Definite orthogonal modular forms: Computations, Excursions and Discoveries
Number Theory
2022-06-07 v2
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.
Cite
@article{arxiv.2203.06405,
title = {Definite orthogonal modular forms: Computations, Excursions and Discoveries},
author = {Eran Assaf and Dan Fretwell and Colin Ingalls and Adam Logan and Spencer Secord and John Voight},
journal= {arXiv preprint arXiv:2203.06405},
year = {2022}
}