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Mathematics of Computation

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

Author: Ariel Pacetti
Journal: Math. Comp. 76 (2007), 2063-2075
MSC (2000): Primary 11F80; Secondary 11F37
Published electronically: April 24, 2007
MathSciNet review: 2336282
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Abstract: 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$.

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Ariel Pacetti
Affiliation: Departamento de Matemática, Universidad de Buenos Aires, Pabellón I, Ciudad Universitaria. C.P:1428, Buenos Aires, Argentina

Keywords: Galois representations, Shimura correspondence
Received by editor(s): July 14, 2005
Received by editor(s) in revised form: March 11, 2006
Published electronically: April 24, 2007
Additional Notes: The author was supported by a CONICET grant
The author would like to thank the “Universitat de Barcelona” where this work was done
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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