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

Abstract

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 PGL2(Fℓ) 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.