Abstract:
In 2000, Darmon described a remarkable program to study the Generalized
Fermat equation Ax^r + By^q = Cz^p using modularity of abelian varieties
of GL2-type over totally real fields. However, this program relies on
various hard conjectures, making it impractical, and until recently it
has been successfully applied only in cases where the abelian varieties
were elliptic curves. In this talk, we will discuss how to combine the
classical modular method using elliptic curves with Frey hyperelliptic
curves due to Kraus and ideas from the Darmon program to study  the
Fermat-type equations of the form x^r + y^r = C z^p.