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.