The Hodge conjecture and the generalized Hodge conjecture for powers of Jacobians of cyclic covers of the line

For the cyclic cover y^ν = ∏(x − z_j)^{a_j} of the projective line with very general branch points, we determine the identity component of the monodromy on all eigenspaces of the cyclic group at once. It is a product of special linear and symplectic groups, one for each class of eigenspaces; two eigenspaces lie in the same class exactly when a Klein four-group symmetry of four branch points links them. From this we obtain the Hodge classes on all powers of the Jacobian J, on products of the Jacobians of independent families, and on the Jacobians of abelian covers of the line. The Hodge conjecture for all these varieties is equivalent to the algebraicity of finitely many explicit Hodge classes on Fermat varieties. Using only the classical algebraic cycles on Fermat varieties, we prove it: - for every exponent vector when ν is a prime, a prime power, 33, 99, or a composite number at most 120 outside thirty exceptional degrees; - for products of powers of the Jacobians of independent covers of prime degrees; - for all (Z/p)^r-covers.In 23 of the remaining 28 exceptional degrees, and in infinitely many further degrees such as 3p for the primes p > 3 with p ≢ 1 mod 12, a parity law proves it for exponent vectors of units with an even number of branch points: at least six, or four with no eigenspace of finite monodromy. For balanced exponent vectors we prove the generalized Hodge conjecture for all powers and many products in every degree up to 22 except 13, 17, 19 and 21, by Abdulali's method of domination, and we show that the method does not extend to the unbalanced exponent vectors of degree 3. Several statements rest on exact computer calculations; code and data are at github.com/sallerk/HC-cyclic-covers. No proof uses the unrefereed claims of the Hodge conjecture for abelian varieties with complex multiplication.