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 GL_2-type over totally real fields. However, his program relies on hard

open conjectures, which has made it difficult to apply in practice, and so

far the only successes were in cases where the Frey varieties are elliptic

curves.

In this talk, we will discuss how using a combination of two

Frey elliptic curves with a Frey hyperelliptic curve

and ideas from the Darmon program, we can give a complete

resolution of the generalized Fermat equation

x 7 + y 7 = 3 z^n for all integers n \ge 2.

This is ongoing work with Billerey, Chen and Dieleufait.

Zoom ID: 919 6497 4060

For password contact Pieter Moree (moree@mpim...).

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