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Geometry-of-numbers methods in the cusp

Arul Shankar, Artane Siad, Ashvin Swaminathan, Ila Varma

TL;DR

The paper introduces two systematic geometry-of-numbers methods to count reducible integral orbits in cusp regions for coregular representations, with a detailed instantiation for the split orthogonal action on quadratic forms. Method I directly analyzes cusp volumes via box-shaped fundamental domains and slicing, producing weighted-volume constants that contribute to the global asymptotics; Method II reduces to a non-reductive parabolic action on a reducible hyperplane and leverages a strong local-to-global principle to assemble p-adic data into a global count. The main results give precise asymptotics for reducible G({\mathbb Z})-orbits on W({\mathbb Z}) and on big families, including explicit finite- and infinite-place constants, and a local-to-global principle for the action of the parabolic subgroup on W_0. These techniques extend the arithmetic-statistics toolkit for reducible orbits, enabling exact asymptotics in infinite families and offering local-global factorization formulas that have potential applications to diverse coregular representations and related counting problems in number theory.

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.

Geometry-of-numbers methods in the cusp

TL;DR

The paper introduces two systematic geometry-of-numbers methods to count reducible integral orbits in cusp regions for coregular representations, with a detailed instantiation for the split orthogonal action on quadratic forms. Method I directly analyzes cusp volumes via box-shaped fundamental domains and slicing, producing weighted-volume constants that contribute to the global asymptotics; Method II reduces to a non-reductive parabolic action on a reducible hyperplane and leverages a strong local-to-global principle to assemble p-adic data into a global count. The main results give precise asymptotics for reducible G({\mathbb Z})-orbits on W({\mathbb Z}) and on big families, including explicit finite- and infinite-place constants, and a local-to-global principle for the action of the parabolic subgroup on W_0. These techniques extend the arithmetic-statistics toolkit for reducible orbits, enabling exact asymptotics in infinite families and offering local-global factorization formulas that have potential applications to diverse coregular representations and related counting problems in number theory.

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.

Paper Structure

This paper contains 39 sections, 35 theorems, 104 equations.

Table of Contents

  1. Introduction
  2. Background on orbit-counting, and relation to earlier work
  3. Notation and setup
  4. Statements of main theorems
  5. Methods of proof
  6. Other applications of our methods
  7. Counting reducible integral orbits on binary quartic forms
  8. Reduction theory for the action of $\mathrm{PGL}_2({\mathbb Z})$ on $V({\mathbb R})$
  9. A box-shaped fundamental domain for $\operatorname{PGL}_2({\mathbb Z}) \curvearrowright \operatorname{PGL}_2({\mathbb R})$
  10. Fundamental sets for $\operatorname{PGL}_2({\mathbb R}) \curvearrowright V({\mathbb R})$ and $\operatorname{PGL}_2({\mathbb Z}) \curvearrowright V({\mathbb R})$
  11. The action of the subgroup $P$ on the reducible hyperplane $V_0$
  12. Reduction theory for the action of $P$ on $V_0$
  13. A Jacobian change-of-variables formula
  14. Counting reducible $\operatorname{PGL}_2({\mathbb Z})$-orbits on $V({\mathbb Z})$
  15. Averaging over fundamental domains
  16. ...and 24 more sections

Key Result

Theorem 1

The number of reducible $G({\mathbb Z})$-orbits on $W({\mathbb Z})^{(r)}$ having height up to $X$ is given by

Theorems & Definitions (63)

  • Theorem 1
  • Remark
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Remark
  • Theorem 6
  • Remark
  • Theorem 7
  • Theorem 8
  • ...and 53 more