Completing the classification of torsion subgroups for rational elliptic curves over sextic fields

TL;DR

This paper completes the classification of possible torsion subgroups for rational elliptic curves over sextic fields by proving that the previously uncertain group $C_3 igoplus C_{18}$ cannot occur, using Galois representations and modular curve analysis.

Contribution

It proves the non-existence of the torsion subgroup $C_3 igoplus C_{18}$ for elliptic curves over sextic fields, completing the classification.

Findings

$C_3 igoplus C_{18}$ torsion subgroup does not occur.

The classification of torsion subgroups over sextic fields is now complete.

Galois representation constraints translate into modular curve rational point analysis.

Abstract

We complete the classification of torsion subgroups $E(K)_{\text{tors}}$ that can occur for an elliptic curve $E/\mathbb{Q}$ over a sextic number field $K$. Previous work determined the complete set of these groups, leaving the existence of only one group in question: $C_3 \oplus C_{18}$. We prove that this group does not occur. Our proof relies on the theory of Galois representations attached to elliptic curves. The assumed existence of a $C_3 \oplus C_{18}$ torsion subgroup would impose strong, simultaneous constraints on the mod-$2$ and $3$-adic Galois representations of the curve. By applying the recent classification of $\ell$-adic Galois images for elliptic curves over $\mathbb{Q}$, we translate these arithmetic constraints into a problem of Diophantine geometry: the $j$-invariant of such a curve must correspond to a rational point on one of the finitely many modular curves. We then analyze these curves using classical methods and show that none have the necessary rational points corresponding to elliptic curves without complex multiplication, thereby proving our main result.