Definite orthogonal modular forms: computations, excursions, and discoveries

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.