Скачать или смотреть Overview of Modularity of Elliptic Curves over Totally Real Fields - Nuno Freitas

Speaker: Nuno Freitas, Instituto de Ciencias Matematicas (ICMAT), Madrid.
Chair for the Talk: Narasimha Kumar, IIT Hyderabad.

Abstract: The modularity of all elliptic curves over Q has been known since the work of Wiles and Breuil, Conrad, Diamond, and Taylor. Essential to this proof are modularity lifting theorems and a 3-5 modularity switching argument.

In this talk, we will overview recent attempts to carry over this approach to totally real fields. More precisely, we will discuss how recent progress on modularity lifting theorems together with a 3-5-7 modularity switching argument allows us to reduce the proof of modularity of elliptic curves over a totally real field K to a calculation of K-points on certain modular curves. This final calculation is often impractical but it has been successfully completed for all quadratic and cubic totally real fields, yielding modularity of all elliptic curves over such fields.