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On the embedding problem for $2^+S_4$ representations

2007
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Mathematics of Computation
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Let 2 + S 4 denote the double cover of S 4 corresponding to the element in H 2 (S 4 , Z/2Z) where transpositions lift to elements of order 2 and the product of two disjoint transpositions to elements of order 4. Given an elliptic curve E, let E[2] denote its 2-torsion points. Under some conditions on E elements in H 1 (Gal Q , E[2])\{0} correspond to Galois extensions N of Q with Galois group (isomorphic to) S 4 . In this work we give an interpretation of the addition law on such fields, and

doi:10.1090/s0025-5718-07-01940-0
fatcat:j3s25fdzbbcldepiawyrjb2ho4