On the embedding problem for 2^+S_4 representations

• Published in: Mathematics of computation 2007-10, Vol.76 (260), p.2063-2076
• Main Author: Pacetti, Ariel
• Format: Article
• Language: English
• ISSN: 0025-5718; 1088-6842
• DOI: 10.1090/S0025-5718-07-01940-0

Let 2^+S_4 denote the double cover of S_4 corresponding to the element in \operatorname{H}^2(S_4,\mathbb Z/2\mathbb Z) 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 \operatorname{H}^1(\operatorname{Gal}_{\mathbb {Q}},E[2])\backslash { 0 \} correspond to Galois extensions N of \mathbb {Q} with Galois group (isomorphic to) S_4. 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 \tilde N with \operatorname{Gal}(\tilde N/ \Q) \simeq 2^+S_4 gives a homomorphism s_4^+:\operatorname{H}^1(\operatorname{Gal}_{\mathbb{Q}},E[2]) \rightarrow \operatorname{H}^2(\operatorname{Gal}_\mathbb {Q}, \mathbb{Z}/2\mathbb {Z}). 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 \mathbb {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.