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

Ariel Pacetti
2007 Mathematics of Computation  
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
more » ... such fields, and prove that the obstruction for N having a Galois extensionÑ with Gal(Ñ/Q) 2 + S 4 gives a homomorphism s + 4 : As a corollary we can prove (if E has conductor divisible by few primes and high rank) the existence of 2-dimensional representations of the absolute Galois group of Q attached to E and use them in some examples to construct 3/2 modular forms mapping via the Shimura map to (the modular form of weight 2 attached to) E.
doi:10.1090/s0025-5718-07-01940-0 fatcat:j3s25fdzbbcldepiawyrjb2ho4