Holomorphic Projection of Mixed Mock Modular Forms Involving Sesquiharmonic Maass Forms

Core Concepts

This research paper presents a method for computing the holomorphic projection of mixed mock modular forms, specifically focusing on those involving sesquiharmonic Maass forms, and explores its applications in number theory, particularly in understanding the coefficients of modular forms and their relationship to arithmetic functions like class numbers.

Abstract

Allen, M., Beckwith, O., & Sharma, V. (2024). Holomorphic Projection for Sesquiharmonic Maass Forms. arXiv:2411.05972v1 [math.NT].

This paper aims to extend the application of holomorphic projection techniques to a broader class of real-analytic modular forms, specifically focusing on sesquiharmonic Maass forms, and to derive explicit formulas for their coefficients in terms of elementary functions and coefficients of the original forms.

Key Insights Distilled From

Holomorphic projection for sesquiharmonic Maass forms

by Michael Alle... at arxiv.org 11-12-2024

https://arxiv.org/pdf/2411.05972.pdf

Deeper Inquiries

How can the techniques presented in this paper be generalized to study the holomorphic projection of products involving other types of real-analytic modular forms beyond sesquiharmonic Maass forms?

This paper focuses on the holomorphic projection of products of a weight 1/2 sesquiharmonic Maass form and a weight 3/2 theta function. Generalizing these techniques to other real-analytic modular forms presents exciting challenges and opportunities:
1. Higher Order Maass Forms:

Challenge: The paper leverages the specific structure of sesquiharmonic Maass forms (whose image under the $\xi_k$ operator is harmonic). Generalizing to forms satisfying higher-order differential equations (e.g., $(\xi_k)^n F = 0$ for some $n > 2$) would require a deeper understanding of the interplay between these operators and holomorphic projection.
Approach:

Develop explicit Fourier expansions for these higher-order forms, analogous to Proposition 2.4.
Investigate how the differential operators transform these expansions.
Carefully analyze the resulting integrals in the holomorphic projection formula.
2. Varying Weights:

Challenge: The weight difference between the sesquiharmonic form and the theta function plays a crucial role in the integral computations. Changing these weights will significantly alter the integrals.
Approach:

Systematically study how the integral formulas in Lemma 4.1, 4.2, 4.3, and 4.4 generalize for different weight combinations.
Explore connections to special functions beyond the hypergeometric functions used in this paper.
3. Beyond Theta Functions:

Challenge: The modularity of the theta function is essential. Using other modular forms will introduce new complexities in the unfolding trick used in Proposition 3.1.
Approach:

Focus on modular forms with well-understood Fourier expansions and transformation properties.
Investigate whether alternative unfolding techniques or Rankin-Selberg convolutions can be applied.
4. Beyond Products:

Challenge: While this paper focuses on products, one could consider more general combinations of real-analytic modular forms (e.g., linear combinations, derivatives).
Approach:

Start with simple cases and gradually increase complexity.
Leverage any known relations between the chosen forms and their derivatives.
Key Takeaway: Successfully generalizing these techniques will likely require a combination of:

Deep knowledge of special functions and their integral representations.
Advanced techniques from the theory of modular forms (e.g., Rankin-Selberg method, Poincaré series).
Careful analysis of the growth properties of the forms involved.

Could there be alternative approaches, perhaps not relying on holomorphic projection, to study the coefficients of sesquiharmonic Maass forms and their relationship to arithmetic functions?

Yes, there are alternative approaches to studying the coefficients of sesquiharmonic Maass forms and their connections to arithmetic functions:
1. Poincaré Series:

Idea: Construct Poincaré series of sesquiharmonic type, analogous to those for harmonic Maass forms. These series would have coefficients directly related to the arithmetic functions of interest.
Challenge: Defining and working with sesquiharmonic Poincaré series is technically demanding. It requires a good understanding of the behavior of the differential operators involved under the action of the modular group.
2. Circle Method:

Idea: The circle method, a powerful tool in analytic number theory, can be adapted to study the asymptotic behavior of coefficients of modular forms.
Challenge: Applying the circle method to sesquiharmonic Maass forms would require overcoming technical hurdles related to their more complicated analytic behavior compared to holomorphic forms.
3. Shimura Lifts:

Idea: Shimura lifts provide a way to relate half-integral weight modular forms to integral weight forms. Exploring whether sesquiharmonic Maass forms have suitable lifts could offer insights into their coefficients.
Challenge: The existence and properties of such lifts for sesquiharmonic forms are not yet established and would require careful investigation.
4. Trace Formulas:

Idea: Trace formulas relate spectral data of automorphic forms to geometric information. Developing trace formulas specifically for sesquiharmonic Maass forms could shed light on their coefficients.
Challenge: Deriving trace formulas for these forms is a highly non-trivial task, often requiring sophisticated techniques from the spectral theory of automorphic representations.
5. p-adic Methods:

Idea: Explore p-adic analogs of sesquiharmonic Maass forms and their connections to p-adic L-functions. This could lead to new perspectives on their coefficients and arithmetic properties.
Challenge: Developing a robust theory of p-adic sesquiharmonic Maass forms is an open area with many challenges.
Key Takeaway: While holomorphic projection is a valuable tool, these alternative approaches offer complementary perspectives and could potentially reveal deeper connections between sesquiharmonic Maass forms and arithmetic.

What are the potential implications of these findings for other areas of mathematics where modular forms play a significant role, such as string theory or cryptography?

While the specific results in the paper focus on the interplay between sesquiharmonic Maass forms and certain arithmetic functions, the techniques and insights have the potential to impact other areas where modular forms are prominent:
String Theory:

Modular Forms and Black Hole Entropy: In string theory, modular forms often appear in the counting of black hole microstates. The study of sesquiharmonic Maass forms and their coefficients could lead to:

New connections between black hole entropy and more refined arithmetic invariants.
A deeper understanding of the modular properties of black hole partition functions.

Mirror Symmetry: Modular forms play a crucial role in mirror symmetry, a duality in string theory. The techniques developed for sesquiharmonic forms might:

Provide new tools for constructing and studying mirror pairs of Calabi-Yau manifolds.
Uncover novel relationships between the geometry of these manifolds and arithmetic objects.
Cryptography:

Lattice-Based Cryptography: Lattices constructed from modular forms have found applications in cryptography. The study of sesquiharmonic Maass forms could:

Lead to the discovery of new families of lattices with desirable cryptographic properties.
Provide insights into the security of existing lattice-based cryptosystems.

Elliptic Curve Cryptography: Modular forms are intimately connected to elliptic curves, which are widely used in cryptography. The techniques in the paper might:

Offer new methods for constructing elliptic curves with special properties relevant to cryptography.
Shed light on the distribution of rational points on elliptic curves, which is crucial for cryptographic applications.
Beyond String Theory and Cryptography:

Number Theory: The results have direct implications for number theory, particularly the study of:

Shifted convolution sums: The bounds obtained in the paper contribute to our understanding of these sums, which are central to many problems in analytic number theory.
Special values of L-functions: The connection between sesquiharmonic Maass forms and shifted convolution L-series suggests potential applications to studying the special values of these L-functions.
Key Takeaway: The techniques and insights from the study of sesquiharmonic Maass forms have the potential to enrich our understanding of modular forms and their applications in various areas of mathematics and theoretical physics. Further exploration of these connections is a promising avenue for future research.