Regulators on some abelian coverings of $\mathbb{P}^1$ minus $n+2$ points

TL;DR

This paper constructs and verifies non-trivial motivic cohomology elements for superelliptic curves derived from abelian coverings of the projective line minus points, using hypergeometric functions to relate to special values of L-functions.

Contribution

It introduces explicit motivic cohomology elements for superelliptic curves and demonstrates their non-triviality and integrality through hypergeometric regulator computations.

Findings

Regulators expressed in terms of Appell-Lauricella hypergeometric functions.

Verification of non-triviality of motivic elements.

Numerical evidence supporting Beilinson's conjecture.

Abstract

In this paper, we construct certain rational or integral elements in the motivic cohomology of superelliptic curves which are quotient curves of abelian coverings of $\mathbb{P}^1$ minus $n+2$ points, and prove that these elements are non-trivial by expressing their regulators in terms of Appell-Lauricella hypergeometric functions. We also check that such elements are integral under a mild assumption. We also give various numerical examples for the Beilinson conjecture on special values of $L$-functions of the superelliptic curves by using hypergeometric expressions.