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Counting integral points on symmetric varieties with applications to arithmetic statistics

Arul Shankar, Artane Siad, Ashvin A. Swaminathan

TL;DR

The paper develops a novel framework that blends Bhargava’s geometry-of-numbers counting with dynamical counting techniques to enumerate integral points on symmetric varieties in fundamental domains, enabling precise averages for arithmetic invariants. It provides upper bounds and conjectural equalities for the average sizes of the $2$-torsion in class groups of cubic fields and the $2$-Selmer groups of elliptic curves within various thin (unit-monogenised and wider) families, revealing violations of Cohen–Lenstra–Martinet predictions under global unit-monogenisation and other local conditions. The analysis hinges on orbit parametrisations by ${ m SL}_3$ acting on pairs of ternary quadratic forms, the treatment of cusps via $ riangle$-distinguished orbits, and the computation of local masses that feed the global averages. The results illuminate how global constraints and local ramification data shape arithmetic statistics in families of number fields and elliptic curves, and they establish a robust, transferable machinery for future investigations in arithmetic statistics of symmetric varieties.

Abstract

In this article, we combine Bhargava's geometry-of-numbers methods with the dynamical point-counting methods of Eskin--McMullen and Benoist--Oh to develop a new technique for counting integral points on symmetric varieties lying within fundamental domains for coregular representations. As applications, we study the distribution of the $2$-torsion subgroup of the class group in thin families of cubic number fields, as well as the distribution of the $2$-Selmer groups in thin families of elliptic curves over $\mathbb{Q}$. For example, our results suggest that the existence of a generator of the ring of integers with small norm has an increasing effect on the average size of the $2$-torsion subgroup of the class group, relative to the Cohen--Lenstra predictions.

Counting integral points on symmetric varieties with applications to arithmetic statistics

TL;DR

The paper develops a novel framework that blends Bhargava’s geometry-of-numbers counting with dynamical counting techniques to enumerate integral points on symmetric varieties in fundamental domains, enabling precise averages for arithmetic invariants. It provides upper bounds and conjectural equalities for the average sizes of the -torsion in class groups of cubic fields and the -Selmer groups of elliptic curves within various thin (unit-monogenised and wider) families, revealing violations of Cohen–Lenstra–Martinet predictions under global unit-monogenisation and other local conditions. The analysis hinges on orbit parametrisations by acting on pairs of ternary quadratic forms, the treatment of cusps via -distinguished orbits, and the computation of local masses that feed the global averages. The results illuminate how global constraints and local ramification data shape arithmetic statistics in families of number fields and elliptic curves, and they establish a robust, transferable machinery for future investigations in arithmetic statistics of symmetric varieties.

Abstract

In this article, we combine Bhargava's geometry-of-numbers methods with the dynamical point-counting methods of Eskin--McMullen and Benoist--Oh to develop a new technique for counting integral points on symmetric varieties lying within fundamental domains for coregular representations. As applications, we study the distribution of the -torsion subgroup of the class group in thin families of cubic number fields, as well as the distribution of the -Selmer groups in thin families of elliptic curves over . For example, our results suggest that the existence of a generator of the ring of integers with small norm has an increasing effect on the average size of the -torsion subgroup of the class group, relative to the Cohen--Lenstra predictions.
Paper Structure (32 sections, 48 theorems, 83 equations, 3 tables)

This paper contains 32 sections, 48 theorems, 83 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Results on unit-monogenised cubic fields
  3. Results on more general thin families of cubic fields
  4. Partitioning into stable families
  5. Results on $2$-Selmer groups of elliptic curves
  6. Discussion of the proof
  7. Orbit parametrisations
  8. The representations $2 \otimes \operatorname{Sym}_2(3)$ and $2 \otimes \operatorname{Sym}^2(3)$ of $\operatorname{SL}_3$
  9. Parametrisation of $2$-torsion subgroups of the class groups of cubic fields
  10. Parametrisation of $2$-Selmer groups of elliptic curves
  11. Distinguished and $\Delta$-distinguished orbits
  12. Counting integral points on symmetric varieties
  13. Notation and setup
  14. Main counting results
  15. Congruence conditions
  16. ...and 17 more sections

Key Result

Theorem 1

When totally real unit-monogenised cubic fields are ordered by balanced height, the average size of the $2$-torsion subgroups of their class groups is at most $2 + 3/56$.

Theorems & Definitions (55)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Remark 6
  • Remark 7
  • Theorem 8
  • Theorem 9
  • Theorem 2.1
  • ...and 45 more