On the embedding problem for 2+S4 representations
Let 2+54 denote the double cover of S4 corresponding to the element in H2(54, 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 elem...
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| Formato: | JOUR |
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| Acceso en línea: | http://hdl.handle.net/20.500.12110/paper_00255718_v76_n260_p2063_Pacetti |
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| Sumario: | Let 2+54 denote the double cover of S4 corresponding to the element in H2(54, 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 H 1(Galℚ, E[2])\\{0} correspond to Galois extensions N of ℚ with Galois group (isomorphic to) 54. 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 Ñ with Gal(Ñ/ℚ) ≃ 2+54 gives a homomorphism s4 +: H1(Galℚ, E[2]) → H 2(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. © 2007 American Mathematical Society. |
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