GALOIS ACTIONS ON FUNDAMENTAL GROUPS OF CURVES AND THE CYCLE $C-C^{-}$

Abstract

Suppose that $K$ is a subfield of $\mathbb{C}$ for which the $\ell$-adic cyclotomic character has infinite image. Suppose that $C$ is a curve of genus $g\geq3$ 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 $(C^{\mathrm{an}},\xi)$ and the absolute Galois group $G_K$ of $K$ on the $\ell$-adic prounipotent fundamental group of $(C^{\mathrm{an}},\xi)$. A close relationship is established between

1. the image of the absolute Galois group of $K$ in the automorphism group of the $\ell$-adic unipotent fundamental group of $C$; and

2. the $\ell$-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 $(C^{\mathrm{an}},\xi)$ if and only if the two classes in (ii) are non-torsion and the Galois image in $\mathrm{GSp}_g(\mathbb{Q}_{\ell})$ 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