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 $\mathrm{rank}\ge n\cdot \phi (N)$, where $\phi (N)$ is Euler's totient function, by computing their images under the transcendental regulator map.

16 pages, 1 figure