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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
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Official URL: https://doi.org/10.1016/j.jalgebra.2018.05.017
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 | ||||||||||||
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| 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 | ||||||||||||
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| 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 | ||||||||||||
| RIOXX Funder/Project Grant: |
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