Galois Representations and Galois Groups Over Q

Sara Arias-de-Reyna; Cécile Armana; Valentijn Karemaker; Marusia Rebolledo; Lara Thomas; Núria Vila

doi:10.1007/978-3-319-17987-2_8

Galois Representations and Galois Groups Over Q

Sara Arias-de-Reyna, Cécile Armana, Valentijn Karemaker, Marusia Rebolledo, Lara Thomas, Núria Vila^*

^*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceeding › Chapter › Academic › peer-review

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 N` W GQ ! GSp.J.C/OE`/ be the Galois
representation attached to the `-torsion 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 ` (if they exist) such that N` is surjective.
In particular we realize GSp6.F`/ as a Galois group over Q for all primes ` 2 OE11; 500;000.

Original language	English
Title of host publication	Women in Numbers Europe
Subtitle of host publication	Research Directions in Number Theory
Editors	Marie José Bertin, Alina Bucur, Brooke Feigon, Leila Schneps
Publisher	Springer
Pages	191-205
Number of pages	15
Edition	1
ISBN (Electronic)	978-3-319-17987-2
ISBN (Print)	978-3-319-17986-5, 978-3-319-36836-8
DOIs	https://doi.org/10.1007/978-3-319-17987-2_8
Publication status	Published - 2015

Publication series

Name	Association for Women in Mathematics Series
Publisher	Springer
Volume	2
ISSN (Print)	2364-5733
ISSN (Electronic)	2364-5741

Bibliographical 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.

Funding

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. 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.

Keywords

Abelian Variety
Characteristic Polynomial
Galois Group
Galois Representation
Modular Form

Access to Document

10.1007/978-3-319-17987-2_8

Arias-de-Reyna2015_Chapter_GaloisRepresentationsAndGaloisFinal published version, 210 KB

Cite this

Arias-de-Reyna, S., Armana, C., Karemaker, V., Rebolledo, M., Thomas, L., & Vila, N. (2015). Galois Representations and Galois Groups Over Q. In M. J. Bertin, A. Bucur, B. Feigon, & L. Schneps (Eds.), Women in Numbers Europe: Research Directions in Number Theory (1 ed., pp. 191-205). (Association for Women in Mathematics Series; Vol. 2). Springer. https://doi.org/10.1007/978-3-319-17987-2_8

@inbook{0c2fd8bec4814ed68d60a1eaf8decdba,

title = "Galois Representations and Galois Groups Over Q",

abstract = "In this paper we generalize results of P. Le Duff to genus n hyperellipticcurves. More precisely, let C=Q be a hyperelliptic genus n curve, let J.C/ bethe associated Jacobian variety, and let N` W GQ ! GSp.J.C/OE`/ be the Galoisrepresentation attached to the `-torsion of J.C/. Assume that there exists a primep such that J.C/ has semistable reduction with toric dimension 1 at p. We providean algorithm to compute a list of primes ` (if they exist) such that N` is surjective.In particular we realize GSp6.F`/ as a Galois group over Q for all primes ` 2 OE11; 500;000.",

keywords = "Abelian Variety, Characteristic Polynomial, Galois Group, Galois Representation, Modular Form",

author = "Sara Arias-de-Reyna and C{\'e}cile Armana and Valentijn Karemaker and Marusia Rebolledo and Lara Thomas and N{\'u}ria Vila",

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 Econom{\'i}a y Competitividad of Spain, C. Armana by a BQR 2013 Grant from Universit{\'e} de Franche-Comt{\'e} and M. Rebolledo by the ANR Project R{\'e}gulateurs ANR-12-BS01-0002. L. Thomas thanks the Laboratoire de Math{\'e}matiques de Besan{\c c}on for its support. Publisher Copyright: {\textcopyright} 2015, Springer International Publishing Switzerland.",

year = "2015",

doi = "10.1007/978-3-319-17987-2_8",

language = "English",

isbn = "978-3-319-17986-5",

series = "Association for Women in Mathematics Series",

publisher = "Springer",

pages = "191--205",

editor = "Bertin, {Marie Jos{\'e} } and Bucur, {Alina } and Feigon, {Brooke } and Schneps, {Leila }",

booktitle = "Women in Numbers Europe",

edition = "1",

}

Arias-de-Reyna, S, Armana, C, Karemaker, V, Rebolledo, M, Thomas, L & Vila, N 2015, Galois Representations and Galois Groups Over Q. in MJ Bertin, A Bucur, B Feigon & L Schneps (eds), Women in Numbers Europe: Research Directions in Number Theory. 1 edn, Association for Women in Mathematics Series, vol. 2, Springer, pp. 191-205. https://doi.org/10.1007/978-3-319-17987-2_8

Galois Representations and Galois Groups Over Q. / Arias-de-Reyna, Sara; Armana, Cécile; Karemaker, Valentijn et al.
Women in Numbers Europe: Research Directions in Number Theory. ed. / Marie José Bertin; Alina Bucur; Brooke Feigon; Leila Schneps. 1. ed. Springer, 2015. p. 191-205 (Association for Women in Mathematics Series; Vol. 2).

Research output: Chapter in Book/Report/Conference proceeding › Chapter › Academic › peer-review

TY - CHAP

T1 - Galois Representations and Galois Groups Over Q

AU - Arias-de-Reyna, Sara

AU - Armana, Cécile

AU - Karemaker, Valentijn

AU - Rebolledo, Marusia

AU - Thomas, Lara

AU - Vila, Núria

N1 - 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.

PY - 2015

Y1 - 2015

N2 - In this paper we generalize results of P. Le Duff to genus n hyperellipticcurves. More precisely, let C=Q be a hyperelliptic genus n curve, let J.C/ bethe associated Jacobian variety, and let N` W GQ ! GSp.J.C/OE`/ be the Galoisrepresentation attached to the `-torsion of J.C/. Assume that there exists a primep such that J.C/ has semistable reduction with toric dimension 1 at p. We providean algorithm to compute a list of primes ` (if they exist) such that N` is surjective.In particular we realize GSp6.F`/ as a Galois group over Q for all primes ` 2 OE11; 500;000.

AB - In this paper we generalize results of P. Le Duff to genus n hyperellipticcurves. More precisely, let C=Q be a hyperelliptic genus n curve, let J.C/ bethe associated Jacobian variety, and let N` W GQ ! GSp.J.C/OE`/ be the Galoisrepresentation attached to the `-torsion of J.C/. Assume that there exists a primep such that J.C/ has semistable reduction with toric dimension 1 at p. We providean algorithm to compute a list of primes ` (if they exist) such that N` is surjective.In particular we realize GSp6.F`/ as a Galois group over Q for all primes ` 2 OE11; 500;000.

KW - Abelian Variety

KW - Characteristic Polynomial

KW - Galois Group

KW - Galois Representation

KW - Modular Form

UR - http://www.scopus.com/inward/record.url?scp=84983687129&partnerID=8YFLogxK

U2 - 10.1007/978-3-319-17987-2_8

DO - 10.1007/978-3-319-17987-2_8

M3 - Chapter

AN - SCOPUS:84983687129

SN - 978-3-319-17986-5

SN - 978-3-319-36836-8

T3 - Association for Women in Mathematics Series

SP - 191

EP - 205

BT - Women in Numbers Europe

A2 - Bertin, Marie José

A2 - Bucur, Alina

A2 - Feigon, Brooke

A2 - Schneps, Leila

PB - Springer

ER -

Galois Representations and Galois Groups Over Q

Abstract

Publication series

Bibliographical note

Funding

Keywords

Access to Document

Other files and links

Fingerprint

Cite this