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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Autor principal: Pacetti, A.
Formato: JOUR
Materias:
Galois representations
Shimura correspondence
Acceso en línea:http://hdl.handle.net/20.500.12110/paper_00255718_v76_n260_p2063_Pacetti
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Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires
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spelling todo:paper_00255718_v76_n260_p2063_Pacetti2023-10-03T14:36:12Z On the embedding problem for 2+S4 representations Pacetti, A. Galois representations Shimura correspondence 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. JOUR info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar http://hdl.handle.net/20.500.12110/paper_00255718_v76_n260_p2063_Pacetti
institution Universidad de Buenos Aires
institution_str I-28
repository_str R-134
collection Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA)
topic Galois representations
Shimura correspondence
spellingShingle Galois representations
Shimura correspondence
Pacetti, A.
On the embedding problem for 2+S4 representations
topic_facet Galois representations
Shimura correspondence
description 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.
format JOUR
author Pacetti, A.
author_facet Pacetti, A.
author_sort Pacetti, A.
title On the embedding problem for 2+S4 representations
title_short On the embedding problem for 2+S4 representations
title_full On the embedding problem for 2+S4 representations
title_fullStr On the embedding problem for 2+S4 representations
title_full_unstemmed On the embedding problem for 2+S4 representations
title_sort on the embedding problem for 2+s4 representations
url http://hdl.handle.net/20.500.12110/paper_00255718_v76_n260_p2063_Pacetti
work_keys_str_mv AT pacettia ontheembeddingproblemfor2s4representations
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