Galois actions on fundamental groups of curves and the cycle

Suppose that [formula omitted] is a subfield of [formula omitted] for which the [formula omitted] -adic cyclotomic character has infinite image. Suppose that [formula omitted] is a curve of genus [formula omitted] defined over [formula omitted], and that [formula omitted] is a [formula omitted] -rational point of [formula omitted]. This paper considers the relation between the actions of the mapping class group of the pointed topological curve [formula omitted] and the absolute Galois group [formula omitted] of [formula omitted] on the [formula omitted] -adic prounipotent fundamental group of [formula omitted]. A close relationship is established between the image of the absolute Galois group of [formula omitted] in the automorphism group of the [formula omitted] -adic unipotent fundamental group of [formula omitted]; andthe [formula omitted] -adic Galois cohomology classes associated to the algebraic [formula omitted] -cycle [formula omitted] in the Jacobian of [formula omitted], and to the algebraic [formula omitted] -cycle [formula omitted] in [formula omitted]. The main result asserts that the Zariski closure of (i) in the automorphism group contains the image of the mapping class group of [formula omitted] if and only if the two classes in (ii) are non-torsion and the Galois image in [formula omitted] is Zariski dense. The result is proved by specialization from the case of the universal curve. AMS 2000 Mathematics subject classification: Primary 11G30. Secondary 14H30; 12G05; 14C25; 14G32. © 2005, Cambridge University Press. All rights reserved.

Duke Scholars

Author Richard Hain Mathematics