arXiv subjects

Hidekazu Furusho

Publications and source records attributed to Hidekazu Furusho.

40 records · Page 3Linked to original sources

arXiv2005

Regularization and generalized double shuffle relations for p-adic multiple zeta values

We will introduce a regularization for $p$-adic multiple zeta values and show that the generalized double shuffle relations hold. This settles a question raised by Deligne, given as a project in Arizona winter school 2002. Our approach is to use the theory of Coleman functions on the moduli space of genus zero curves with marked points and its compactification. The main ingredients are the analytic continuation of Coleman functions to the normal bundle of divisors at infinity and definition of a special tangential base point on the moduli space.

math.NT↗

arXiv2005

Algebraic cycles and motivic generic iterated integrals

Following the work of Gangl, Goncharov and Levin in [GGL], we will give a combinatorial framework for motivic study of iterated integrals on the affine line. We will show that under a certain genericity condition these combinatorial objects yield to elements in the motivic Hopf algebra constructed in Bloch-Kriz [BK]. It will be shown that the Hodge realization of these elements coincides with the Hodge structure induced from the fundamental torsor of path of punctured affine line.

math.NT↗

arXiv2003

p-adic multiple zeta values I -- p-adic multiple polylogarithms and the p-adic KZ equation

Our main aim in this paper is to give a foundation of the theory of $p$-adic multiple zeta values. We introduce (one variable) $p$-adic multiple polylogarithms by Coleman's $p$-adic iterated integration theory. We define $p$-adic multiple zeta values to be special values of $p$-adic multiple polylogarithms. We consider the (formal) $p$-adic KZ equation and introduce the $p$-adic Drinfel'd associator by using certain two fundamental solutions of the $p$-adic KZ equation. We show that our $p$-adic multiple polylogarithms appear as coefficients of a certain fundamental solution of the $p$-adic KZ equation and our $p$-adic multiple zeta values appear as coefficients of the $p$-adic Drinfel'd associator. We show various properties of $p$-adic multiple zeta values, which are sometimes analogous to the complex case and are sometimes peculiar to the $p$-adic case, via the $p$-adic KZ equation.

math.NT↗

arXiv2003

The multiple zeta value algebra and the stable derivation algebra

The MZV algebra is the graded algebra over ${\bold Q}$ generated by all multiple zeta values. The stable derivation algebra is a graded Lie algebra version of the Grothendieck-Teichmüller group. We shall show that there is a canonical surjective $\bold Q $-linear map from the graded dual vector space of the stable derivation algebra over $\bold Q$ to the new-zeta space, the quotient space of the sub-vector space of the MZV algebra whose grade is greater than 2 by the square of the maximal ideal. As a corollary, we get an upper-bound for the dimension of the graded piece of the MZV algebra at each weight in terms of the corresponding dimension of the graded piece of the stable derivation algebra. If some standard conjectures by Y. Ihara and P. Deligne concerning the structure of the stable derivation algebra hold, this will become a bound conjectured in Zagier's talk at 1st European Congress of Mathematics. Via the stable derivation algebra, we can compare the new-zeta space with the $l$-adic Galois image Lie algebra which is associated with so the Galois representation on the pro-$l$ fundamental group of $\bold P ^1_ {\bar{\bold Q}}-{0,1,\infty}$ .

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