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On the embedding problem for representations
Authors:Ariel Pacetti.
Affiliation:Departamento de Matemática, Universidad de Buenos Aires, Pabellón I, Ciudad Universitaria. C.P:1428, Buenos Aires, Argentina
Abstract:Let $ 2^+S_4$ denote the double cover of $ S_4$ corresponding to the element in $ operatorname{H}^2(S_4,mathbb{Z}/2mathbb{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}/2mathbb{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$.

Keywords:Galois representations   Shimura correspondence
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