On 12-congruences of elliptic curves

Abstract

We construct infinite families of pairs of (geometrically non-isogenous) elliptic curves defined over [Formula: see text] with [Formula: see text]-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 [Formula: see text]-torsion subgroups respects the Weil pairing. Our approach is to compute explicit birational models for the modular diagonal quotient surfaces which parametrize such pairs of elliptic curves. A key ingredient in the proof is to construct simple (algebraic) conditions for the [Formula: see text], [Formula: see text] or [Formula: see text]-torsion subgroups of a pair of elliptic curves to be isomorphic as Galois modules. These conditions are given in terms of the [Formula: see text]-invariants of the pair of elliptic curves.