The generalized Fermat equation x¹¹ + y¹¹ = z⁵

We prove that x¹¹ + y¹¹ = z⁵ has no solution in nonzero coprime integers, and that for every prime p ≥ 11 the solutions of xᵖ + yᵖ = z⁵ in nonzero coprime integers with xy prime to 10 have xy ≡ −1 (mod 5). To a solution Darmon attaches a Frey abelian surface over Q(√5), a twist of the member at t = xᵖ/z⁵ of a one-parameter family. For p = 11, results of Chen and Koutsianas and of Billerey, Chen, Dieulefait and Freitas leave only the solutions with 10 | xy and 11 ∤ z. For these, t gives a point on a quotient of Darmon's modular curve X₀(𝔭), 𝔭 | 11, that is an elliptic curve of rank 1 over Q(√5), and Chabauty's method in the restriction of scalars excludes such points. For general p and xy prime to 10, Chen and Koutsianas show that the Frey surface is congruent modulo p, up to a quadratic twist, to one of four surfaces with complex multiplication (or, for p = 19, to one more modular form); we test this congruence at the prime √5. When xy ≡ ±2 (mod 5), the fields cut out by inertia there differ or, in one subcase, a Frobenius trace does. When xy ≡ 1 (mod 5), the reduction at 2 determines the twist, and Frobenius traces exclude the congruence: at an auxiliary prime, and on the special fiber over F₅ that the genus-2 curves defining these surfaces share over a totally ramified extension of Q₅(√5). For 10 | xy, a uniform bound on rational points of order p on abelian surfaces over Q(√5) with real multiplication would exclude this class for large p.