Companion forms and explicit computation of PGL2 number fields with very little ramification
| Autor: | Nicolas Mascot |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2016 |
| Předmět: |
Pure mathematics
Algebra and Number Theory Degree (graph theory) Mathematics - Number Theory 010102 general mathematics Modular form Galois group 010103 numerical & computational mathematics Algebraic number field Galois module 01 natural sciences Discriminant Closure (mathematics) FOS: Mathematics 11F80 11R21 11Y40 (Primary) 11F11 11F30 (Secondary) Number Theory (math.NT) 0101 mathematics Representation (mathematics) QA Mathematics |
| ISSN: | 0021-8693 |
| Popis: | 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. Version 2 update, 31 pages. Removed tables, added a section on the computation of discriminants |
| Databáze: | OpenAIRE |
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