Galois Representations and Galois Groups Over ℚ
Arias-de-Reyna, Sara; Armana, Cécile; Karemaker, Valentijn; Rebolledo, Marusia; Thomas, Lara; Vila, Núria; Bertin, Marie José; Bucur, Alina; Feigon, Brooke; Schneps, Leila
(2015) Women in Numbers Europe, pp. 191 - 205
Association for Women in Mathematics Series, pp. 191 - 205
(Part of book)
Abstract
In this paper we generalize results of P. Le Duff to genus n hyperelliptic curves. More precisely, let C∕ ℚ be a hyperelliptic genus n curve, let J(C) be the associated Jacobian variety, and let ρ̄ℓ: Gℚ→ GSp (J(C) [ ℓ] ) be the Galois representation attached to the ℓ -torsion
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of J(C). Assume that there exists a prime p such that J(C) has semistable reduction with toric dimension 1 at p. We provide an algorithm to compute a list of primes ℓ (if they exist) such that ρ̄ℓ is surjective. In particular we realize GSp6(ℓ) as a Galois group over ℚ for all primes ℓ∈ [ 11, 500, 000 ].
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Keywords: Abelian Variety, Characteristic Polynomial, Galois Group, Galois Representation, Modular Form, Taverne, Mathematics(all), Gender Studies
ISSN: 2364-5733
ISBN: 978-3-319-17986-5
978-3-319-36836-8
978-3-319-17987-2
Publisher: Springer Cham, Springer
Note: Funding Information: The authors would like to thank Marie-Jos? Bertin, Alina Bucur, Brooke Feigon, and Leila Schneps for organizing the WIN-Europe conference which initiated this collaboration. Moreover, we are grateful to the Centre International de Rencontres Math?matiques, the Institut de Math?matiques de Jussieu, and the Institut Henri Poincar? for their hospitality during several short visits. The authors are indebted to Irene Bouw, Jean-Baptiste Gramain, Kristin Lauter, Elisa Lorenzo, Melanie Matchett Wood, Frans Oort, and Christophe Ritzenthaler for several insightful discussions. We also want to thank the anonymous referee for her/his suggestions that helped us to improve this paper. S. Arias-de-Reyna and N. Vila are partially supported by the project MTM2012-33830 of the Ministerio de Econom?a y Competitividad of Spain, C. Armana by a BQR 2013 Grant from Universit? de Franche-Comt? and M. Rebolledo by the ANR Project R?gulateurs ANR-12-BS01-0002. L. Thomas thanks the Laboratoire de Math?matiques de Besan?on for its support. Funding Information: S. Arias-de-Reyna and N. Vila are partially supported by the project MTM2012-33830 of the Ministerio de Economía y Competitividad of Spain, C. Armana by a BQR 2013 Grant from Université de Franche-Comté and M. Rebolledo by the ANR Project Régulateurs ANR-12-BS01-0002. L. Thomas thanks the Laboratoire de Mathématiques de Besançon for its support. Publisher Copyright: © 2015, Springer International Publishing Switzerland.
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