Geometry-of-numbers methods in the cusp

TL;DR

This paper introduces novel geometric methods for counting integral orbits within cusps of fundamental domains, applied specifically to quadratic forms under split orthogonal group actions, advancing number theory techniques.

Contribution

It develops new geometry-of-numbers methods for counting integral orbits in cusps, with applications to quadratic forms and orthogonal group representations.

Findings

New counting techniques for integral orbits in cusps

Application to quadratic forms and orthogonal groups

Enhanced understanding of orbit distribution in fundamental domains

Abstract

In this article, we develop new methods for counting integral orbits having bounded invariants that lie inside the cusps of fundamental domains for coregular representations. We illustrate these methods for a representation of cardinal interest in number theory, namely that of the split orthogonal group acting on the space of quadratic forms.