On the Darmon program for the Generalized Fermat equation

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 first discuss the limitations of the
classical modular method and some of the main ideas and issues of the
Darmon program. Finally, we will discuss an application to
the equation x^7 + y^7 = 3 z^p.