Faithful actions of the absolute Galois group on connected components of moduli spaces

TL;DR

This paper demonstrates that the absolute Galois group acts faithfully on the connected components of moduli spaces of certain algebraic surfaces, leading to examples where Galois conjugates have nonisomorphic fundamental groups.

Contribution

It establishes the faithful action of the Galois group on moduli space components and constructs explicit examples of surfaces with nonisomorphic Galois conjugates.

Findings

Galois group acts faithfully on connected components of moduli spaces.

Existence of surfaces with Galois conjugates having nonisomorphic fundamental groups.

Explicit examples using polynomials with two critical values.

Abstract

We give a canonical procedure associating to an algebraic number a first a hyperelliptic curve C_a, and then a triangle curve (D_a, G_a) obtained through the normal closure of an associated Belyi function. In this way we show that the absolute Galois group Gal(\bar{\Q} /\Q) acts faithfully on the set of isomorphism classes of marked triangle curves, and on the set of connected components of marked moduli spaces of surfaces isogenous to a higher product (these are the free quotients of a product C_1 x C_2 of curves of respective genera g_1, g_2 >= 2 by the action of a finite group G). We show then, using again the surfaces isogenous to a product, first that it acts faithfully on the set of connected components of moduli spaces of surfaces of general type (amending an incorrect proof in a previous ArXiv version of the paper); and then, as a consequence, we obtain that for every element \sigma \in \Gal(\bar{\Q} /\Q), not in the conjugacy class of complex conjugation, there exists a surface of general type X such that X and the Galois conjugate surface X^{\sigma} have nonisomorphic fundamental groups. Using polynomials with only two critical values, we can moreover exhibit infinitely many explicit examples of such a situation.