Motivic interpretations for iterated integrals on some specific algebraic curves

Abstract Multiple zeta values (MZVs for short) can be represented as iterated integrals of $${\mathbb {Q}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Q</mml:mi> </mml:math> -rational algebraic differential forms on $${\mathbb {P}}^1({\mathbb {C}})\setminus \{0, 1, \infty \}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>P</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>C</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>\</mml:mo> <mml:mrow> <mml:mo>{</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mi>∞</mml:mi> <mml:mo>}</mml:mo> </mml:mrow> </mml:mrow> </mml:math> . This interpretation allows us to consider MZVs geometrically, and this is one of the motivations for Deligne–Goncharov, Terasoma et al. to give motivic interpretations of MZVs by using the theory of mixed Tate motives and the motivic fundamental groups. In this paper, we consider the iterated integrals on some rational curves over $${\mathbb {Q}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Q</mml:mi> </mml:math> and study their arithmetic properties. They are an extension of MZVs and also include some other known special values such as multiple $${\widetilde{T}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>T</mml:mi> <mml:mo>~</mml:mo> </mml:mover> </mml:math> -values. Furthermore, we give motivic interpretations of them by investigating a relationship with motivic iterated integrals given by Goncharov. At this point, it is important to consider the base expansion and the Galois invariant part of the space of motivic iterated integrals. Finally, we denote that a motivic interpretation of the alternating multiple mixed values can be given by the same method. Our results also extend a part of author’s previous work.