Papers › On $12$-congruences of elliptic curves
On $12$-congruences of elliptic curves
Sam Frengley
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We construct infinite families of pairs of (geometrically non-isogenous) elliptic curves defined over 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.
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