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Companion forms and explicit computation of PGL2 number fields with very little ramification
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Mascot, Nicolas (2018) Companion forms and explicit computation of PGL2 number fields with very little ramification. Journal of Algebra, 509 . pp. 476-506. doi:10.1016/j.jalgebra.2017.10.027 ISSN 0021-8693.
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Official URL: https://doi.org/10.1016/j.jalgebra.2017.10.027
Abstract
In previous works, we described algorithms to compute the number field cut out by the mod ell representation attached to a modular form of level N=1. In this article, we explain how these algorithms can be generalised to forms of higher level N.
As an application, we compute the Galois representations attached to a few forms which are supersingular or admit a companion mod ell with ell=13 (and soon ell=41), and we obtain previously unknown number fields of degree ell+1 whose Galois closure has Galois group PGL(2,ell) and a root discriminant that is so small that it beats records for such number fields.
Finally, we give a formula to predict the discriminant of the fields obtained by this method, and we use it to find other interesting examples, which are unfortunately out of our computational reach.
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): | Forms, Modular, Galois theory | ||||||
Journal or Publication Title: | Journal of Algebra | ||||||
Publisher: | Academic Press Inc Elsevier Science | ||||||
ISSN: | 0021-8693 | ||||||
Official Date: | 1 September 2018 | ||||||
Dates: |
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Volume: | 509 | ||||||
Page Range: | pp. 476-506 | ||||||
DOI: | 10.1016/j.jalgebra.2017.10.027 | ||||||
Status: | Peer Reviewed | ||||||
Publication Status: | Published | ||||||
Access rights to Published version: | Restricted or Subscription Access | ||||||
Date of first compliant deposit: | 20 September 2019 | ||||||
Date of first compliant Open Access: | 20 September 2019 | ||||||
RIOXX Funder/Project Grant: |
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