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Definite orthogonal modular forms: computations, excursions, and discoveries.

Eran Assaf1, Dan Fretwell2, Colin Ingalls3

  • 1Department of Mathematics, Dartmouth College, 6188 Kemeny Hall, Hanover, NH 03755 USA.

Research in Number Theory
|October 25, 2022
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Summary

This study explores modular forms for orthogonal groups, using Kneser neighbours and theta series to investigate endoscopy. New results connect Kneser neighbour counts to modular form coefficients and prove Eisenstein congruences.

Keywords:
LatticesModular formsOrthogonal groupsQuadratic formsTheta series

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Area of Science:

  • Number Theory
  • Representation Theory
  • Algebraic Geometry

Background:

  • Modular forms are central objects in number theory, connecting to various fields.
  • Orthogonal groups and their associated modular forms are rich areas of study.
  • Understanding Hecke operators and their action is crucial for analyzing these spaces.

Purpose of the Study:

  • To investigate modular forms attached to definite orthogonal groups of low even rank and nontrivial level.
  • To explore the concept of endoscopy using theta series and Rallis' theorem.
  • To establish connections between Kneser neighbour counts and coefficients of classical/Siegel modular forms.

Main Methods:

  • Algorithms for computing spaces of modular forms.
  • Endoscopy theory applied to orthogonal groups.
  • Utilizing theta series and Rallis' theorem for theoretical investigation.
  • Explicit examples and computational approaches.

Main Results:

  • New expressions for counts of Kneser neighbours in terms of modular form coefficients.
  • Proof of new instances of Eisenstein congruences of Ramanujan and Kurokawa-Mizumoto type.
  • Development of computational tools and algorithms for these spaces.

Conclusions:

  • The study provides novel insights into the structure and properties of modular forms for orthogonal groups.
  • Connections are established between different areas of number theory, including modular forms, group theory, and L-functions.
  • The results open avenues for further research into automorphic forms and related conjectures.