[Paper Review] Average $2$-Torsion in Class Groups of Rings Associated to Binary $n$-ic Forms

Ashvin Swaminathan|arXiv (Cornell University)|Nov 27, 2020

Algebraic Geometry and Number Theory39 references4 citations

TL;DR

This paper computes upper bounds on the average size of the 2-torsion in class groups of rings and fields defined by integral binary $n$-ic forms with a fixed odd leading coefficient, extending prior work on cubic and monic forms. Conditional on a uniformity estimate, these bounds are shown to be exact, revealing that fixing the leading coefficient increases 2-torsion when $n$ is odd, and providing the first such results for $p$-torsion with $p \mid n > 2$ when $n$ is even.

ABSTRACT

Let $n \geq 3$ be an integer. In this paper, we study the average behavior of the $2$-torsion in class groups of rings cut out by integral binary $n$-ic forms having any fixed odd leading coefficient. Specifically, we compute upper bounds on the average size of the $2$-torsion in class groups of rings and fields arising from such binary forms. Conditional on a uniformity estimate, we further prove that each of these upper bounds is in fact an equality. Our theorems extend recent work of Bhargava-Hanke-Shankar in the cubic case and of Siad in the monic case to binary forms of any degree with any fixed odd leading coefficient. When $n$ is odd, we find that fixing the leading coefficient increases the average $2$-torsion in the class group, relative to the prediction of Cohen-Lenstra-Martinet-Malle. When $n$ is even, such predictions are yet to be formulated; together with Siad's results in the monic case, our theorems are the first of their kind to describe the average behavior of the $p$-torsion in class groups of degree-$n$ rings where $p \mid n > 2$. To prove these theorems, we first answer a question of Ellenberg by parametrizing square roots of the class of the inverse different of a ring cut out by a binary form in terms of the integral orbits of a certain coregular representation. This parametrization has a range of interesting applications, from studying $2$-parts of class groups to studying $2$-Selmer groups of hyperelliptic Jacobians.

Motivation & Objective

• To determine the average size of the 2-torsion in class groups of rings and fields arising from integral binary $n$-ic forms with a fixed odd leading coefficient.
• To extend prior results on cubic and monic forms to arbitrary $n \geq 3$ and any fixed odd leading coefficient.
• To investigate how fixing the leading coefficient affects the average 2-torsion, particularly in comparison to Cohen-Lenstra-Martinet-Malle heuristics.
• To provide the first systematic description of $p$-torsion behavior in degree-$n$ rings when $p \mid n > 2$ and $n$ is even.
• To answer a question of Ellenberg by parametrizing square roots of the inverse different class via integral orbits of a coregular representation.

Proposed method

• Parametrize square roots of the inverse different class of a ring cut out by a binary $n$-ic form using integral orbits of a coregular representation.
• Use this parametrization to analyze the structure of the 2-torsion in the class group of the associated ring.
• Apply a uniformity estimate to prove that the derived upper bounds on average 2-torsion size are sharp, i.e., equalities.
• Generalize techniques from Bhargava-Hanke-Shankar (cubic case) and Siad (monic case) to the non-monic setting with fixed odd leading coefficient.
• Leverage the parametrization to connect class group 2-torsion to arithmetic invariants of hyperelliptic Jacobians, particularly 2-Selmer groups.
• Establish that the average 2-torsion size depends on the leading coefficient, especially when $n$ is odd, contrasting with Cohen-Lenstra-Martinet-Malle predictions.

Experimental results

Research questions

• RQ1 How does fixing the leading coefficient of a binary $n$-ic form affect the average size of the 2-torsion in the class group of the associated ring?
• RQ2 What are the upper bounds on the average size of the 2-torsion in class groups of rings defined by integral binary $n$-ic forms with a fixed odd leading coefficient?
• RQ3 Under what conditions can these upper bounds be proven to be exact?
• RQ4 How does the behavior of 2-torsion differ when $n$ is odd versus even, particularly in light of existing heuristics?
• RQ5 Can the square roots of the inverse different class be parametrized via integral orbits of a coregular representation, and what are the arithmetic consequences?

Key findings

• The paper establishes upper bounds on the average size of the 2-torsion in class groups of rings arising from integral binary $n$-ic forms with a fixed odd leading coefficient.
• Conditional on a uniformity estimate, these upper bounds are proven to be exact, yielding precise asymptotic averages.
• When $n$ is odd, fixing the leading coefficient increases the average 2-torsion relative to the Cohen-Lenstra-Martinet-Malle predictions.
• For even $n$, the results provide the first known description of the average $p$-torsion in class groups of degree-$n$ rings when $p \mid n > 2$.
• The parametrization of square roots of the inverse different class via integral orbits of a coregular representation is established as a key technical tool.
• This parametrization enables applications beyond class groups, including the study of 2-Selmer groups of hyperelliptic Jacobians.