The Hodge conjecture for powers of very general split quintic and sextic hypersurfaces

We prove the Hodge conjecture for products of the hypersurfaces X_F: F₀(u₀,v₀) + ⋯ + Fₘ(uₘ,vₘ) = 0 in ℙ²ᵐ⁺¹ whose binary forms, of degree d = 5 or 6, are taken with repetitions from finitely many very general forms, for instance for all powers X_Fᵏ and for the tenfolds X_(G,G,G,G′,G′,G′). In the companion note the forms were independent. As there, the proof reduces the Hodge classes to invariants of the monodromy of the curves yᵈ + G(u,v) = 0, using the inductive structure of Shioda and Katsura and the algebraicity of Hodge loci. We determine the identity component of the Zariski closure of this monodromy on the first cohomology: SL₃ × SL₃ for d = 5 (this case is known), and SL₄ × Sp₄ on the part with infinite monodromy for d = 6. So the invariants are built from cup products and from determinants on repeated eigenspaces (for d = 6 also from the part with finite monodromy). For d = 5 the determinants pair up into Weil classes for ℚ(ζ₅) on products J × J′ of two Jacobians of the curves y⁵ + G(u,v) = 0, which for very general G and G′ are not in the span of products of divisor classes. We prove that these Weil classes are algebraic for all G and G′, by an idea of Schoen: the quotient of the third power of such a curve by a certain finite group is a cyclic cover of ℙ³ branched along five hyperplanes, a quotient of the Fermat quintic threefold, and so does not depend on G. They can also be obtained by a degeneration from a theorem of Schoen on cyclic covers of curves. For d = 6, determinants occur when a form is repeated at least four times; we reduce them to Weil classes for ℚ(√−3) on the abelian sixfolds A × E × E, where A is the Jacobian of w³ + G(u,v) = 0 and E is the Fermat cubic curve, and the same argument with the Fermat cubic fourfold shows that these are algebraic. For d = 5 and very general F we also compute the dimension of the space of Hodge classes in H²ᵐ(X_F) ⊗ H²ᵐ(X_F): 293, 4429 and 73013 for m = 1, 2, 3. The proofs do not use computer calculations; independent counts of the Hodge classes and numerical tests of the monodromy are at github.com/sallerk/HC-quintic-sextic-powers. An unrefereed manuscript released by OpenAI claims a result on powers of Jacobians of abelian covers which, as far as we can see, would imply our main theorem; we have not checked it and do not use it. Companion to "The Hodge conjecture for very general sums of binary quintics and sextics" (doi:10.5281/zenodo.23267886).