On $12$-congruences of elliptic curves

TL;DR

This paper constructs infinite families of elliptic curve pairs over Q with isomorphic 12-torsion Galois modules, extending prior work by relaxing Weil pairing assumptions through explicit models and algebraic conditions based on j-invariants.

Contribution

It introduces new explicit models for modular surfaces parametrizing such elliptic pairs without Weil pairing restrictions, and provides algebraic criteria for torsion subgroup isomorphisms.

Findings

Constructed infinite families of elliptic curves with isomorphic 12-torsion Galois modules.

Derived explicit birational models for the relevant modular diagonal quotient surfaces.

Established algebraic conditions based on j-invariants for torsion subgroup isomorphisms.

Abstract

We construct infinite families of pairs of (geometrically non-isogenous) elliptic curves defined over $\mathbb{Q}$ with $12$-torsion subgroups that are isomorphic as Galois modules. This extends previous work of Chen and Fisher where it is assumed that the underlying isomorphism of $12$-torsion subgroups respects the Weil pairing. Our approach is to compute explicit birational models for the modular diagonal quotient surfaces which parametrise such pairs of elliptic curves. A key ingredient in the proof is to construct simple (algebraic) conditions for the $2$, $3$, or $4$-torsion subgroups of a pair of elliptic curves to be isomorphic as Galois modules. These conditions are given in terms of the $j$-invariants of the pair of elliptic curves.