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In this paper we extend the techniques and the basic results of the classical Galois theory of the fields extension Open image in new window is an algebraic closure of Q, to the algebras extension ℝ ⊂ ℂsym(G), where this last is the ℝ— algebra of all the continuous symmetric functions ƒ defined on the absolute Galois group Open image in new window with values in ℂ.

In this paper we continue to study the spectral norms and their completions ([4]) in the case of the algebraic closure of ℚ in ℂ. Let be the completion of relative to the spectral norm. We prove that can be identified with the R-subalgebra of all symmetric functions of C(G), where C(G) denotes the ℂ-Banach algebra of all continuous functions defined on the absolute Galois group G = Gal. We prove that any compact, closed to conjugation subset of ℂ is the pseudo-orbit of a suitable element of . We also prove that the topological closure of any algebraic number field in is of the form with x in .

Fix a prime number l. We prove a conjecture stated by Ihara, which he attributes to Deligne, about the action of the absolute Galois group on the pro-l 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 completion of a profinite group with respect to a reductive representation (and other appropriate data).

A central pair over a field k of characteristic 0 consists of a finite Abelian group which is equipped with a central 2-cocycle with values in the multiplicative group k* of k. In this paper we use specific central pairs to construct a class of projective representations of the absolute Galois group Gk of k and if k is a number field we investigate the liftings of these projective representations to linear representations of Gk. In particular we relate these linear representations to automorphic representations. It turns out that some of these automorphic representations correspond to certain indefinite modular forms already constructed by E. Hecke.

Let C be a smooth projective absolutely irreducible curve over a finite field F q , F its function field and A the subring of F of functions which are regular outside a fixed point ∞ of C. For every place l of A, we denote the completion of A at l by A l . In [Pi2], Pink proved the Mumford–Tate conjecture for Drinfeld modules. Let φ be a Drinfeld module of rank r defined over a finitely generated field K containing F. For every place l of A, we denote by Γl the image of the representation ρ l : Γ K → Aut A l (T l (φ))≅ GL r ( A l ) of the absolute Galois group ΓK of K on the Tate module Tl(φ). The Mumford–Tate conjecture states that some subgroup of finite index of Γl is open inside a prescribed algebraic subgroup Hl of GL r, A l . In fact, he proves this result for representations of ΓK on a finite product of distinct Tate modules. A τ-module over AK is a projective A⊗K-module of finite type endowed with a 1⊗ϕ-semilinear injective homomorphism τ, where ϕ denotes the Frobenius morphism on K. Such a τ-module is said to have dimension 1, if the K-vector space M/K·τ(M) has dimension 1. Drinfeld showed how to associate, in a functorial way, to every Drinfeld module over K a τ-module M(φ) over AK of dimension 1, called the t-motive of φ. In this paper, we generalize Pink's theorem to representations of Tate modules Tl(M) of τ-modules M of dimension 1 over AK. The key result can be formulated as follows: if we suppose that EndK(M)=A, then for every finite place l of F, the image Γl of the representation ρ l : Γ K → Aut A l (T l (M)) is open in GL r ( A l ) , where r denotes the rank of M. As already demonstrated in the proof of the Tate conjecture for Drinfeld modules by Taguchi and Tamagawa, the relation between τ-modules over AK and Galois representations with coefficients in A l is more natural and direct than that between Drinfeld modules (or, more generally, abelian t-modules) and their Tate modules. By this philosophy, the assumption that a τ-module M is pure, or, equivalently, is the t-motive of a Drinfeld module φ, should be and, indeed, is superfluous in proving a qualitative statement like the above Mumford–Tate conjecture. The main result of this paper is the corresponding statement for τ-modules of dimension 1, i.e. whose maximal exterior power is the t-motive of a Drinfeld module. We stick to the basic outline of Pink's proof: reducing ourselves to the case where the absolute endomorphism ring of M equals A, we first show that Γl is Zariski dense in GL r, A l and we use his results on compact Zariski dense subgroups of algebraic groups to conclude that Γl if open in GL r ( A l ) . After a reduction to the case where K has transcendence degree 1 over F q , the essential tools we will use are the Tate and semisimplicity theorem for simple τ-modules, Serre's Frobenius tori and the tori given by inertia at places of good reduction for M.

We prove the finiteness of Selmer groups attached to lifts of certain 2-dimensional mod p representations of the absolute Galois group of Q. The mod p representation can be either even or odd. The lifts considered are the ones that were proven to exist by the seond named author.

It is shown that the natural action of the absolute Galois group on the ideals defining principal congruence subgroups of certain nonarithmetic Fuchsian triangle groups is compatible with its action on the algebraic curves that these congruence groups uniformize.

Let K be a number field which contains Open image in new window the absolute Galois group of K. Let Open image in new window be the completion of Open image in new window relative to the spectral norm Open image in new window. A trace series and a trace function are associated to an element Open image in new window. A measure preserving function H:G{inK}→[0,1] is constructed and a Fourier type theory is considered for elements of Open image in new window.

We prove that some new infinite families of odd two dimensional icosahedral representations of the absolute Galois group of [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /] are modular and hence satsify the Artin conjecture. We also give an account of work of Ramakrishna on lifting mod l Galois representations to characteristic zero.

Neukirch-Uchida gave a certain characterization of number fields using absolute Galois groups. In this article, a new characterization of some infinite extension fields using absolute Galois groups is given.

Langlands has conjectured the existence of a universal group, an extension of the absolute Galois group, which would play a fundamental role in the classification of automorphic representations. We shall describe a possible candidate for this group. We shall also describe a possible candidate for the complexification of Grothendieck's motivic Galois group.

We prove the following result: Theorem.Let K be a countable Hilbertian field, S a finite set of local primes of K, and e ≥ 0 an integer. Then, for almost allσ ∈ G (K)e, the field Ks [σ] ∩ Ktot,Sis PSC. Here a local prime is an equivalent class 𝔭 of absolute values of K whose completion is a local field, 𝔭. Then K𝔭 = Ks ∩ 𝔭 and Ktot,S = ∩𝔭 ∈ S∩σ ∈ G(K)Kσ𝔭. G(K) stands for the absolute Galois group of K. For each σ = (σ1, …, σe ) ∈ G(K)e we denote the fixed field of σ1, …, σe in Ks by Ks(σ). The maximal Galois extension of K in Ks(σ) is Ks[σ]. Finally “almost all” means “for all but a set of Haar measure zero”.

Classically the ramification filtration of the Galois group of a complete discrete valuation field is defined in the case where the residue field is perfect. In this paper, we define without any assumption on the residue field, two ramification filtrations and study some of their properties., Extended and corrected version

DAHA-Jones polynomials of torus knots T(r, s) are studied systematically for reduced root systems and in the case of $$C^\vee C_1$$ . We prove the polynomiality and evaluation conjectures from the author’s previous paper on torus knots and extend the theory by the color exchange and further symmetries. The DAHA-Jones polynomials for $$C^\vee C_1$$ depend on five parameters. Their surprising connection to the DAHA-superpolynomials (type A) for the knots $$T(2p+1,2)$$ is obtained, a remarkable combination of the color exchange conditions and the author’s duality conjecture (justified by Gorsky and Negut). The uncolored DAHA-superpolynomials of torus knots are expected to coincide with the Khovanov–Rozansky stable polynomials and the superpolynomials defined via rational DAHA and/or in terms of certain Hilbert schemes. We end the paper with certain arithmetic counterparts of DAHA-Jones polynomials for the absolute Galois group in the case of $$C^\vee C_1$$ , developing the author’s previous results for $$A_1$$ .

We study modulo p reduction of compatible systems of p-adic representations of the absolute Galois group of Q , arising from an algebraic automorphic representation. In particular, we prove that there is a field extension of Q with the Galois group G 2 ( p ) , ramified at 5 and p only, for a set of primes p of density one.

We show that a two dimensional $\ell $-adic representation of the absolute Galois group of a number field which is locally potentially equivalent to a $GL(2)$-$\ell$-adic representation $\rho$ at a set of places of $K$ of positive upper density is potentially equivalent to $\rho$. For an elliptic curver \( E \) defined over a number field \( K \) and a finite place \( v \) of \( K \) of good reduction for \( E \), let \( F(E,v) \) denote the Frobenius field of \( E \) at \( v \), given by the splitting field of the characteristic polynomial of the Frobenius automorphism at \( v \) acting on the Tate module of \( E \). As an application, suppose \( E_1 \) and \( E_2 \) defined over a number field \( K \), with at least one of them without complex multiplication. We prove that the set of places \( v \) of \( K \) of good reduction such that the corresponding Frobenius fields are equal has positive upper density if and only if \( E_1 \) and \( E_2 \) are isogenous over some extension of \( K \). We show that for an elliptic curve \( E \) defined over a number field \( K \), the set of finite places of \( K \) such that the Frobenius field \( F(E, v) \) at $v$ equals a fixed imaginary quadratic field \( F \) has positive upper density if and only if \( E \) has complex multiplication by \( F \)., Comment: 15 pages. This is a revised, corrected and expanded version. A new author has been added. There are also changes to the title and the abstract

Let p denote an odd prime. For all p-admissible conductors c over a quadratic number field \(K=\mathbb{Q}(\sqrt{d})\), p-ring spaces \(V_p(c)\) modulo c are introduced by defining a morphism \(\psi:\,f\mapsto V_p(f)\) from the divisor lattice \(\mathbb{N}\) of positive integers to the lattice S of subspaces of the direct product \(V_p\) of the p-elementary class group \(C/C^p\) and unit group \(U/U^p\) of K. Their properties admit an exact count of all extension fields N over K, having the dihedral group of order 2p as absolute Galois group \(Gal(N | \mathbb{Q})\) and sharing a common discriminant \(d_N\) and conductor c over K. The number \(m_p(d,c)\) of these extensions is given by a formula in terms of positions of p-ring spaces in S, whose complexity increases with the dimension of the vector space \(V_p\) over the finite field \(\mathbb{F}_p\), called the modified p-class rank \(\sigma_p\) of K. Up to now, explicit multiplicity formulas for discriminants were known for quadratic fields with \(0\le\sigma_p\le 1\) only. Here, the results are extended to \(\sigma_p=2\), underpinned by concrete numerical examples., Comment: 27 pages, 6 figures, 11 tables, presented at the 122nd Annual DMV Meeting 2012, University of the Saarland, Sarrebruck, FRG, 18 September 2012

The starting point of Fontaine theory is the possibility of translating the study of a p-adic representation of the absolute Galois group of a finite extension K of Qp into the investigation of a (φ, Γ)-module. This is done by decomposing the Galois group along a totally ramified extension of K, via the theory of the field of norms: the extension used is obtained by means of the cyclotomic tower which, in turn, is associated to the multiplicative Lubin-Tate group. It is known that one can insert different Lubin-Tate groups into the "Fontaine theory" machine to obtain equivalences with new categories of (φ, Γ)-modules (here φ may be iterated). This article uses only (φ, Γ)-theoretical terms to compare the different (φ, Γ) modules arising from various Lubin-Tate groups.

— Given a global fieldK and a rational function φ ∈ K(x), one may take pre-images of 0 under successive iterates of φ, and thus obtain an infinite rooted tree T∞ by assigning edges according to the action of φ. The absolute Galois group of K acts on T∞ by tree automorphisms, giving a subgroup G∞(φ) of the group Aut(T∞) of all tree automorphisms. Beginning in the 1980s with work of Odoni, and developing especially over the past decade, a significant body of work has emerged on the size and structure of this Galois representation. These inquiries arose in part because knowledge of G∞(φ) allows one to prove density results on the set of primes of K that divide at least one element of a given orbit of φ. Following an overview of the history of the subject and two of its fundamental questions, we survey in Section 2 cases where G∞(φ) is known to have finite index in Aut(T∞). While it is tempting to conjecture that such behavior should hold in general, we exhibit in Section 3 four classes of rational functions where it does not, illustrating the difficulties in formulating the proper conjecture. Fortunately, one can achieve the aforementioned density results with comparatively little information about G∞(φ), thanks in part to a surprising application of probability theory, as we discuss in Section 4. Underlying all of this analysis are results on the factorization into irreducibles of the numerators of iterates of φ, which we survey briefly in Section 5. We find that for each of these matters, the arithmetic of the forward orbits of the critical points of φ proves decisive, just as the topology of these orbits is decisive in complex dynamics. 2010 Mathematics Subject Classification. — 11R32, 37P15, 11F80.

This is a survey of recent progress on Beauville surfaces, concentrating almost entirely on the group-theoretic and combinatorial problems associated with them. A Beauville surface \(\mathcal{S}\) is a complex surface formed from two orientably regular hypermaps of genus at least 2 (viewed as compact Riemann surfaces and hence as algebraic curves), with the same automorphism group G acting freely on their product. The following questions are discussed: Which groups G (called Beauville groups) have this property? What can be said about the automorphism group and the fundamental group of \(\mathcal{S}\)? Beauville surfaces are defined (as algebraic varieties) over the field \(\overline{\mathbb{Q}}\) of algebraic numbers, so how does the absolute Galois group \(\mathrm{Gal}\,\overline{\mathbb{Q}}/\mathbb{Q}\) act on them?

Let $F$ be a non-archimedean local field and $G={\bf{G}}(F)$ the group of $F$-rational points of a connected reductive $F$-group. Then we have the Langlands classification of complex irreducible admissible representations $\pi$ of $G$ in terms of triples $(P,\sigma,\nu)$ where $P\subset G$ is a standard $F$-parabolic subgroup, $\sigma$ is an irreducible tempered representation of the standard Levi-group $M_P$ and $\nu \in \Bbb{R}\otimes X^*(M_P)$ is regular with respect to $P.$ Now we consider Langlands' L-parameters $[\phi]$ which conjecturally will serve as a system of parameters for the representations $\pi$ and which are (roughly speaking) equivalence classes of representations $\phi$ of the absolute Galois group $\Gamma=\text{Gal}(\overline{F}|F)$ with image in Langlands' L-group $\,^LG$, and we classify the possible $[\phi]$ in terms of triples $(P,[\,^t\phi],\nu)$ where the data $(P,\nu)$ are the same as in the Langlands classification of representations and where $[\,^t\phi]$ is a tempered L-parameter of $M_P.$, Comment: 39 pages

The action of the absolute Galois group $\text{Gal}(K^{\text{ksep}}/K)$ of a global field $K$ on a tree $T(\phi, \alpha)$ of iterated preimages of $\alpha \in \mathbb{P}^1(K)$ under $\phi \in K(x)$ with $\text{deg}(\phi) \geq 2$ induces a homomorphism $\rho: \text{Gal}(K^{\text{ksep}}/K) \to \text{Aut}(T(\phi, \alpha))$, which is called an arboreal Galois representation. In this paper, we address a number of questions posed by Jones and Manes about the size of the group $G(\phi,\alpha) := \text{im} \rho = \underset{\leftarrow n}\lim\text{Gal}(K(\phi^{-n}(\alpha))/K)$. Specifically, we consider two cases for the pair $(\phi, \alpha)$: (1) $\phi$ is such that the sequence $\{a_n\}$ defined by $a_0 = \alpha$ and $a_n = \phi(a_{n-1})$ is periodic, and (2) $\phi$ commutes with a nontrivial Mobius transformation that fixes $\alpha$. In the first case, we resolve a question posed by Jones about the size of $G(\phi, \alpha)$, and taking $K = \mathbb{Q}$, we describe the Galois groups of iterates of polynomials $\phi \in \mathbb{Z}[x]$ that have the form $\phi(x) = x^2 + kx$ or $\phi(x) = x^2 - (k+1)x + k$. When $K = \mathbb{Q}$ and $\phi \in \mathbb{Z}[x]$, arboreal Galois representations are a useful tool for studying the arithmetic dynamics of $\phi$. In the case of $\phi(x) = x^2 + kx$ for $k \in \mathbb{Z}$, we employ a result of Jones regarding the size of the group $G(\psi, 0)$, where $\psi(x) = x^2 - kx + k$, to obtain a zero-density result for primes dividing terms of the sequence $\{a_n\}$ defined by $a_0 \in \mathbb{Z}$ and $a_n = \phi(a_{n-1})$. In the second case, we resolve a conjecture of Jones about the size of a certain subgroup $C(\phi, \alpha) \subset \text{Aut}(T(\phi, \alpha))$ that contains $G(\phi, \alpha)$, and we present progress toward the proof of a conjecture of Jones and Manes concerning the size of $G(\phi, \alpha)$ as a subgroup of $C(\phi, \alpha)$., Comment: 21 pages

We present in this note a definition of zeta function of the field \(\mathbb{Q}\) which incorporates all p-adic L-functions of Kubota-Leopoldt for all p and also so called Soule classes of the field \(\mathbb{Q}\). This zeta function is a measure, which we construct using the action of the absolute Galois group \(G_{\mathbb{Q}}\) on fundamental groups.

We prove the Breuil–Mézard conjecture for two-dimensional potentially Barsotti–Tate representations of the absolute Galois group $G_{K}$ , $K$ a finite extension of $\mathbb{Q}_{p}$ , for any $p>2$ (up to the question of determining precise values for the multiplicities that occur). In the case that $K/\mathbb{Q}_{p}$ is unramified, we also determine most of the multiplicities. We then apply these results to the weight part of Serre’s conjecture, proving a variety of results including the Buzzard–Diamond–Jarvis conjecture.

We construct and study fields F with the property that F has infinitely many extensions of some fixed degree, but E*/(E*)^n is finite for every finite extension E of F and every n>0., 9 pages

LetK be a perfect pseudo-algebraically closed field and letF be an extension ofK of relative transcendence degree 1. It is shown that the restriction map Res: Br(F)→Πp Br(Fph) is injective, where p ranges over all non-trivialK-places ofF, andFph is the corresponding henselization. Conversely, the validity of this Hasse principle for all such extensionsF implies a weaker version of pseudo-algebraic closedness. As an application we determine the finitely generated pro-p closed subgroups of the absolute Galois group ofK(t).

Let p ≥ 5 be a prime, and℘ a place ofQ above it.We denote byGQ the absolute Galois group Gal(Q/Q). For a finite field k of characteristic p, an absolutely irreducible representation ρ : GQ → GL2(k) is said to arise from a newform of weight k and level a positive integer M if ρ is isomorphic to the reduction mod ℘ of the ℘-adic representation associated to a newform of weight k and level M . Note that then it is known that ρ also arises from a newform of weight 2 and level dividingMp2. Thus we assume without loss of generality that ρ arises from a newform of weight 2 and level a positive integer N . The question of “raising” the level of ρ, i.e., determining when a ρ arising from a newform of weight 2 and level N as above also arises from a newform of weight 2 and levelN ′, whereN dividesN ′, has been widely studied by numerous mathematicians (cf. [10,8,5,2]). In the first paper that considers this kind of question [10], as well as in most of the papers dealing with similar questions (e.g., [2]), the cases considered correspond to N ′/N being coprime to p. Some discussion for the case where

We show that the absolute Galois group acts faithfully on the set of Hubbard trees. Hubbard trees are finite planar trees, equipped with self-maps, which classify postcritically finite polynomials as holomorphic dynamical systems on the complex plane. We establish an explicit relationship between certain Hubbard trees and the trees known as “dessins d'enfants” introduced by Grothendieck.

We examine which representations of the absolute Galois group of a field of finite characteristic with image over a finite field of the same characteristic may be constructed by the Galois group’s action on the division points of an appropriate Drinfeld module.

We prove that the functor from the category of all complete discrete valuation fields with finite residue fields of characteristic ≠2 to the category of profinite filtered groups given by taking the Galois group of corresponding field together with its filtration by higher ramification subgroups is fully faithful. If [K; ℚp]K(p)/ΓK(p)(a), where a>0, ΓK(p) is the absolute Galois group of the maximal p-extension of K and filtration is induced by ramification filtration.

If $\mathcal S$ is a a scheme of finite type over a local field F, and $\mathcal X \longrightarrow \mathcal S$ is a proper smooth family, then to each rational point $s \in \mathcal S$ one can assign an extension of the absolute Galois group of F by the geometric fundamental group G of the fibre $\mathcal{X}_s$ . If F has uniformiser $\pi$ , and residue characteristic p, we show that the corresponding extension of the absolute Galois group of $\mathcal S$ by the maximal prime to p quotient of G is locally constant in the $\pi$ -adic topology of $\mathcal S$ . We give a similar result in the case of non-proper families, and families over $\pi$ -adic analytic spaces.

Soit F un corps ayant une valuation complete et discrete, et de caracteristique residuelle p. Si U est une variete sur F, notons π (p') 1,geom (u) le quotient maximal du groupe etale fondamental de U qui est premier a p. Nous considerons l'application ρ: Gal(F sep /F) → Out(π (p') 1,geom (u)) au groupe des automorphismes exterieurs, et nous montrons qu'elle applique le groupe de ramification sauvage sur un groupe fini. Nous montrons que sous certaines conditions p depend seulement de la reduction de U modulo une puissance de l'ideal maximal de F. Les preuves utilisent la theorie des schemas logarithmiques.

Let X be an algebraic curve of genus g, n -punctured, defined over a number field K . Then, the profinite or the pro- l completion of the topological fundamental group of X admits two actions: the action of the profinite completion of the mapping class group of the orientable surface of topological type ( g, n ) and the action of the absolute Galois group of K . This paper compares these two. In the profinite case, it is shown that the intersection of the images of these two actions is trivial if X is affine and its fundamental group is nonabelian. On the contrary, in the pro- l case, there are many curves such that the image of the Galois action contains the image of the mapping-class-group action. It is proved that the set of points corresponding to such curves is dense in the moduli space of ( g, n )-curves.

For non-archimedean local field K and a prime number p we compute the finitely generated pro-p (closed) subgroups of the absolute Galois group of K(t). In addition, we characterize the finitely generated pro-p groups which occur as the maximal pro-p Galois group of algebraic extensions of K(t) containing a primitive pth root of unity.

This paper is principally concerned with the action of the absolute Galois group on a family of dessins d'enfants i.e. isomorphism classes of coverings of the projective line unramified outside three points. More precisely, we prove a generalisation of a conjecture proposed by Yu. Kotchetkov in 1997. The main tool used in this work is a correspondence between dessins d'enfants and ribbon graphs arising from the theory of Strebel differentials.

This article is concerned with arithmetic properties of totally real closures of formally real fields. We generalize previous results of Fried, V\"olklein and Pop to show that if an algebraic extension K/ℚ is formally real and hilbertian then the absolute Galois group of the cyclotomic closure of the totally real closure of K is pro-free. In addition, we give a precise description of the Brauer group of : it is always an elementary abelian 2-group. Finally, using a result of Glass and Ribenboim, we show that an automorphism of the group , where K is a formally real number field, is necessarily the identity.

Let X be a proper smooth variety over a local field K of mixed characteristics and let l be a prime number different from the characteristic of the residue field of K. Let IK be the inertia subgroup of GK := Gal(K/K). Our main result is the l-independence of the alternating sum of traces of g ∈ IK on H(X,Ql) and its comparison with the traces on p -adic cohomology. 0.Introduction Let K be a complete discrete valuation field with finite residue field Fph , GK the absolute Galois group of K, IK the inertia subgroup of GK and WK the Weil group of K. Recall that the Weil group is a subgroup of GK defined by the following exact sequence: 0 −→ IK −→ WK u −→ Z −→ Z/hZ −→ 0 ∥ ∩ ∩ ∥ 0 −→ IK −→ GK −→ Gal(Fp/Fp) −→ Gal(Fph/Fp) −→ 0, where the map u is defined by g 7−→ f for the geometric Frobenius f : x 7−→ x in Gal(Fp/Fp) ∼= Ẑ and F denotes the separable closure of F for any field F . We define a subset W K of WK to be W K := {g ∈ WK |u(g) ≥ 0}. Let X be a variety over K (Throughout the paper, a variety X over a field K means a reduced irreducible scheme X separated and of finite type over K). Throughout this paper, X means the scalar extension X ⊗ K K. We denote by l a prime number = p. We consider the traces of the action of elements of W K or WK on the compact support etale cohomology H c(X,Ql) := lim ←− n H c(X,Z/lZ) ⊗ Zl Ql. Let us recall the following classical conjecture. Conjecture([S-T]). For any variety X over K and g ∈ W K , Tr(g∗;Hi c(X,Ql)) is a rational integer which is independent of the choice of l. Remark. If X is a d-dimensional proper smooth variety, the conjecture above holds for i = 0, 1, 2d−1, 2d [SGA7-1]. If X has good reduction, the above conjecture is true for any i due to the Weil conjecture proved by P. Deligne [De]. In this paper, we shall prove the following weak versions of the conjecture: Typeset by AMS-TEX 1

Given a global field F and a prime number p we characterize the finitely generated pro- p closed subgroups of the absolute Galois group of F .

We show that the action of the absolute Galois group on dessins d’enfants of given genus g is faithful, a result that had been previously established for g = 0 and g = 1.

We consider a question of describing the one-dimensional P -adic representations that lift a given representation over a finite field of the absolute Galois group of a function field. In this case, the characterization of abelian p -power extensions of fields of characteristic p can be extended to abelian pro- p -extensions, and refined to allow only restricted ramification at the places of K , and can be a tool for analyzing one-dimension P -adic representations. We then turn to the problem of classifying those representations which can be realized as the action of the Galois group on the division points of a rank one Drinfeld module, discussing both results and a conjecture about the form of the representations that arise in this manner.

n=1 Ln(A) be the universal graded Lie algebra associated to A (see §2 for exact definitions). Any homomorphism φ of A into G/G(2) gives rise to a homomorphism φ∗ of L(A) into L(G). In this paper we study the special situation that A is the profinite completion K× of the multiplicative group K× of a local field K, i.e. a field which is complete with respect to a discrete valuation with finite residue class field. The group G is the absolute Galois group GK of K and φ is the Artin isomorphism of K× onto GK/G (2) K . The surjectivity of φ implies the same for φ∗. The goal of this paper is the determination of the kernel of φ∗. This is equivalent to the determination of the kernel of component homomorphisms

In this paper we study local indices of systems of p-adic linearly differential equations which arise from p-adic representations of the absolute Galois group of local field of characteristic p with finite monodromy. We show the induction formula of the local index of p-adic differential equations and prove the equality between the local index of differential equations and the Swan conductor of p-adic Galois representations by inductive methods.

Let S \mathcal {S} be a finite set of rational primes. We denote the maximal Galois extension of Q \mathbb {Q} in which all p ∈ S p\in \mathcal {S} totally decompose by N N . We also denote the fixed field in N N of e e elements σ 1 , … , σ e \sigma _{1},\ldots , \sigma _{e} in the absolute Galois group G ( Q ) G( \mathbb {Q}) of Q \mathbb {Q} by N ( σ ) N( {\boldsymbol \sigma }) . We denote the ring of integers of a given algebraic extension M M of Q \mathbb {Q} by Z M \mathbb {Z}_{M} . We also denote the set of all valuations of M M (resp., which lie over S S ) by V M \mathcal {V}_{M} (resp., S M \mathcal {S}_{M} ). If v ∈ V M v\in \mathcal {V}_{M} , then O M , v O_{M,v} denotes the ring of integers of a Henselization of M M with respect to v v . We prove that for almost all σ ∈ G ( Q ) e {\boldsymbol \sigma }\in G( \mathbb {Q})^{e} , the field M = N ( σ ) M=N( {\boldsymbol \sigma }) satisfies the following local global principle: Let V V be an affine absolutely irreducible variety defined over M M . Suppose that V ( O M , v ) ≠ ∅ V(O_{M,v})\not =\varnothing for each v ∈ V M ∖ S M v\in \mathcal {V}_{M}\backslash \mathcal {S}_{M} and V s i m ( O M , v ) ≠ ∅ V_{\mathrm {sim}}(O_{M,v})\not =\varnothing for each v ∈ S M v\in \mathcal {S}_{M} . Then V ( O M ) ≠ ∅ V(O_{M})\not =\varnothing . We also prove two approximation theorems for M M .

In this article we interpret the relations defining the Grothendieck-Teichmuller group % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaecaaeaaca% WGhbGaamivaaGaayPadaaaaa!3854! $$\widehat{GT}$$ as cocycle relations for certain non-commutative co-homology sets, which we compute using a result due to Brown, Serre and Scheiderer. This interpretation allows us to give a new description of the elements of % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaecaaeaaca% WGhbGaamivaaGaayPadaaaaa!3854! $$\widehat{GT}$$ , as well as a new proof of the Drinfel’d-Ihara theorem stating that % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaecaaeaaca% WGhbGaamivaaGaayPadaaaaa!3854! $$\widehat{GT}$$ contains the absolute Galois group . From the same methods we deduce other properties of % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaecaaeaaca% WGhbGaamivaaGaayPadaaaaa!3854! $$\widehat{GT}$$ analogous to known properties of , such as the self-centralizing of the complex conjugation element.

This paper is devoted to the last unsolved case of the Artin Conjecture in two dimensions. Given an irreducible 2-dimensional complex representation of the absolute Galois group of a number fieldF, the Artin Conjecture states that the associatedL-series is entire. The conjecture has been proved for all cases except the icosahedral one. In this paper we construct icosahedral representations of the absolute Galois group of ℚ(√5) by means of 5-torsion points of an elliptic curve defined over ℚ. We compute the L-series explicitely as an Euler product, giving algorithms for determining the factors at the difficult primes. We also prove a formula for the conductor of the elliptic representation.

In this Note we introduce a certain subgroup Г of the Grothendieck-Teichmuller group GT, obtained by adding two new relations to the definition of GT. We show that Г gives an automorphism group of the profinite completions of certain surface mapping class groups with geometric compatibility conditions, and that the absolute Galois group (ℚ/ℚ) is embedded into Г.

RESUME. On caracterise les corps K satisfaisant le theoreme de l’axe principal a l’aide de proprietes des formes trace des extensions finies de K. Grâce a la caracterisation de ces memes corps due a Waterhouse, on retrouve a partir de la, de facon elementaire, un resultat de Becker selon lequel un pro-2-groupe qui se realise comme groupe de Galois absolu d’un tel corps K est engendre par des involutions. ABSTRACT We characterizegeneral fields K, satisfying the Principal Axis Theorem, by means of properties of trace forms of the finite extensions of K. From this and Waterhouse’s characterization of the same fields, we rediscover, in quite an elementary way, a result of Becker according to which a pro-2-group which occurs as the absolute Galois group of such a field K, is generated by involutions.

Let K be a field of characteristic 0, t a transcendental over K, and Ir be the absolute Galois group of K(t). Then two non-constant polynomials f, g E K[X] are said to be Kronecker conjugate if an element of F fixes a root of f (X) t if and only if it fixes a root of g(X) t. If K is a number field, and f, g E OK [X] where OK is the ring of integers of K, then f and g are Kronecker conjugate if and only if the value set f((OK) equals g(OK) modulo all but finitely many non-zero prime ideals of O9K. In 1968 H. Davenport suggested the study of this latter arithmetic property. The main progress is due to M. Fried, who showed that under certain assumptions the polynomials f and g differ by a linear substitution. Further, he found non-trivial examples where Kronecker conjugacy holds. Until now there were only finitely many known such examples. This paper provides the first infinite series. The main part of the construction is group theoretic.