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Black box Galois representations

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Argaez-Garcia, Alejandro and Cremona, J. E. (2018) Black box Galois representations. Journal of Algebra, 512 . pp. 526-565. doi:10.1016/j.jalgebra.2018.05.017 ISSN 0021-8693.

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Official URL: https://doi.org/10.1016/j.jalgebra.2018.05.017

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Abstract

We develop methods to study 2-dimensional 2-adic Galois representations ρ of the absolute Galois group of a number field K, unramified outside a known finite set of primes S of K, which are presented as Black Box representations, where we only have access to the characteristic polynomials of Frobenius automorphisms at a finite set of primes. Using suitable finite test sets of primes, depending only on K and S, we show how to determine the determinant , whether or not ρ is residually reducible, and further information about the size of the isogeny graph of ρ whose vertices are homothety classes of stable lattices. The methods are illustrated with examples for K= Q, and for K imaginary quadratic, ρ being the representation attached to a Bianchi modular form.

Item Type: Journal Article
Subjects: Q Science > QA Mathematics
Divisions: Faculty of Science, Engineering and Medicine > Science > Mathematics
Library of Congress Subject Headings (LCSH): Galois theory, Curves, Elliptic, Automorphic forms
Journal or Publication Title: Journal of Algebra
Publisher: Academic Press Inc Elsevier Science
ISSN: 0021-8693
Official Date: 15 October 2018
Dates:
DateEvent
15 October 2018Published
21 May 2018Available
3 May 2018Accepted
Volume: 512
Page Range: pp. 526-565
DOI: 10.1016/j.jalgebra.2018.05.017
Status: Peer Reviewed
Publication Status: Published
Access rights to Published version: Open Access (Creative Commons)
Date of first compliant deposit: 10 May 2018
Date of first compliant Open Access: 4 October 2018
RIOXX Funder/Project Grant:
Project/Grant IDRIOXX Funder NameFunder ID
Chancellor’s ScholarshipUniversity of Warwickhttp://dx.doi.org/10.13039/501100000741
EP/K034383/1[EPSRC] Engineering and Physical Sciences Research Councilhttp://dx.doi.org/10.13039/501100000266
#676541H2020 European Research Councilhttp://dx.doi.org/10.13039/100010663
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