STNB2020(34è any)

The Darmon program for the Generalized Fermat equation


Luis Dieulefait


In 2000, Darmon described a program to study the Generalized Fermat equation~$Ax^r + By^q = Cz^p$ using modularity of abelian varieties of $\GL_2$-type over totally real fields. This highly ambitious program is still completely open as it relies on very hard open conjectures. In this lecture we will discuss the core ideas of Darmon's program which will be used in the final two lectures to obtain new unconditional Diophantine applications.


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