A result on the equation $x^p + y^p = z^r$ using Frey abelian varieties

Abstract:

We prove a Diophantine result on generalized Fermat equations of the form $x^p + y^p = z^r$ which for the first time requires the use of Frey abelian varieties of dimension $\geq 2$ in Darmon’s program. More precisely, for $r \ge 5$ a regular prime we prove that there exists a constant $C(r)$ such that for every prime number $p > C(r)$ the equation $x^p + y^p = z^r$ has no non-trivial primitive integer solutions $(a,b,c)$ satisfying $r \mid ab$ and $2 \nmid ab$.

For the proof, we complement Darmon’s ideas in a particular case by providing an irreducibility criterion for the mod $\mathfrak {p}$ representations attached to certain families of abelian varieties of $\operatorname {GL}_2$-type over totally real fields.