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A new parametrization for ideal classes in rings defined by binary forms, and applications

Ashvin Swaminathan

Abstract

We give a parametrization of square roots of the ideal class of the inverse different of rings defined by binary forms in terms of the orbits of a coregular representation. This parametrization, which can be construed as a new integral model of a ``higher composition law'' discovered by Bhargava and generalized by Wood, was the missing ingredient needed to solve a range of previously intractable open problems concerning distributions of class groups, Selmer groups, and related objects. For instance, in this paper, we apply the parametrization to bound the average size of the $2$-class group in families of number fields defined by binary $n$-ic forms, where $n \geq 3$ is an arbitrary integer, odd or even; in the paper [41], we applied it to prove that most integral odd-degree binary forms fail to primitively represent a square; and in the paper [11], joint with Bhargava and Shankar, we applied it to bound the second moment of the size of the $2$-Selmer group of elliptic curves.

A new parametrization for ideal classes in rings defined by binary forms, and applications

Abstract

We give a parametrization of square roots of the ideal class of the inverse different of rings defined by binary forms in terms of the orbits of a coregular representation. This parametrization, which can be construed as a new integral model of a ``higher composition law'' discovered by Bhargava and generalized by Wood, was the missing ingredient needed to solve a range of previously intractable open problems concerning distributions of class groups, Selmer groups, and related objects. For instance, in this paper, we apply the parametrization to bound the average size of the -class group in families of number fields defined by binary -ic forms, where is an arbitrary integer, odd or even; in the paper [41], we applied it to prove that most integral odd-degree binary forms fail to primitively represent a square; and in the paper [11], joint with Bhargava and Shankar, we applied it to bound the second moment of the size of the -Selmer group of elliptic curves.

Paper Structure

This paper contains 41 sections, 37 theorems, 120 equations, 1 table.

Table of Contents

  1. Introduction
  2. The parametrization
  3. Relation to earlier work
  4. Summary of applications
  5. Impact of $f_0$-monogenicity on $2$-class group distributions
  6. Solubility of superelliptic equations
  7. $2$-Selmer groups of hyperelliptic Jacobians
  8. Main results on class group distributions
  9. Notation and setup
  10. Statements of results, part $(i)$$:$ odd degree
  11. Statements of results, part $(ii)$$:$ even degree
  12. Comparison with class group heuristics
  13. Method of proof
  14. Orbit parametrization
  15. Rings associated to binary forms
  16. ...and 26 more sections

Key Result

Theorem 1

Square roots of the class of the inverse different of $R_F$ naturally give rise to $G_n(\mathbb Z)$-orbits of pairs $(A,B) \in \mathbb Z^2 \otimes_\mathbb Z \operatorname{Sym}_2 \mathbb Z^n$ of symmetric $n \times n$ integer matrices such that where $g \in G_n(\mathbb Z)$ acts on $(A,B)$ via $g \cdot (A,B) = (g A g^T, g B g^T)$. The locus of pairs $(A,B)$ that arise in this way is cut out of t

Theorems & Definitions (78)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Corollary 7
  • Conjecture 8: Cohen--Lenstra--Martinet--Malle
  • Conjecture 9
  • Theorem 10
  • ...and 68 more