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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 abelianvarieties of dimension $\geq 2$ in Darmon's program. For that, we provide anirreducibility criterion for the mod $\mathfrak{p}$ representations attached tocertain abelian varieties of $\text{GL}_2$-type over totally real fields.

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