Torsion of elliptic curves with rational $j$-invariant over quartic number fields

TL;DR

The paper classifies all possible torsion subgroups $E(K)_{\mathrm{tors}}$ for elliptic curves $E$ defined over quartic number fields $K$ with rational $j$-invariant, showing that only the groups listed in $\Phi_j^{\mathbb{Q}}(4)$ can occur. It combines quadratic-twist techniques for curves with $j(E)\in\mathbb{Q}\setminus\{0,1728\}$, Galois-theoretic analysis of quartic and quartic-closure extensions, and the known $\Phi(4)$ classification (Derickx–Najman) to eliminate several candidate torsion structures, notably excluding $\mathbb{Z}/N\mathbb{Z}$ with $N\in\{11,14,18,22\}$, $\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/2N\mathbb{Z}$ with $N\in\{7,9\}$, and $\mathbb{Z}/3\mathbb{Z}\oplus\mathbb{Z}/9\mathbb{Z}$. The authors then exhibit explicit curves realizing each permitted structure, including nontrivial cases such as $\mathbb{Z}/17\mathbb{Z}$ and $\mathbb{Z}/21\mathbb{Z}$ in quartic fields, thereby completing the classification for this setting. The work extends the framework of $\Phi(d)$ classifications to the special case of rational $j$-invariants and provides concrete examples for computational and theoretical applications.

Abstract

Let $E$ be an elliptic curve, defined over a quartic extension $K$ of $\mathbb{Q}$, with $j(E) \in \mathbb{Q}$. In this paper, we classify the possible torsion subgroup structures $E(K)_{\text{tors}}$.

Theorems & Definitions (7)

• proof
• proof : Proof of Lemma \ref{['quartic_quadratic_cor 2']}
• proof
• proof
• proof
• proof
• proof : Proof of Lemma \ref{['9-torsion']}