On the embedding problem for $2^{+}{S}_4$ representations

Abstract

Let $2^{+}{S}_4$ denote the double cover of $S_4$ corresponding to the element in $H^2(S_4,\mathbb{Z}/2\mathbb{Z})$ where transpositions lift to elements of order 2 and the product of two disjoint transpositions to elements of order 4 (denoted ${\tilde{S}}_4$ in \cite{Serre}). Given an elliptic curve $E$, let $E[2]$ denote its 2-torsion points. Under some conditions on $E$ (as in \cite{Bayer}) elements in $H^1(\Gal_\Q,E[2])\backslash \{0 \}$ correspond to Galois extensions $N$ of $\Q$ with Galois group (isomorphic to) $S_4$. On 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 $\Gal(\tilde N/ \Q) \simeq 2^+S_4$ gives an homomorphism $s_4^+:H^1(\Gal_\Q,E[2]) \to H^2(\Gal_\Q,\Z/2\Z)$. As a Corollary we can prove (if $E$ has conductor divisible by few primes and high rank) the existence of 1$-dimensionalrepresentationsattachedto$E$andusetheminsomeexamplestoconstruct3/2modularformsmappingviatheShimuramapto(themodularformattachedto)$E$.

11 pages