[Paper Review] Singular invariants and coefficients of weak harmonic Maass forms of weight 5/2

Nickolas Andersen|arXiv (Cornell University)|Oct 27, 2014

Advanced Mathematical Identities22 references3 citations

TL;DR

This paper establishes that the coefficients of weak harmonic Maass forms of weight 5/2 on the full modular group are given by traces of singular invariants—singular moduli for imaginary quadratic fields and cycle integrals of the j-function over geodesics for real quadratic fields. It unifies and extends previous results by Bruinier–Ono (on partition functions) and Duke–Imamoğllu–Töth (on cycle integrals), showing that these coefficients arise from arithmetic traces of non-holomorphic modular functions at CM points or their real quadratic analogues.

ABSTRACT

We study the coefficients of a natural basis for the space of weak harmonic Maass forms of weight $5/2$ on the full modular group. The non-holomorphic part of the first element of this infinite basis encodes the values of the partition function $p(n)$. We show that the coefficients of these harmonic Maass forms are given by traces of singular invariants. These are values of non-holomorphic modular functions at CM points or their real quadratic analogues: cycle integrals of such functions along geodesics on the modular curve. The real quadratic case relates to recent work of Duke, Imamoglu, and Tóth on cycle integrals of the $j$-function, while the imaginary quadratic case recovers the algebraic formula of Bruinier and Ono for the partition function.

Motivation & Objective

• To understand the arithmetic nature of coefficients of mock modular forms of weight 5/2 whose shadows are weakly holomorphic modular forms.
• To extend the known connection between coefficients of harmonic Maass forms and singular moduli to include real quadratic analogues via cycle integrals.
• To unify the algebraic formula for the partition function (Bruinier–Ono) with the cycle integral formula (Duke–Imamoğllu–Töth) in a single framework.
• To provide a complete characterization of coefficients of a natural basis for the space of weak harmonic Maass forms of weight 5/2 on SL₂(ℤ).

Proposed method

• Constructs a natural basis for the space of weak harmonic Maass forms of weight 5/2 on SL₂(ℤ), with the non-holomorphic part encoding the partition function p(n).
• Uses the differential operator ξₖ to relate the holomorphic part (mock modular form) to its shadow, a weakly holomorphic modular form of weight -1/2.
• Applies trace formulas involving generalized genus characters χ_D and stabilizer orders w_Q to express coefficients as weighted sums over equivalence classes of quadratic forms.
• Derives explicit formulas for coefficients via cycle integrals of the j-function over geodesics in the upper half-plane, using the theory of Poincaré series and Bessel functions.
• Establishes a connection between coefficients and special values of L-functions through the use of K-Bessel functions and modular forms of half-integral weight.
• Applies regularization techniques and analytic continuation to handle divergent cycle integrals when dD is a perfect square, extending results from prior work.

Experimental results

Research questions

• RQ1 How are the coefficients of weak harmonic Maass forms of weight 5/2 on SL₂(ℤ) related to arithmetic invariants such as singular moduli and cycle integrals?
• RQ2 Can the algebraic formula for the partition function p(n) be recovered as a trace of singular invariants in the imaginary quadratic case?
• RQ3 How do cycle integrals of the j-function over geodesics on the modular curve relate to coefficients of harmonic Maass forms in the real quadratic case?
• RQ4 What is the unified arithmetic interpretation of coefficients in the basis of weight 5/2 harmonic Maass forms that includes both imaginary and real quadratic contributions?
• RQ5 How can divergent cycle integrals (when dD is a square) be regularized to yield well-defined coefficients in the basis?

Key findings

• The coefficients of the first element in the natural basis of weight 5/2 weak harmonic Maass forms are given by the values of the partition function p(n), which are encoded in the non-holomorphic part of the form.
• For negative discriminants, the coefficients are traces of singular moduli—algebraic integers J(τ_Q) at CM points τ_Q associated with positive definite quadratic forms.
• For positive discriminants, the coefficients are given by weighted sums of cycle integrals of the j-function over closed geodesics on the modular curve, corresponding to indefinite quadratic forms.
• The paper provides a unified formula that interpolates between the Bruinier–Ono formula (imaginary quadratic) and the Duke–Imamoğllu–Töth formula (real quadratic), with the same underlying trace structure.
• The coefficients are expressed via a trace formula involving K-Bessel functions and generalized genus characters, valid even when dD is a perfect square through regularization.
• The method establishes that the coefficients arise from arithmetic traces of non-holomorphic modular functions at CM points or their real quadratic analogues, completing a bridge between mock modular forms and arithmetic geometry.