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

• 成果类型: Article
• 署名作者: Bauer, Ingrid; Catanese, Fabrizio; Grunewald, Fritz
• 署名单位: University of Bayreuth
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-014-0531-2
• 发表日期: 2015
• 页码: 859-888
• 关键词: surfaces; real

摘要

We use 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((Q) over bar /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 the moduli space of surfaces of general type (amending an incorrect proof in a previous arXiv version of the paper); and then, as a consequence, we obtain our main result: for each element sigma is an element of Gal((Q) over bar /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.