On the Darmon program for the Generalized Fermat equationMPIM

by Nuno Freitas (MPIM)

Thursday Feb 13, 2025, 3:00 PM → 4:00 PM Europe/Berlin

MPIM, Vivatsgasse, 7 - Lecture Hall (Max Planck Institute for Mathematics)

MPI-Oberseminar

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 ta lk, 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.