Algebraicity of Weil Classes on an Explicit Cyclic-Sextic Prym Sixfold

Abstract We investigate the algebraicity of Weil classes on an explicit abelian sixfold of Weil type associated with the imaginary quadratic field K = Q(√−3). The sixfold is constructed as the faithful cyclotomic Prym factor of the cyclic sextic curve y⁶ = ∏(x − j)^{eⱼ}, j = 0,…,7, with exponent vector (1, 2, 2, 3, 3, 3, 5, 5). The resulting abelian variety has Weil signature (3,3) and Hermitian discriminant class [−2], placing it outside the split-discriminant setting of previously established general algebraicity results. We present a geometric construction of algebraic classes using divisors on a cyclic surface cover, two Gysin morphisms, finite-quotient transfer, and an Abel–Prym correspondence. A rational algebraic projector is then used to obtain classes spanning the two-dimensional rational Weil space. The manuscript includes exact computational checks of the integral polarization, reproducibility materials, and an account of the construction's verification and limitations. This result concerns one explicitly constructed abelian sixfold and does not establish the general Hodge conjecture. The manuscript is an unrefereed research preprint, submitted for independent mathematical scrutiny. Author: Miyoboku Sage Keywords: Hodge conjecture; Weil classes; abelian sixfolds; cyclic covers; Prym varieties; algebraic cycles; Hermitian discriminant; Gysin morphisms.