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

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Abstract
We prove a Diophantine result on generalized Fermat equations of the form x p + y p = z r x^p + y^p = z^r which for the first time requires the use of Frey abelian varieties of dimension ≥ 2 \geq 2 in Darmon’s program. More precisely, for r ≥ 5 r \ge 5 a regular prime we prove that there exists a constant C ( r ) C(r) such that for every prime number p > C ( r ) p > C(r) the equation x p + y p = z r x^p + y^p = z^r has no non-trivial primitive integer solutions ( a , b , c ) (a,b,c) satisfying r ∣ a b r \mid ab and 2 ∤ a b 2 \nmid ab . For the proof, we complement Darmon’s ideas in a particular case by providing an irreducibility criterion for the mod p \mathfrak {p} representations attached to certain families of abelian varieties of GL 2 \operatorname {GL}_2 -type over totally real fields.
Funding Information
  • Agence Nationale de la Recherche (ANR-14-CE25-0015)

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