Real multiplication and the generalized Fermat equation x⁵ + y³ = zᵖ

Let (X, Y, Z) be a primitive solution of x⁵ + y³ = zᵖ with p ≥ 7 prime. Pacetti and Villagra Torcomian attach to it a genus-2 Frey curve whose Jacobian Jₜ, t = −X⁵/Y³, has real multiplication by the integers of Q(√5). In a preprint they show that, if p lies outside an explicit finite set and 𝔭 is a prime above p, the mod-𝔭 representation of Jₜ is reducible, or congruent to a Hilbert newform with complex multiplication, or congruent to the Jacobian at t = −1/8 or t = 9/8, the parameters of 1⁵ + 2³ = 3² and (−3)⁵ + 6³ = (−3)³. Comparing the fields cut out by inertia at √5, we show that this last congruence forces 5 ∤ XYZ and restricts t to eight classes modulo 25, and, by comparing traces of Frobenius at an auxiliary prime, to four classes when p ≥ 17. A congruence multiplies the Weil pairings by a scalar, and by comparing this scalar at 3 and at √5 we show that the congruence cannot occur when 3 | Y and p ≡ 19, 29, 31, 41 (mod 60). Assuming statements of the preprint, we deduce that, for every prime p outside that explicit finite set, there are no solutions in several families of classes, among them those with 3 ∤ XYZ.