Mock Theta Functions: Modular Insights

• Mock theta functions are q-hypergeometric series introduced by Ramanujan that display modular-like q-expansions but lack complete modular transformation properties.
• Their non-holomorphic corrections via explicit period integrals transform them into harmonic Maass forms, bridging classical modular theory with modern applications.
• Connections with indefinite theta series and Jacobi forms provide deep insights into their analytic structure and the origins of their modular anomalies.

Mock theta functions are special $q$-hypergeometric series introduced by Ramanujan in his final letter to Hardy and further developed in later manuscripts. Unlike classical modular forms and theta functions, mock theta functions display modular-like $q$-expansions but fail to transform as modular forms under the modular group. Their true nature and modularity properties were elucidated by Sander Zwegers, who showed that the addition of explicit non-holomorphic correction terms yields real-analytic modular forms known as harmonic Maass forms. Mock theta functions are central to areas spanning modular forms, partition theory, indefinite theta series, and mathematical physics.

1. Definitions and q-Hypergeometric Structure

Mock theta functions are typically defined as $q$-hypergeometric series that resemble modular (theta) functions in their $q$-expansions but differ crucially in analytic and transformation properties. A canonical example is Ramanujan's third-order mock theta function: $$f(q) = 1 + \sum_{n=1}^{\infty}\frac{q^{n^2}}{(-q;q)_n^2},$$ where $(a;q)_n$ denotes the $q$-Pochhammer symbol. This type of series appears as a unilateral partial theta series: the summation runs only over $n \geq 0$ (a "half-lattice") rather than $n \in \mathbb{Z}$ as is the case for modular theta functions (Bringmann et al., 2011).

Multiple families of mock theta functions exist—classified by order (e.g., 2nd, 3rd, 5th, 7th, 10th)—each associated with distinct $q$-hypergeometric expansions. Generalizations such as the "universal mock theta functions" $g_2(w;q),\, g_3(w;q)$, and $K(w;q)$ parametrize all classical examples and demonstrate the flexibility and richness of these series.

2. Modular Completion and Real-Analytic Modular Forms

While mock theta functions are not modular forms, they admit a "completion" to real-analytic modular forms of weight $1/2$. Zwegers established that each mock theta function $f(q)$ can be completed by adding a non-holomorphic correction, typically a period integral of a weight $3/2$ unary theta function—referred to as the "shadow"—to produce

$\widehat{f}(z) = f(q) + R(z),$

where $q = e^{2\pi i z}$, $R(z)$ is real-analytic, and the resulting function $\widehat{f}$ transforms as a modular form and is annihilated by the weight $1/2$ Laplace operator (the hyperbolic Laplacian) (0807.4834).

For example, given a meromorphic Jacobi form, the Fourier coefficients can be shown to yield mock theta functions as their holomorphic part. Their modular completion, via appropriate non-holomorphic terms, ensures full modularity.

A foundational result: for $8$ of Ramanujan's $10$ fifth-order and all $3$ seventh-order mock theta functions, explicit correction terms can be constructed as period integrals of unary theta functions. The corrected sum produces a real-analytic weight $1/2$ modular object that satisfies the hyperbolic Laplacian equation (0807.4834).

3. Analytic Theory of Indefinite Theta Series

Zwegers' analysis also introduced the role of indefinite theta series, notably those of signature $(r-1, 1)$, in describing mock theta functions. Lerch sums—also called Appell functions or generalized Lambert series—play a central role as analytic building blocks for mock theta functions. These sums feature in interpretations where mock theta functions arise as holomorphic parts of indefinite theta series. The indefinite character, as opposed to the positive-definite lattices of classical theta functions, is crucial for the emergence of mock modularity and the need for non-holomorphic corrections.

In practice, the associated non-holomorphic period integrals often take the form of Eichler integrals of weight $3/2$ unary theta series.

4. Explicit Results for Mock Theta Functions

Through results for indefinite theta series and analysis of meromorphic Jacobi forms, explicit decompositions and identities are obtained for classical mock theta functions:

• For each original fifth- and seventh-order function, there exists a period integral correction yielding a non-holomorphic modular form.
• The "shadow"—the non-holomorphic integral—is always a unary theta function of weight $3/2$.
• The resulting completed function is a real-analytic modular form of weight $1/2$, annihilated by the hyperbolic Laplacian.

These results are achieved via analytic tools such as Fourier expansion analysis, manipulation of indefinite theta series, and explicit evaluation of the corrective period integral terms.

5. Connection to Jacobi Forms and Fourier Coefficients

A deep link is established between mock theta functions and the Fourier coefficients of meromorphic Jacobi forms. Specifically, for meromorphic forms with simple poles, the Fourier expansion yields a decomposition: $$\text{Fourier coefficient} = \text{mock theta function (holomorphic part)} + \text{explicit real-analytic correction}.$$ This perspective clarifies the modularity anomaly of mock theta functions and situates their paper within the broader context of Jacobi forms and the theory of automorphic forms.

6. Broader Impact and Modern Applications

The reinterpretation of mock theta functions as holomorphic parts of harmonic Maass forms has broad implications:

• It unifies combinatorial $q$-series, classical partition theory, and the analytic theory of modular objects.
• Mock theta functions and their real-analytic completions now play a central role in understanding connections with indefinite theta series, automorphic representations, and the spectral theory of Maass forms.
• Their properties influence areas such as quantum topology (notably in the paper of invariants of $3$-manifolds), representation theory (particularly of affine Lie superalgebras), and string theory.
• Modern developments repeatedly use the modular completion strategy for both classical and generalizations of mock theta and partial theta functions.

These conceptual advances have made the structure, symmetry, and arithmetic properties of mock theta functions accessible to the powerful toolkit of modern automorphic form theory.

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In summary, mock theta functions are $q$-hypergeometric series that, after analytic continuation and the addition of non-holomorphic period integrals (shadows), become real-analytic modular forms of weight $1/2$ annihilated by the hyperbolic Laplacian. The underlying structure and properties are intimately governed by connections to indefinite theta series, Appell functions, and Jacobi forms, and are central to contemporary research in modular forms, combinatorics, and mathematical physics (0807.4834).