Companion forms and explicit computation of PGL2 number fields with very little ramification

JOURNAL OF ALGEBRA

卷:509

Article

Mascot, Nicolas¹ 

[1] Univ Warwick, Coventry CV4 7AL, W Midlands, England

关键词: Galois representation;    Modular form;    Number field tabulation;    Ramification;    Discriminant;    Inverse Galois problem;   

DOI  :  10.1016/j.jalgebra.2017.10.027

来源: Elsevier

【 摘 要 】

In previous works [18] and [19], we described algorithms to compute the number field cut out by the mod l 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 l with l = 13 and l = 41, and we obtain previously unknown number fields of degree l + 1 whose Galois closure has Galois group PGL(2)(F-l) 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. (C) 2017 Published by Elsevier Inc.

【 授权许可】

Free