The generalized Fermat equation xᵖ + yᵖ = z⁵ and relative class numbers

We prove that the equation xᵖ + yᵖ = z⁵ has no solution in nonzero coprime integers for every prime p ≥ 13 such that p ≢ 1 (mod 5) and 5 ∤ h⁻(ℚ(ζ₅ₚ)), using results of Chen and Koutsianas (to appear in Algebra & Number Theory) on Darmon's Frey surfaces. These conditions hold at 41 primes 13 ≤ p < 1000 and at 2432 primes 13 ≤ p < 10⁵. For every prime p ≥ 7 with p ≢ 1 (mod 5) and 5 ∤ h⁻(ℚ(ζ₅ₚ)) we prove, without these results, that no solution (a, b, c) has 10p ∣ ab. Such solutions are close, at each of 2, 5 and p, to one of the trivial solutions (1, 0, 1), (0, 1, 1), which survive every local test. To exclude such solutions we attach to a solution a curve of genus 2(p − 1) whose Jacobian has multiplication by ℤ[ζ₅, ζₚ + ζₚ⁻¹]. It belongs to a family of quotients, by an involution, of cyclic covers in a hypergeometric family of the kind used in Darmon's program. The member of this family at the trivial solution is Y⁵ = Rᵖ − 1, which has complex multiplication, and we compare the 5-adic Tate modules of the two curves one power of 1 − ζ₅ at a time. The obstructions to extending this comparison lie in Selmer groups that vanish when 5 ∤ h⁻(ℚ(ζ₅ₚ)), as in Kummer's theorem for regular primes; the local condition at 5 comes from Breuil–Kisin modules over a tamely ramified extension. The generalized Riemann hypothesis is not used.