Nagell-Lutz Theorem for Imaginary Quadratic Fields and class groups of quadratic fields

TL;DR

This work extends the Nagell-Lutz theorem to imaginary quadratic fields with class number one, proving that for elliptic curves $E: y^2=x^3+Ax+B$ with $A,B$ in the ring of integers $\mathcal{O}_K$ of $K=\mathbb{Q}(\sqrt{D})$, any torsion point of finite order which is not $2$-torsion defined over $K$ has coordinates in the base ring $\mathcal{O}_K$ (and, in many discriminant cases, in $\mathbb{Z}[\sqrt{D}]$). The authors develop a normalization to ensure coefficients lie in $\mathcal{O}_K$, perform a $p$-adic denominator analysis using an $E_r$-filtration and a transform to a curve $E'$, and derive structural constraints that force the coordinates into the desired ring. Building on this extended Nagell-Lutz, they construct connections between $2$-torsion on certain curves over $\mathbb{Q}(\sqrt{-d})$ and the class groups of quadratic fields $\mathbb{Q}(\sqrt{-p^3+4mp})$ for infinitely many primes $p$, yielding $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}$ subgroups in those class groups. The paper also analyzes torsion for a family of curves over $\mathbb{Q}(i)$, showing a baseline $\mathbb{Z}/6\mathbb{Z}$ torsion structure and situating it within known bounds, thereby illustrating the broader arithmetic impact of the extended theorem. Overall, the results connect elliptic curve torsion phenomena to explicit class-group constructions via norm-based and Soleng-type techniques, with potential applications to producing nontrivial class group elements in families of quadratic fields.

Abstract

We prove the Nagell-Lutz theorem for the imaginary quadratic fields of class number one.

Theorems & Definitions (44)

• Remark 1
• Theorem 1
• Theorem 2
• Definition
• Remark 2
• Lemma 1
• proof
• Remark 3
• Lemma 2
• proof