Galois actions on fundamental groups of curves and the cycle C-C-

Suppose that K is a subfield of C for which the l-adic cyclotomic character has infinite image. Suppose that C is a curve of genus g >= 3 defined over K, and that xi is a K-rational point of C. This paper considers the relation between the actions of the mapping class group of the pointed topological curve (C-an, xi) and the absolute Galois group G(K) of K on the l-adic prounipotent fundamental group of (C-an, xi). A close relationship is established between (i) the image of the absolute Galois group of K in the automorphism group of the l-adic unipotent fundamental group of C; and (ii) the l-adic Galois cohomology classes associated to the algebraic 1-cycle C - C- in the Jacobian of C, and to the algebraic 0-cycle (2g - 2)xi - K-C in C. The main result asserts that the Zariski closure of (i) in the automorphism group contains the image of the mapping class group of (C-an, xi) if and only if the two classes in (ii) are non-torsion and the Galois image in GSp(g)(Q(l)) is Zariski dense. The result is proved by specialization from the case of the universal curve.