M. Rovinsky
Let $k$ be an algebraically closed field of characteristic zero, $F$ be an algebraically closed extension of $k$ of transcendence degree one, and $G$ be the group of automorphisms over $k$ of the field $F$. The purpose of this note is to calculate the group of continuous automorphisms of $G$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We show how to speed up the computation of isomorphisms of hyperelliptic curves by using covariants. We also obtain new theoretical and practical results concerning models of these curves over their field of moduli.
Let ${\mathcal L}/{\mathcal K}$ be a finite Galois extension and let $X$ be an affine algebraic variety defined over ${\mathcal L}$. Weil's Galois descent theorem provides necessary and sufficient conditions for $X$ to be definable over ${\mathcal K}$, that is, for the existence of an algebraic variety $Y$ defined over ${\mathcal K}$ together with a birational isomorphism $R:X \to Y$ defined over ${\mathcal L}$. Weil's proof does not provide a method to construct the birational isomorphism $R.$ The aim of this paper is to give an explicit construction of $R$.
The article provides a sufficient condition for a locally finite module over the absolute Galois group of a finite field F to satisfy the Riemann Hypothesis Analogue with respect to the projective line. The condition holds for all smooth irreducible projective curves of positive genus, defined over F. By construction of an explicit example we establish that the scope of our main result is larger than the class of the smooth irreducible projective varieties, defined over F.
In characteristic 0 there are essentially two approaches to the conjectural theory of mixed motives, one due to Nori and the other one due to, independently, Hanamura, Levine, and Voevodsky. Although these approaches are apriori quite different it is expected that ultimately they can be reduced to one another. In this article we provide some evidence for this belief by proving that their associated motivic Galois groups are canonically isomorphic.
We prove absolute purity for the rational motivic sphere spectrum. The main ingredient is the construction of an analogue of the Chern character, where algebraic K-theory is replaced by hermitian K-theory, and motivic cohomology by the plus and minus parts of the rational sphere spectrum. Another ingredient is absolute purity for hermitian K-theory.
In this article, we develop the geometry of canonical stratifications of the spaces $\mathcal{M}_{0,n}$ and prepare ground for studying the action of the Galois group $Gal (\overline{\mathbb{Q}} /\mathbb{Q})$ upon strata. We define and introduce a version of a gravity operad constructed for a class of moduli spaces $\mathcal{M}_{0,n}$, equipped with a hidden holomorphic involution. This additional symmetry is associated to a split quaternionic structure. We introduce a categorical framework to present this object. Interaction between the geometry, physics and the arithmetics are discussed. An …
We show that the triviality of the differential Galois cohomologies over a partial differential field K of a linear differential algebraic group is equivalent to K being algebraically, Picard-Vessiot, and linearly differentially closed. This former is also known to be equivalent to the uniqueness up to an isomorphism of a Picard-Vessiot extension of a linear differential equation with parameters.
We prove that certain fields have the property that their absolute Galois groups are free as profinite groups: the function field of a real curve with no real points; the maximal abelian extension of a 2-variable Laurent series field over a separably closed field; and the maximal abelian extension of the function field of a curve over a finite field. These results are related to generalizations of Shafarevich's conjecture.
We present possible extensions of the quantum statistical mechanical formulation of class field theory to the non-abelian case, based on the action of the absolute Galois group on Grothendieck's dessins d'enfant, the embedding in the Grothendieck-Teichmüller group, and the Drinfeld-Ihara involution.
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…
Let $G$ denote a linear algebraic group over $\mathbf{Q}$ and $K$ and $L$ two number fields. Assume that there is a group isomorphism of points on $G$ over the finite adeles of $K$ and $L$, respectively. We establish conditions on the group $G$, related to the structure of its Borel groups, under which $K$ and $L$ have isomorphic adele rings. Under these conditions, if $K$ or $L$ is a Galois extension of $\mathbf{Q}$ and $G(\mathbf{A}_{K,f})$ and $G(\mathbf{A}_{L,f})$ are isomorphic, then $K$ and $L$ are isomorphic as fields. We use this result to show that if for two number fields $K$ and $L$…
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…
Our aim of this and subsequent papers is to enlighten (a part of, presumably) arithmetic structures of knots. This paper introduces a notion of profinite knots which extends topological knots and shows its various basic properties. Particularly an action of the absolute Galois group of the rational number field on profinite knots is rigorously established, which is attained by our extending the action of Drinfeld's Grothendieck-Teichmüller group on profinite braid groups into on profinite knots.
Geometric properties of homomorphisms between the absolute Galois groups of mixed-characteristic complete discrete valuation fields with perfect residue fields
本文将 Mochizuki 与 Hoshi 关于 $p$-进局部域绝对伽罗瓦群几何同态的判别条件推广到具有完美剩余域的一般混合特征完备离散赋值域，并得到弱 Isom 远阿贝尔结果。
The existence of a good theory of Thom isomorphisms in some rational category of mixed Tate motives would permit a nice interpolation between ideas of Kontsevich on deformation quantization, and ideas of Connes and Kreimer on a Galois theory of renormalization, mediated by Deligne's ideas on motivic Galois groups.
Let $G$ be a finite group, $Λ$ an absolutely irreducible $\Z[G]$-module and $w$ a weight of $Λ$. To any Galois covering with group $G$ we associate two correspondences, the Schur and the Kanev correspondence. We work out their relation and compute their invariants. Using this, we give some new examples of Prym-Tyurin varieties.
In this article we consider outer Galois actions on a free profinite group of rank two, induced by the étale fundamental group of a projective line minus three points or of a pointed elliptic curve over a number field. Under mild technical assumptions their respective images uniquely determine the curves and the number fields.
We review some ideas of Grothendieck and others on actions of the absolute Galois group Γ Q of Q (the automorphism group of the tower of finite extensions of Q), related to the geometry and topology of surfaces (mapping class groups, Teichm{ü}ller spaces and moduli spaces of Riemann surfaces). Grothendieck's motivation came in part from his desire to understand the absolute Galois group. But he was also interested in Thurston's work on surfaces, and he expressed this in his Esquisse d'un programme, his R{é}coltes et semailles and on other occasions. He introduced the notions of dessin d'enfant…
We study the relationship between the local and global Galois theory of function fields over a complete discretely valued field. We give necessary and sufficient conditions for local separable extensions to descend to global extensions, and for the local absolute Galois group to inject into the global absolute Galois group. As an application we obtain a local-global principle for the index of a variety over such a function field. In this context we also study algebraic versions of van Kampen's theorem, describing the global absolute Galois group as a pushout of local absolute Galois groups.
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…
A main problem in Galois theory is to characterize the fields with a given absolute Galois group. We apply a K-theoretic method for constructing valuations to study this problem in various situations. As a first application we obtain an algebraic proof of the 0-dimensional case of Grothendieck's anabelian conjecture (proven by Pop), which says that finitely generated infinite fields are determined up to purely inseparable extensions by their absolute Galois groups. As a second application (which is a joint work with Fesenko) we analyze the arithmetic structure of fields with the same absolute …
Let $F$ be a field of characteristic $0$ containing all roots of unity. We construct a functorial compact Hausdorff space $X_F$ whose profinite fundamental group agrees with the absolute Galois group of $F$, i.e. the category of finite covering spaces of $X_F$ is equivalent to the category of finite extensions of $F$. The construction is based on the ring of rational Witt vectors of $F$. In the case of the cyclotomic extension of $\mathbb{Q}$, the classical fundamental group of $X_F$ is a (proper) dense subgroup of the absolute Galois group of $F$. We also discuss a variant of this constructio…
We prove a generalization of Shafarevich's Conjecture for fields of Laurent series in two variables over an arbitrary field. While not projective, the absolute Galois group of such a field is shown to be semi-free. We also show that the function field of a smooth projective curve over a large field has semi-free absolute Galois group. In the first edition of this paper it was shown that these groups are quasi-free, which is weaker.
We construct orbits of the absolute Galois group, of explicit unbounded size, consisting of surfaces with mutually non-isomorphic fundamental groups. These are Beauville surfaces with Beauville group PGL_2(p).
For a character of the absolute Galois group of a complete discrete valuation field, we define a lifting of the refined Swan conductor, using higher dimensional class field theory.
This is a revision of the paper that was previously entitled "Weighted Completion of Galois Groups and Some Conjectures of Deligne". Fix a prime number $ł$. We prove a conjecture stated by Ihara, which he attributes to Deligne, about the action of the absolute Galois group on the pro-$ł$ completion of the fundamental group of the thrice punctured projective line. Similar techniques are also used to prove part of a conjecture of Goncharov, also about the action of the absolute Galois group on the fundamental group of the thrice punctured projective line. The main technical tool is the weighted …
In this paper we develop a theory of class invariants associated to $p$-adic representations of absolute Galois groups of number fields. Our main tool for doing this involves a new way of describing certain Selmer groups attached to $p$-adic representations in terms of resolvends associated to torsors of finite group schemes.
Let k be a field of characteristic 0 and let X be a smooth geometrically integral k-variety. In our previous paper we defined the extended Picard complex UPic(X) as a certain complex of Galois modules in degrees 0 and 1. We computed the isomorphism class of UPic(G) in the derived category of Galois modules for a connected linear k-group G. In this paper we assume that X is a homogeneous space of a connected linear k-group G with geometric stabilizer H. We compute the isomorphism class of UPic(X) in the derived category of Galois modules in terms of the character groups of G and H. The proof is…
Let $k$ be a totally real field, and let $A/k$ be an absolutely irreducible, polarized Abelian variety of odd, prime dimension whose endomorphisms are all defined over $k$. Then the only strictly compatible families of abstract, absolutely irreducible representations of $\gal(\ov{k}/k)$ coming from $A$ are tensor products of Tate twists of symmetric powers of two-dimensional $λ$-adic representations plus field automorphisms. The main ingredients of the proofs are the work of Borel and Tits on the `abstract' homomorphisms of almost simple algebraic groups, plus the work of Shimura on the fields…
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 …
The fundamental group of Fermat and generalized Fermat curves is computed. These curves are Galois ramified covers of the projective line with abelian Galois groups $H$. We provide a unified study of the action of both cover Galois group $H$ and the absolute Galois group $\mathrm{Gal}(\bar{\Q}/\Q)$ on the pro-$\ell$ homology of the curves in study. Also the relation to the pro-$\ell$ Burau representation is investigated.
The main result of the paper is a reciprocity law which proves that compatible systems of semisimple, abelian mod $p$ representations (of arbitrary dimension) of absolute Galois groups of number fields, arise from Hecke characters. In the last section of the paper analogs for Galois groups of function fields of these results are explored, and a question is raised whose answer will require developments in transcendence theory in characteristic $p$.
Past studies of the Brauer group of a scheme tells us the importance of the interrelationship among Brauer groups of its finite étale coverings. In this paper, we consider these groups simultaneously, and construct an integrated object "Brauer-Mackey functor". We realize this as a {\it cohomological Mackey functor} on the Galois category of finite étale coverings. For any finite étale covering of schemes, we can associate two homomorphisms for Brauer groups, namely the pull-back and the norm map. These homomorphisms make Brauer groups into a bivariant functor ($=$ Mackey functor) on the Galois…
The object of this paper is the study of a class of dessins d'enfants, the so-called diameter four trees. These objects, first introduced by G. Shabat, can be considered as the simplest non trivial example of etale covers of the projective line minus three points. Their arithmetic properties are still mysterious, and their study can inspire the understanding of more general situations. Here, the main interest is devoted to the action of the absolute Galois group on these (isomorphism classes of) coverings. In particular, in many cases, we are able to distinguish Galois orbits and to describe t…
In previous work, the authors constructed and studied a lift of the Galois correspondence to stable homotopy categories. In particular, if $L/k$ is a finite Galois extension of fields with Galois group $G$, there is a functor $c_{L/k}^*$ from the $G$-equivariant stable homotopy category to the stable motivic homotopy category over $k$ such that $c_{L/k}^*(G/H_+) = Spec(L^H)_+$. We proved that when $k$ is a real closed field and $L=k[i]$, the restriction of $c_{L/k}^*$ to the $η$-complete subcategory is full and faithful. Here we "uncomplete" this theorem so that it applies to $c_{L/k}^*$ itsel…
We establish the faithfulness of a geometric action of the absolute Galois group of the rationals that can be defined on the discriminantal variety associated to a finite complex reflection group, and review some possible connections with the profinite Grothendieck-Teichmüller group.
We prove the equality of two non-logarithmic ramification filtrations defined by Matsuda and Abbes-Saito for the abelianized absolute Galois group of a complete discrete valuation field in positive characteristic. We also compute the refined Swan conductor and the characteristic form of a character of the fundamental group of a smooth separated scheme over a perfect field of positive characteristic by using sheaves of Witt vectors.
Resolution of non-singularities and the absolute anabelian conjecture
本文针对p进局部域上双曲曲线的绝对远阿贝尔猜想，将S. Mochizuki针对准Belyi型曲线的证明结果推广到包含双曲Mumford曲线在内的更广泛曲线族。
From its early beginnings up to nowadays, algebraic number theory has evolved in symbiosis with Galois theory: indeed, one could hold that it consists in the very study of the absolute Galois group of the field of rational numbers. Nothing like that can be said of transcendental number theory. Nevertheless, couldn't one associate conjugates and a Galois group to transcendental numbers such as $π$? Beyond, can't one envision an appropriate Galois theory in the field of transcendental number theory? In which role? The aim of this text is to indicate what Grothendieck's theory of motives has to s…
Let $n \geq 6$ be an integer. We prove that the number of number fields with Galois group $A_n$ and absolute discriminant at most $X$ is asymptotically at least $X^{1/8 + O(1/n)}$. For $n \geq 8$ this improves upon the previously best known lower bound of $X^{(1 - \frac{2}{n!})/(4n - 4) - ε}$, due to Pierce, Turnage-Butterbaugh, and Wood.
In this paper we characterise the action of the absolute Galois group on the geometric finite cyclic groups without étale factorization of stack inertia of the profinite geometric fundamental group of moduli spaces of marked curves. As a complementary result, we give the same action on prime order profinite elements in genus 2.
Let $C$ be a complex algebraic curve uniformised by a Fuchsian group $Γ$. In the first part of this paper we identify the automorphism group of the solenoid associated with $Γ$ with the Belyaev completion of its commensurator $\mathrm{Comm}(Γ)$ and we use this identification to show that the isomorphism class of this completion is an invariant of the natural Galois action of $\mathrm{Gal}(\mathbb C/\mathbb Q)$ on algebraic curves. In turn this fact yields a proof of the Galois invariance of the arithmeticity of $Γ$ independent of Kazhhdan's. In the second part we focus on the case in which $Γ$…
In this paper the transcendental Galois extensions of a field will be introduced as counterparts to algebraic Galois ones. There exist several types of transcendental Galois extensions of a given field, from the weakest one to the strongest one, such as Galois, tame Galois, strong Galois, and absolute Galois. The four Galois extensions are distinct from each other in general but coincide with each other for cases of algebraic extensions. The transcendental Galois extensions arise from higher relative dimensional Galois covers of arithmetic schemes. In the paper we will obtain several propertie…
Motivic stable stems and Galois approximations of cellular motivic categories
本文证明小上同调维数域上的胞腔动机谱范畴可由绝对 Galois 群刻画，并据此确定几乎所有域在特征外的动机稳定茎。
We consider Galois/monodromy groups arising in computer vision applications, with a view towards building more efficient polynomial solvers. The Galois/monodromy group allows us to decide when a given problem decomposes into algebraic subproblems, and whether or not it has any symmetries. Tools from numerical algebraic geometry and computational group theory allow us to apply this framework to classical and novel reconstruction problems. We consider three classical cases--3-point absolute pose, 5-point relative pose, and 4-point homography estimation for calibrated cameras--where the decomposi…
For an odd prime $p$ satisfying Vandiver's conjecture, we give explicit formulae for the action of the absolute Galois group $G_{\mathbb{Q}(ζ_p)}$ on the homology of the degree $p$ Fermat curve, building on work of Anderson. Further, we study the invariants and the first Galois cohomology group which are associated with obstructions to rational points on the Fermat curve.
We finish the proof of the conjecture of F. Bogomolov and F. Pop: Let $F_{1}$ and $F_{2}$ be fields finitely-generated and of transcendence degree $\geq 2$ over $k_{1}$ and $k_{2}$, respectively, where $k_{1}$ is either $\bar{\mathbb{Q}}$ or $\bar{\mathbb{F}}_{p}$, and $k_{2}$ is algebraically closed. We denote by $G_{F_1}$ and $G_{F_2}$ their respective absolute Galois groups. Then the canonical map $φ_{F_{1}, F_{2}}: \Isom^i(F_1, F_2)\rightarrow \Isom^{\Out}_{\cont}(G_{F_2}, G_{F_1})$ from the isomorphisms, up to Frobenius twists, of the inseparable closures of $F_1$ and $F_2$ to continuous …
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。