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

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