# Higher Chow cycles on Jacobian of Fermat curves and Hypergeometric functions

## TL;DR

This paper constructs higher Chow cycles on Fermat curve Jacobians, generalizes Collino's work, and computes their regulators using hypergeometric functions, extending previous regulator calculations for Fermat varieties.

## Contribution

It introduces new higher Chow cycles on Fermat Jacobians and expresses their regulators via hypergeometric functions, expanding on prior regulator computations.

## Key findings

- Constructed higher Chow cycles in K_1 of Fermat Jacobians.
- Expressed regulators in terms of special hypergeometric values.
- Extended regulator calculations to higher K-theory of Fermat varieties.

## Abstract

In this paper we construct certain higher Chow cycles in the $K_{1}$ of the Jacobian of Fermat curves, generalising a construction of Collino. We further compute the regulator of these elements in terms of special values of hypergeometric functions. Otsubo has computed the regulator of certain elements of $K_0$ and $K_2$ of Fermat varieties and this paper is along the same lines.