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Flipping operators and locally harmonic Maass forms
Kathrin Bringmann1, Andreas Mono2, Larry Rolen2
1Department of Mathematics and Computer Science, University of Cologne, Weyertal 86-90, 50931 Cologne, Germany.
The Ramanujan Journal
|August 11, 2025
Summary
The flipping operator acts similarly on locally harmonic Maass forms as it does on harmonic Maass forms. This study extends the understanding of the flipping operator
Area of Science:
- Number Theory
- Automorphic Forms
Background:
- Harmonic Maass forms of manageable growth are crucial in number theory.
- Key operators include the Bol, shadow, and flipping operators, each controlling specific aspects of these forms.
- Maass-Poincaré series provide a basis for negative weight harmonic Maass forms.
Purpose of the Study:
- To investigate the behavior of the flipping operator on locally harmonic Maass forms.
- To establish a similar property for the flipping operator on these forms as observed for harmonic Maass forms.
- To connect the properties of the flipping operator across different types of Maass forms.
Main Methods:
- Analysis of differential operators (Bol, shadow, flipping) in the context of harmonic Maass forms.
- Study of Maass-Poincaré series as a basis for negative weight harmonic Maass forms.
- Extension of these concepts to locally harmonic Maass forms and their relation to Poincaré series of hyperbolic type.
Main Results:
- The flipping operator's action on locally harmonic Maass forms is shown to be analogous to its action on harmonic Maass forms.
- A similar property is proven for the flipping operator applied to Poincaré series of hyperbolic type.
- The study confirms the interplay between the flipping operator and these advanced number theoretic objects.
Conclusions:
- The flipping operator plays a consistent role in manipulating components of both harmonic and locally harmonic Maass forms.
- This research deepens the understanding of the structure and transformations within the theory of Maass forms.
- The findings contribute to the ongoing study of Shimura and Shintani lifts and related automorphic forms.
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