Given an object over the algebraic closure Qbar of Q, there is often no reason for invariants of the corresponding holomorphic object to be preserved by the absolute Galois group Gal(Qbar/Q), and in general this is not true, although it is sometimes surprising to observe in practice. The case of covers of the projective line branched only over the points 0, 1, and infinity, through Belyi's theorem, leads to Grothendieck's dessins d'enfants program for understanding the absolute Galois group through its faithful action on such covers. This note is motivated by Catanese's question about a higher-dimensional analogue: does the absolute Galois group act faithfully on the deformation equivalence classes of smooth surfaces? (These equivalence classes are of course by definition the strongest deformation invariants.) We give a short proof of a weaker result: the absolute Galois group acts faithfully on the irreducible components of the moduli space of smooth surfaces (of general type, canonically polarized). Bauer, Catanese, and Grunewald have recently answered Catanese's original question using a different construction.
We study the absolute Galois group by looking for invariants and orbits of its faithful action on Grothendieck's dessins d'enfants. We define a class of functions called Belyi-extending maps, which we use to construct new Galois invariants of dessins from previously known invariants. Belyi-extending maps are the source of the ``new-type'' relations on the injection of the absolute Galois group into the Grothendieck-Teichmuller group. We make explicit how to get from a general Belyi-extending map to formula for its associated invariant which can be implemented in a computer algebra package. We …
We show that the Galois group $Gal(\bar{\Q} /\Q)$ operates faithfully on the set of connected components of the moduli spaces of surfaces of general type, and also that for each element $σ\in Gal(\bar{\Q} /\Q)$ different from the identity and from complex conjugation, there is a surface of general type such that $X$ and the Galois conjugate variety $X^σ$ have nonisomorphic fundamental groups. The result was announced by the second author at the Alghero Conference 'Topology of algebraic varieties' in september 2006. Before the present paper was actually written, we received a very interesting p…
We study the action of $\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ on the category of Belyi functions (finite, étale covers of $\mathbb{P}^1_{\overline{\mathbb{Q}}}\setminus \{0,1,\infty\}$). We describe a new combinatorial $\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$-invariant for whose monodromy cycle types above $0$ and $\infty$ are the same. We use a version of our invariant to prove that $\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ acts faithfully on the set of Belyi functions whose monodromy cycle types above 0 and $\infty$ are the same; the proof of this r…
We start discussing the group of automorphisms of the field of complex numbers, and describe, in the special case of polynomials with only two critical values, Grothendieck's program of 'Dessins d' enfants', aiming at giving representations of the absolute Galois group. We describe Chebycheff and Belyi polynomials, and other explicit examples. As an illustration, we briefly treat difference and Schur polynomials. Then we concentrate on a higher dimensional analogue of the triangle curves, namely, Beauville surfaces and varieties isogenous to a product. We describe their moduli spaces, and show…
The Mathieu group $M_{23}$ is a Galois group over $\mathbb{Q}$
Temple University（天普大学）；Carnegie Mellon University（卡内基梅隆大学）；University of Connecticut（康涅狄格大学）；Korea Institute for Advanced Study（韩国高等研究院）；Massachusetts Institute of Technology（麻省理工学院）；Colorado State University（科罗拉多州立大学）；California Institute of Technology（加州理工学院）
This paper initiates a research program that seeks to recover algebro-geometric Galois representations from combinatorial data. We study tropicalizations equipped with symmetries coming from the Galois-action present on the lattice of $1$-parameter subgroups inside ambient Galois-twisted toric varieties. Over a Henselian field, the resulting tropicalization maps become Galois-equivariant. We call their images Galois-equivariant tropicalizations, and use them to construct a large supply of Galois representations in the tropical cellular cohomology groups of Itenberg, Katzarkov, Mikhalkin, and Z…
Teichmueller curves are geodesic discs in Teichmueller space that project to algebraic curves $C$ in the moduli space $M_g$. Some Teichmüller curves can be considered as components of Hurwitz spaces. We show that the absolute Galois group $G_\QQ$ acts faithfully on the set of these embedded curves. We also compare the action of $G_\QQ$ on $π_1(C)$ with the one on $π_1(M_g)$ and obtain a relation in the Grothendieck-Teichmueller group, seemingly independent of the known ones.
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 t…
Let f: S' --> S be a finite and faithfully flat morphism of locally noetherian schemes of constant rank n and let G be a smooth, commutative and quasi-projective S-group scheme with connected fibers. For every r>0, let Res_{G}^{(r)}: H^{r}(S_{et},G)---> H^{r}(S'_{et},G) and Cores_{G}^{(r)}: H^{r}(S'_{et},G)---> H^{r}(S_{et},G) be, respectively, the restriction and corestriction maps in etale cohomology induced by f. For certain pairs (f, G), we construct maps α_{r}: Ker Cores_{G}^{(r)}---> Coker Res_{G}^{(r)} and β_{r}: Coker Res_{G}^{(r)}---> Ker Cores_{G}^{(r)} such that α_{r}oβ_{r}=β_{r}oα_…
We consider a rather special class of translation surfaces (called M-Origamis in this work) that are obtained from dessins by a construction introduced by Martin Möller. We give a new proof with a more combinatorial flavour of Möller's theorem that $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ acts faithfully on the Teichmüller curves of M-Origamis and extend his result by investigating the Galois action in greater detail. We determine the Strebel directions and corresponding cylinder decompositions of an M-Origami, as well as its Veech group, which contains the modular group $Γ(2)$ and is …
We introduce dessins d'enfants from the various existing points of view: As topological covering spaces, as surfaces with triangulations, and as algebraic curves with functions ramified over three points. We prove Belyi's theorem that such curves are defined over number fields, and define the action of the absolute Galois group Gal($\overline Q/Q$) on dessins d'enfants. We prove that several kinds of dessins d'enfants are defined over their field of moduli: regular dessins, dessins with no nontrivial automorphisms and dessins with one face. In the last part, we give two examples of regular des…
Let $X$ be a smooth projective geometrically irreducible curve over a perfect field $k$ of positive characteristic $p$. Suppose $G$ is a finite group acting faithfully on $X$ such that $G$ has non-trivial cyclic Sylow $p$-subgroups. We show that the decomposition of the space of holomorphic differentials of $X$ into a direct sum of indecomposable $k[G]$-modules is uniquely determined by the lower ramification groups and the fundamental characters of closed points of $X$ that are ramified in the cover $X\to X/G$. We apply our method to determine the $\mathrm{PSL}(2,\mathbb{F}_\ell)$-module stru…
Inoue constructed the first examples of smooth minimal complex surfaces of general type with $p_g=0$ and $K^2=7$.These surfaces are finite Galois covers of the $4$-nodal cubic surface with the Galois group, the Klein group $\mathbb{Z}_2\times \mathbb{Z}_2$. For such a surface $S$, the bicanonical map of $S$ has degree $2$ and it is composed with exactly one involution in the Galois group. The divisorial part of the fixed locus of this involution consists of two irreducible components:one is a genus $3$ curve with self-intersection number $0$ and the other is a genus $2$ curve with self-interse…
Given a relative faithfully flat pointed scheme over the spectrum of a discrete valuation ring $X \to S$ this paper is motivated by the study of the natural morphism from the fundamental group scheme of the generic fiber $X_η$ to the generic fiber of the fundamental group scheme of $X$. Given a torsor $T \to X_η$ under an affine group scheme $G$ over the generic fiber of $X$, we address the question to find a model of this torsor over $X$, focusing in particular on the case where $G$ is finite. We obtain partial answers to this question, showing for instance that, when $X$ is integral and regu…
The quotient space of a $K3$ surface by a finite group is an Enriques surface or a rational surface if it is smooth. Finite groups where the quotient space are Enriques surfaces are known. In this paper, by analyzing effective divisors on smooth rational surfaces, we will study finite groups which act faithfully on $K3$ surfaces such that the quotient space are smooth. In particular, we will completely determine effective divisors on Hirzebruch surfaces such that there is a finite Abelian cover from a $K3$ surface to a Hirzebrunch surface such that the branch divisor is that effective divisor.…
In this article we prove that the numerical Grothendieck group of every smooth proper dg category is invariant under primary field extensions, and also that the mod-n algebraic K-theory of every dg category is invariant under extensions of separably closed fields. As a byproduct, we obtain an extension of Suslin's rigidity theorem, as well as of Yagunov-Ostvaer's equivariant rigidity theorem, to singular varieties. Among other applications, we show that base-change along primary field extensions yields a faithfully flat morphism between noncommutative motivic Galois groups. Finally, along the …
School of Mathematical Sciences, Key Laboratory of Intelligent Computing and Applications (Tongji University), Ministry of Education（同济大学数学科学学院，智能计算与应用教育部重点实验室）
