Construction of higher Chow cycles on cyclic coverings of $\mathbb{P}^1 \times \mathbb{P}^1$, Part II

Abstract

In this paper, we construct higher Chow cycles of type $(2, 1)$ on a family of surfaces related to a product of curves, which are certain degree $N$ abelian covers of $\mathbb{P}^1$ branched over $n+2$ points. We prove that for a very general member, these cycles generate a subgroup of the indecomposable part of $\operatorname{rank} \ge n\cdot \varphi(N)$, where $\varphi(N)$ is Euler's totient function, by computing their images under the transcendental regulator map.