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

Abstract In this paper, we will define analogues of multiple zeta values by replacing the differential forms defining multiple zeta values with some $$\mathbb {Q}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Q</mml:mi> </mml:math> -rational differential forms on the Fermat curve $$F_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>F</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> 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 $$\mathcal {H}_4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>H</mml:mi> <mml:mn>4</mml:mn> </mml:msub> </mml:math> of level 4 and its Galois invariant part.