
• The coefficients of square-index terms are interpreted via the extended inner product $\langle h_d, F \rangle_1$, which is shown to exist and be well-defined even when $d$ is a square.
• The formula $\sqrt{24} \langle h_d, F \rangle_1 = -\operatorname{Tr}_d(f) + \chi_{12}(\sqrt{d})\left(\operatorname{Tr}_1(f) - 12 \frac{h^*(d)}{\sqrt{d}} - i\right)$ provides a complete interpretation of the square-index coefficients.
• The regularized inner product $\langle h_d, F \rangle_{\operatorname{reg}}$ exists only when $d$ is not a square, and in that case, $\sqrt{24} \langle h_d, F \rangle_{\operatorname{reg}} = -\operatorname{Tr}_d(f)$.
• The extended inner product $\langle h_d, F \rangle_1$ is defined via analytic continuation of the integral $I(h_d, F; w)$ to $w = 0$, resolving the divergence issue for square $d$.
• The result establishes a precise link between the coefficients of the square-index Fourier modes and the traces of the modular function $f$ on $\Gamma_0(6)$, with correction terms involving $h^*(d)/\sqrt{d}$ and $i$.