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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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:	Date Event 15 October 2018 Published 21 May 2018 Available 3 May 2018 Accepted
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 ID RIOXX Funder Name Funder ID Chancellor’s Scholarship University of Warwick http://dx.doi.org/10.13039/501100000741 EP/K034383/1 [EPSRC] Engineering and Physical Sciences Research Council http://dx.doi.org/10.13039/501100000266 #676541 H2020 European Research Council http://dx.doi.org/10.13039/100010663
Related URLs:	Publisher
Open Access Version:	ArXiv

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