Construction of Higher Chow cycles on cyclic coverings of ${\mathbb{P}}^1\times {\mathbb{P}}^1$

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 $\mathbb{Z}$ and generate a subgroup of $\mathrm{rank}\ge 36\cdot \phi (N)$, where $\phi (N)$ is Euler’s totient function, by computing the image of the transcendental regulator map.