Construction of Higher Chow cycles on cyclic coverings of P¹ × P¹

In this paper, we construct higher Chow cycles of type $(2, 1)$ on a certain family of surfaces, which are constructed by a product of certain hypergeometric curves of degree N. We prove that for a very general member, these cycles are linearly independent over Z and generate a subgroup of rank ≥ 36 · φ(N), where $φ(N)$ is Euler's totient function, by computing the image of the transcendental regulator map.