Geometry-of-numbers methods in the cusp

Shankar, Arul; Siad, Artane; Swaminathan, Ashvin A.; Varma, Ila

doi:10.2140/ant.2025.19.1099

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Geometry-of-numbers methods in the cusp

Arul Shankar, Artane Siad, Ashvin A. Swaminathan and Ila Varma

Vol. 19 (2025), No. 6, 1099–1145

DOI: 10.2140/ant.2025.19.1099

Abstract

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.

Keywords

geometry-of-numbers, coregular representations, fundamental domain, cusps, irreducibility

Mathematical Subject Classification

Primary: 11R29, 11R45

Secondary: 11H55, 11E76

Milestones

Received: 24 August 2022

Revised: 30 July 2024

Accepted: 3 September 2024

Published: 14 May 2025

Authors

Arul Shankar
Department of Mathematics University of Toronto Toronto, ON Canada

Artane Siad
Department of Mathematics Princeton University Princeton, NJ United States	School of Mathematics Institute for Advanced Study Princeton, NJ United States

Ashvin A. Swaminathan
Department of Mathematics Harvard University Cambridge, MA United States

Ila Varma
Department of Mathematics University of Toronto Toronto, ON Canada

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Vol. 19, No. 6, 2025