Galois group G_ℚ, singularity E₇, and moduli M₃

Summary

An Artin group is a generalization, defined for any graph, of the usual braid groups. The Artin group of E₇ Dynkin diagram has a natural surjection to the mapping class group of genus three curves, defined by sending the canonical generators to a set of Dehn twists, called Humphries generators.

We first show that this group homomorphism comes from a geometric morphism between moduli spaces by taking π₁.

Next, we describe the notion of tangential morphisms. Let U be an affine variety and D₁,..., D_l be normal crossing divisors which are principal. The tangential morphism from D₁∩ … ∩D_l to ∪ — (D₁ ∪ … ∪ D _l) is defined when we specify the generator of the defining ideal of D_i for each i. This yields a morphism between the algebraic fundamental groups. A tangential morphism from a point Spec k coincides with the notion of k–rational tangential base point.

Using this, we describe the action of the absolute Galois group on the profmite completion of Artin groups of type A_n, B_n, C_n, D_n, and E₇. (The first type gives the Braid group investigated before.) Description uses only the Belyi's coordinates, i.e. the action of the Galois group on the algebraic fundamental group of ℙ₁ minus 0,1,∞.

As a corollary, we give an explicit description of the Galois action on the profinite completion of the mapping class group of genus 3 curve.