On the embedding problem for 2+s4 representations

Abstract

Let 2+S4 denote the double cover of S4 corresponding to the element in H2(S4, 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 H1(GalQ, E[2])\{0} correspond to Galois extensions N of Q with Galois group (isomorphic to) S4. In this work we give an interpretation of the addition law on such fields, and prove that the obstruction for N having a Galois extension N˜ with Gal(N/˜ Q) 2+S4 gives a homomorphism s+ 4 : H1(GalQ, E[2]) → H2(GalQ, Z/2Z). 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.