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p-Adic coupling of mock modular forms and shadows.

Pavel Guerzhoy1, Zachary A Kent, Ken Ono

  • 1Department of Mathematics, University of Hawaii, Honolulu, HI 96822-2273, USA.

Proceedings of the National Academy of Sciences of the United States of America
|March 24, 2010
PubMed
Summary
This summary is machine-generated.

Researchers developed a p-adic method to connect mock modular forms and their cusp form shadows, specifically for integer weight newforms. This advances understanding of harmonic Maass forms and their arithmetic-geometric mean applications.

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Area of Science:

  • Number Theory
  • Algebraic Geometry

Background:

  • Mock modular forms are the holomorphic parts of harmonic Maass forms.
  • The nonholomorphic part relates to period integrals of cusp form "shadows."
  • Direct methods for coefficient relations between shadows and mock modular forms are lacking.

Purpose of the Study:

  • To establish a direct method for relating coefficients of mock modular forms and their shadows.
  • To solve this problem specifically when the shadow is an integer weight newform.
  • To introduce and utilize the concept of algebraic "regularized mock modular forms."

Main Methods:

  • Development of a p-adic method.
  • Definition of algebraic "regularized mock modular forms."
  • Application to the modular solution of the cubic arithmetic-geometric mean.

Main Results:

  • A novel p-adic method is presented for relating mock modular forms and cusp form shadows.
  • The method is effective for integer weight newform shadows.
  • The approach provides insights into the structure of harmonic Maass forms.

Conclusions:

  • The study provides a breakthrough in understanding the relationship between mock modular forms and their shadows.
  • The developed p-adic method offers a new tool for number theoretic investigations.
  • Applications include advancements in computing the cubic arithmetic-geometric mean.