Solvability of Galois embedding problems arising from 3-torsion of elliptic curves over Q is equivalent to the existence of infinitely many elliptic curves with the corresponding 3-division fields, for all possible images of the mod 3 representation.
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On Galois Embedding Problems Arising from 3-Torsion of Elliptic Curves
Solvability of Galois embedding problems arising from 3-torsion of elliptic curves over Q is equivalent to the existence of infinitely many elliptic curves with the corresponding 3-division fields, for all possible images of the mod 3 representation.