### A multi-Frey approach to the Darmon program for Fermat-type equations

Heilbronn Number Theory Seminar

22nd January 2020, 4:00 pm – 5:00 pm

Fry Building, 2.04

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 ≥ 2.

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