Some convolution identities for mock modular forms arising from the theory of holomorphic projection

Technical Summary: Some Convolution Identities for Mock Modular Forms Arising from the Theory of Holomorphic Projection

Problem Statement
Convolution identities and recursive formulas are fundamental tools in the theory of holomorphic modular forms and their applications to combinatorics. While classical examples (such as those involving divisor sums and the partition function) are well-established, there is renewed interest in identities arising from non-holomorphic modular forms, specifically harmonic Maass forms. Previous work by Imamo˘glu, Raum, and Richter (2014) established a theory of holomorphic projection for products of harmonic Maass forms and holomorphic modular forms, yielding significant results such as generalizations of Hurwitz-Kronecker class number relations and formulas essential to the proof of the Umbral Moonshine Conjecture.

However, the existing framework by Imamo˘glu, Raum, and Richter is optimized for cases where the associated generalized Pell equation $m^2 - Dn^2 = N$ has a square value for $D$, resulting in only finitely many solutions for a given $N$. Many natural cases involving mock theta functions, particularly those related to Ramanujan's third-order mock theta functions, correspond to cases where $D$ is not a perfect square, leading to infinitely many solutions. This paper addresses the gap in developing a convenient and computationally feasible holomorphic projection theory for these general cases.

Methodology
The authors employ the theory of holomorphic projection to derive convolution identities. The core methodology involves the following steps:

1. Tensor Product Construction: The authors form the tensor product of a specific vector-valued harmonic Maass form, $H^{(3)}$, with a vector-valued holomorphic modular form, $N$. The form $H^{(3)}$ is constructed from Ramanujan's mock theta functions ($\phi, \psi, \nu$) and their non-holomorphic completions (defined via unary theta functions and Eichler integrals) as described by Klein and Kupka. The form $N$ is constructed from eta-quotients.
2. Representation Theory: The tensor product $H^{(3)} \otimes N$ yields a vector of dimension 18. The authors decompose the associated representation $\rho^{(3)} \otimes \rho$ into irreducible components. Using dimension formulas for spaces of modular forms, they demonstrate that the space of quasimodular forms of weight 2 for this representation has dimension 3.
3. Holomorphic Projection: By applying the holomorphic projection operator $\pi_{hol}$ to the tensor product, the authors relate the non-holomorphic parts of the harmonic Maass form to holomorphic modular forms. The projection of the holomorphic part ($F^{(3)} \otimes N$) yields the left-hand sides of the identities, while the projection of the non-holomorphic part ($G^{(3)} \otimes N$) yields the convolution sums on the right-hand sides.
4. Generalized Projection Formula: A critical methodological contribution is the development of Theorem 4.2, a general formula for weight-2 holomorphic projections. Unlike previous formulations that assumed $D$ was a square, this theorem handles the case where the generalized Pell equation $m^2 - Dn^2 = 1$ has infinitely many solutions (non-square $D$). The proof involves:
   • Analyzing the growth conditions of the functions.
   • Utilizing the fundamental solution of the Pell equation to classify integer solutions into equivalence classes.
   • Summing over these classes using Dirichlet characters and specific parameters ($\epsilon_1, \epsilon_2$) to simplify the resulting double series.
   • Employing Lemma 4.3 to perform unimodular substitutions that reduce complex double sums into more compact forms.

Key Contributions and Results
The primary result of the paper is Theorem A, which establishes five distinct convolution-type $q$-series identities involving Ramanujan's third-order mock theta functions $\phi(q)$, $\psi(q)$, and $\nu(q)$.

• Resolution of a Conjecture: Identity (a) in Theorem A resolves a recent conjecture posed by the second author (Garvan) regarding a specific convolution identity for $\psi(q)$.
• Reduction of Identities: The holomorphic projection of the 18-component tensor product initially yields 18 identities. Through classical relations between mock theta functions (specifically identities relating $\phi, \psi, \nu$ and $\omega$), the authors demonstrate that these 18 identities reduce to the five distinct relations stated in Theorem A.
• Explicit Formulas: The identities take the form of a product of a modular form (involving infinite products like $(q;q)_\infty$) and a mock theta function equaling a linear combination of a modular form and a double sum involving Kronecker symbols and quadratic exponents. For example, identity (a) relates $(q;q)_\infty^3 \psi(q)$ to a double sum involving $\binom{-4}{m}\binom{n}{6}$.
• Alternative Proofs: The appendices provide direct $q$-series proofs for several identities, confirming the results obtained via the more abstract holomorphic projection method.

Significance
The paper claims significance in several specific areas:

1. Extension of Holomorphic Projection: It extends the theory of holomorphic projection to cases where the underlying Pell equation has infinitely many solutions, a scenario that was previously difficult to handle computationally due to the divergence of integrals and the complexity of the solution sets.
2. Automated Discovery: The authors emphasize that the utility of the holomorphic projection formulas lies in their ability to serve as a tool for the "automatic discovery and verification" of such formulas. The method provides a systematic way to generate identities that might be difficult to find via direct $q$-series manipulation alone.
3. Unification: The work unifies the derivation of 18 identities arising from a single tensor product, reducing them to a core set of five, thereby clarifying the structural relationships between different mock theta function identities.
4. Foundation for Future Work: The authors note that the methods developed here are designed to extend to $p$-order mock theta functions and the Dyson rank function modulo $p$, suggesting a pathway for further exploration of Mock Theta Conjectures, though the current paper focuses strictly on the third-order case.

The paper concludes that while the computational details of vector-valued harmonic Maass forms are tedious, the establishment of Theorem 4.2 and the subsequent derivation of Theorem A provide a robust framework for generating new convolution identities in the theory of mock modular forms.