In the present paper, we study a new kind of anabelian phenomenon concerning the smooth pointed stable curves in positive characteristic. It shows that the topology of moduli spaces of curves can be understood from the viewpoint of anabelian geometry. We formulate some new anabelian-geometric conjectures concerning tame fundamental groups of curves over algebraically closed fields of characteristic $p>0$ from the point of view of moduli spaces. The conjectures are generalized versions of the Weak Isom-version of the Grothendieck conjecture for curves over algebraically closed fields of characteristic $p>0$ which was formulated by Tamagawa. Moreover, we prove that the conjectures hold for certain points lying in the moduli space of curves of genus $0$.

Hyperelliptic mapping class groups are defined either as the centralizers of hyperelliptic involutions inside mapping class groups of oriented surfaces of finite type or as the inverse images of these centralizers by the natural epimorphisms between mapping class groups of surfaces with marked points. We study these groups in a systematic way. An application of this theory is a counterexample to the genus $2$ case of a conjecture by Putman and Wieland on virtual linear representations of mapping class groups. In the last section, we study profinite completions of hyperelliptic mapping class groups: we extend the congruence subgroup property to the general class of hyperelliptic mapping class groups introduced above and then determine the centralizers of multitwists and of open subgroups in their profinite completions.

We explain three methods for showing that the $p$-adic monodromy of a modular family of abelian varieties is ‘as large as possible', and illustrate them in the case of the ordinary locus of the moduli space of $g$-dimensional principally polarized abelian varieties over a field of characteristic $p$. The first method originated from Ribet's proof of the irreducibility of the Igusa tower for Hilbert modular varieties. The second and third methods both exploit Hecke correspondences near a hypersymmetric point, but in slightly different ways. The third method was inspired by work of Hida, plus a group theoretic argument for the maximality of $\ell$-adic monodromy with $\ell\neq p$.

Let $\mathbbm{k}$ be a (topological) field of characteristic 0. To any representation of a given Hopf algebra $\mathfrak{B}_n(\mathbbm{k})$, one can associate (using a Drinfeld associator) a representation $\widehat{\Phi}(\rho)$ of the braid group over the field $\mathbbm{k}((h))$ of Laurent series. We investigate the dependence on $\Phi$ of $\widehat{\Phi}(\rho)$ for a certain class of representations (so-called GT-rigid representations) and from this dependence deduce (continuous) projective representations of the Grothendieck–Teichmüller group $GT_1(\mathbbm{k})$; in particular, for $\mathbbm{k} = \mathbbm{Q}_l$ we obtain representations of the absolute Galois group of $\mathbbm{Q}(\mu_{l^{\infty}})$. In most situations, these projective representations can be decomposed into linear characters, as we do for the representations of the Iwahori–Hecke algebra of type A. In this case, moreover, we express $\widehat{\Phi}(\rho)$ when $\Phi$ is even and obtain unitary matrix models for the representations of the Iwahori–Hecke algebra. The representations of this algebra corresponding to hook diagrams have noteworthy properties under the action of $GT_1(\mathbbm{k})$.

We continue to study l-adic iterated integrals introduced in the first part. We shall show that the l-adic iterated integrals satisfy essentially the same functional equations as the classical complex iterated integrals. Next we are studying l-adic analogs of classical polylogarithms.

We continue to study l-adic iterated integrals introduced in the first part. We shall calculate explicitly l-adic logarithm and l-adic polylogarithms. Next we shall use these results to study Galois representations on the fundamental group of .

We are studying some aspects of the action of Galois groups on the torsor of paths connecting two (possibly tangential) points on a projec-tive line minus a finite number of points. We obtain objects which formally behave like classical iterated integrals and polylogarithms. We formulate an analog of Zagier conjecture for these l-adic analogs of iterated integrals and polylogarithms.