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Flipping operators and locally harmonic Maass forms.

Kathrin Bringmann1, Andreas Mono2, Larry Rolen2

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Summary
This summary is machine-generated.

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

Keywords:
Eisenstein seriesFlipping operatorHarmonic maass formsIntegral binary quadratic formsModular formsPoincaré series

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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.