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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
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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