Definite orthogonal modular forms: Computations, Excursions and Discoveries

TL;DR

This paper explores the computation and properties of definite orthogonal modular forms, revealing new connections with classical modular forms, endoscopy, and Eisenstein congruences through algorithms, examples, and conjectures.

Contribution

It introduces algorithms for computing definite orthogonal modular forms, investigates endoscopy via theta series, and establishes new instances of Eisenstein congruences.

Findings

Expressed counts of Kneser neighbours in terms of classical modular form coefficients

Proved new Eisenstein congruences of Ramanujan and Kurokawa-Mizumoto types

Provided numerous examples and posed several conjectures

Abstract

We consider spaces of modular forms attached to definite orthogonal groups of low even rank and nontrivial level, equipped with Hecke operators defined by Kneser neighbours. After reviewing algorithms to compute with these spaces, we investigate endoscopy using theta series and a theorem of Rallis. Along the way, we exhibit many examples and pose several conjectures. As a first application, we express counts of Kneser neighbours in terms of coefficients of classical or Siegel modular forms, complementing work of Chenevier-Lannes. As a second application, we prove new instances of Eisenstein congruences of Ramanujan and Kurokawa-Mizumoto type.