On arithmetic properties of periods for some rational differential forms over $\mathbb{Q}$ on the Fermat curve $F_2$ of degree 2

TL;DR

This paper introduces analogues of multiple zeta values using rational differential forms on the Fermat curve F_2, exploring their arithmetic properties and motivic structures, and extending the theory of motivic periods.

Contribution

It defines new period analogues on Fermat curves and investigates their motivic and Galois structures, extending existing theories.

Findings

Defined analogues of multiple zeta values on F_2

Analyzed the motivic structure of these periods

Studied the Galois invariants of the extended motivic period space

Abstract

In this paper, we will define analogues of multiple zeta values by replacing the differential forms defining multiple zeta values with some $\mathbb{Q}$-rational differential forms on the Fermat curve $F_2$ of degree 2 and discuss their arithmetic properties. We also investigate a motivic structure of the motivic periods corresponding to our periods. However, in order to study them, the current theory for motivic zeta elements is insufficient, and it leads us to study the base extension of the space of the motivic periods $\mathcal{H}_4$ of level 4 and its Galois invariant part.