Odd quadratic orders and real $j$-invariants

TL;DR

This paper explicitly describes the 2-torsion subgroup of the class group of odd discriminant orders in imaginary quadratic fields, revealing its size depends on the number of prime divisors of the discriminant.

Contribution

It provides an explicit description of the 2-torsion subgroup of class groups for orders with odd discriminant, including a formula for its size.

Findings

The order of the 2-torsion subgroup is 2^{s_D-1}.

The size of Cl(O)[2] depends on the number of prime divisors of D.

Explicit structure of Cl(O)[2] for odd discriminant orders.

Abstract

Let $O$ be an order of odd discriminant $D$ in an imaginary quadratic field $K$. Let $Cl(O)$ be the group of proper $O$-ideals and $Cl(O)[2]$ the kernel of multiplication by $2$ in $Cl(O)$. We describe explicitly the group $Cl(O)[2]$. In particular, we prove that its order is $2^{s_D-1}$ where $s_D$ is the number of prime divisors of $D$.