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Lifting cusp forms to Maass forms with an application to partitions.

Kathrin Bringmann¹, Ken Ono

¹Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA.

Proceedings of the National Academy of Sciences of the United States of America

|March 16, 2007

Summary

Researchers define lifts of Poincaré series to harmonic weak Maass forms, explaining Ramanujan

Area of Science:

Number Theory
Harmonic Analysis
Algebraic Geometry

Background:

Ramanujan's mock theta functions remain a significant area of study in number theory.
The properties of cuspidal Poincaré series and their connections to modular forms are well-established.
The relationship between different types of modular forms, such as cusp forms and Maass forms, is a key area of research.

Purpose of the Study:

To define lifts of cuspidal Poincaré series to weight 2-k harmonic weak Maass forms.
To provide a general framework that explains Ramanujan's mock theta functions.
To explore the application of these lifts to number theory, specifically partition functions.

Main Methods:

Construction of lifts from S(k)(Gamma(0)(N)) to weight 2-k harmonic weak Maass forms.

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Utilizing the framework to analyze Ramanujan's mock theta functions.

Applying the theory of Maass forms and singular moduli to partition theory.

Main Results:

A general framework is established for lifting cuspidal Poincaré series to harmonic weak Maass forms.
This construction provides an explanation for Ramanujan's mock theta functions, answering a question posed by Dyson.
The number of partitions of an integer n is shown to be the trace of singular moduli of a specific Maass form.

Conclusions:

The study successfully defines lifts of Poincaré series, offering a new perspective on mock theta functions.
The connection between modular forms and partition theory is further elucidated through this construction.
The results have implications for understanding deep connections between number theory and algebraic geometry (Calabi-Yau threefolds).