The author will prove that Drinfel'd's pentagon equation implies his two hexagon equations in the Lie algebra, pro-unipotent, pro-l and pro-nilpotent contexts.

It is proved that Drinfel'd's pentagon equation implies the generalized double shuffle relation. As a corollary, an embedding from the Grothendieck-Teichmüller group GRT 1 into Racinet's double shuffle group DM R 0 is obtained, which settles the project of Deligne-Terasoma. It is also proved that the gamma factorization formula follows from the generalized double shuffle relation.

Abstract. We establish a tannakian formalism of p-adic multiple polylogarithms and p-adic multiple zeta values introduced in our previous paper via a comparison isomorphism between a de Rham fundamental torsor and a rigid fundamental torsor of the projective line minus three points and also discuss its Hodge andétale analogues. As an application we give a way to erase log poles of p-adic multiple polylogarithms and introduce overconvergent p-adic multiple polylogarithms which might be p-adic multiple analogue of Zagier's single-valued complex polylogarithms.

We introduce the method of desingularization of multi-variable multiple zetafunctions (of the generalized Euler-Zagier type), under the motivation of finding suitable rigorous meaning of the values of multiple zeta-functions at non-positive integer points. We reveal that multiple zeta-functions (which are known to be meromorphic in the whole space with infinitely many singular hyperplanes) turn to be entire on the whole space after taking the desingularization. The desingularized function is given by a suitable finite 'linear' combination of multiple zeta-functions with some arguments shifted. It is shown that specific combinations of Bernoulli numbers attain the special values at their non-positive integers of the desingularized ones. We also discuss twisted multiple zeta-functions, which can be continued to entire functions, and their special values at non-positive integer points can be explicitly calculated.

The MZV algebra is the graded algebra over Q generated by all multiple zeta values. The stable derivation algebra is a graded Lie algebra version of the GrothendieckTeichmüller group. We shall show that there is a canonical surjective Q-linear map from the graded dual vector space of the stable derivation algebra over 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 the Galois representation on the pro-l fundamental group of P 1 Q − {0, 1, ∞}.