arXiv subjects
Publications and source records attributed to Eisuke Otsuka.
K.T. Chen showed that iterated integrals give comparison isomorphisms between the cohomologies of bar complexes and fundamental group rings. This led to the development of an algebraic-geometric approach to studying periods given by iterated integrals. In this paper we consider an analogue of this comparison isomorphism theorem for iterated integrals on the Legendre family.
Multiple zeta values (MZVs for short) can be represented as iterated integrals of $\mathbb{Q}$-rational algebraic differential forms on $\mathbb{P}^1(\mathbb{C})\setminus\{0, 1, \infty\}$. 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 $\mathbb{Q}$ and study their arithmetic properties. They are an extension of MZVs and also include some other known special values such as multiple $\widetilde{T}$-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.
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.