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We compute all signatures of [Formula: see text] and [Formula: see text] which classify all orientation preserving actions of the groups [Formula: see text] and [Formula: see text] on compact, connected, orientable surfaces with orbifold genus [Formula: see text]. This classification is well-grounded in the other branches of Mathematics like topology, smooth and conformal geometry, algebraic categories, and it is also directly related to the inverse Galois problem.

This paper focuses on a refinement of the inverse Galois problem. We explore what finite groups appear as the Galois group of an extension of the rational numbers in which only a predetermined set of primes may ramify. After presenting new results regarding extensions in which only a single finite prime ramifies, we move on to studying the more complex situation in which multiple primes from a finite set of arbitrary size may ramify. We then continue by examining a conjecture of Harbater that the minimal number of generators of the Galois group of a tame, Galois extension of the rational numbers is bounded above by the sum of a constant and the logarithm of the product of the ramified primes. We prove the validity of Harbater's conjecture in a number of cases, including the situation where we restrict our attention to finite groups containing a nilpotent subgroup of index 1 , 2 , or 3. We also derive some consequences that are implied by the truth of this conjecture.

We study the inverse Galois problem with local conditions. In particular, we ask whether every finite group occurs as the Galois group of a Galois extension of $\mathbb{Q}$ all of whose decomposition groups are cyclic (resp., abelian). This property is known for all solvable groups due to Shafarevich's solution of the inverse Galois problem for those groups. It is however completely open for nonsolvable groups. In this paper, we provide general criteria to attack such questions via specialization of function field extensions, and in particular give the first infinite families of Galois realizations with only cyclic decomposition groups and with nonsolvable Galois group. We also investigate the analogous problem over global function fields., Comment: Updated to the published version. Additionally, inserted corrigendum addressing a minor error on p.17

This paper deals with the Weak Inverse Galois Problem which, for a given field $k$, states that, for every finite group $G$, there exists a finite separable extension $L/k$ such that ${\rm{Aut}}(L/k)=G$. One of its goals is to explain how one can generically produce families of fields which fulfill this problem, but which do not fulfill the usual Inverse Galois Problem. We show that this holds for, e.g., the fields $\mathbb{Q}^{{{\rm{sol}}}}$, $\mathbb{Q}^{{{\rm{tr}}}}$, $\mathbb{Q}^{{{\rm{pyth}}}}$, and for the maximal pro-$p$-extensions of $\mathbb{Q}$. Moreover, we show that, for every finite non-trivial group $G$, there exists many fields fulfilling the Weak Inverse Galois Problem, but over which $G$ does not occur as a Galois group. As a further application, we show that every field fulfills the regular version of the Weak Inverse Galois Problem.

Let $H$ be a skew field of finite dimension over its center $k$. We solve the Inverse Galois Problem over the field of fractions $H(X)$ of the ring of polynomial functions over $H$ in the variable $X$, if $k$ contains an ample field.

This article is on the inverse Galois problem in Galois theory of linear iterative differential equations in positive characteristic. We show that it has an affirmative answer for reduced algebraic group schemes over any iterative differential field which is finitely generated over its algebraically closed field of constants. We also introduce the notion of equivalence of iterative derivations on a given field - a condition which implies that the inverse Galois problem over equivalent iterative derivations are equivalent., Comment: 13 pages

In this paper we show how to explicitly write down equations of hyperelliptic curves over Q such that for all odd primes l the image of the mod l Galois representation is the general symplectic group. The proof relies on understanding the action of inertia groups on the l-torsion of the Jacobian, including at primes where the Jacobian has non-semistable reduction. We also give a framework for systematically dealing with primitivity of symplectic mod l Galois representations. The main result of the paper is the following. Suppose n=2g+2 is an even integer that can be written as a sum of two primes in two different ways, with none of the primes being the largest primes less than n (this hypothesis appears to hold for all g different from 0,1,2,3,4,5,7 and 13). Then there is an explicit integer N and an explicit monic polynomial $f_0(x)\in \mathbb{Z}[x]$ of degree n, such that the Jacobian $J$ of every curve of the form $y^2=f(x)$ has $Gal(\mathbb{Q}(J[l])/\mathbb{Q})\cong GSp_{2g}(\mathbb{F}_l)$ for all odd primes l and $Gal(\mathbb{Q}(J[2])/\mathbb{Q})\cong S_{2g+2}$, whenever $f(x)\in\mathbb{Z}[x]$ is monic with $f(x)\equiv f_0(x) \bmod{N}$ and with no roots of multiplicity greater than $2$ in $\overline{\mathbb{F}}_p$ for any p not dividing N., Comment: 24 pages, minor corrections

In this article, we show that the Inverse Galois Problem has a positive answer over some non-trivial skew fields of fractions $$H(t,\alpha ,\delta )$$ .

Solving the equation P a ( X ) : = X q + 1 + X + a = 0 over the finite field F Q , where Q = p n , q = p k and p is a prime, arises in many different contexts including finite geometry, the inverse Galois problem [2] , the construction of difference sets with Singer parameters [8] , determining cross-correlation between m-sequences [9] , [15] and the construction of error-correcting codes [5] , as well as speeding up the index calculus method for computing discrete logarithms on finite fields [11] , [12] and on algebraic curves [18] . Subsequently, in [3] , [13] , [14] , [6] , [4] , [16] , [7] , [19] , the F Q -zeros of P a ( X ) have been studied. It was shown in [3] that their number is 0, 1, 2 or p gcd ⁡ ( n , k ) + 1 . Some criteria for the number of the F Q -zeros of P a ( x ) were found in [13] , [14] , [6] , [16] , [19] . However, while the ultimate goal is to identify all the F Q -zeros, even in the case p = 2 , it was solved only under the condition gcd ⁡ ( n , k ) = 1 [16] . We discuss this equation without any restriction on p and gcd ⁡ ( n , k ) . Criteria for the number of the F Q -zeros of P a ( x ) are proved by a new methodology. For the cases of one or two F Q -zeros, we provide explicit expressions for these rational zeros in terms of a. For the case of p gcd ⁡ ( n , k ) + 1 rational zeros, we provide a parametrization of such a's and express the p gcd ⁡ ( n , k ) + 1 rational zeros by using that parametrization.

Fix a positive integer $g$ and rational prime $p$. We prove the existence of a genus $g$ curve $C/\mathbb{Q}$ such that the mod $p$ representation of its Jacobian is tame by imposing conditions on the endomorphism ring. As an application, we consider the tame inverse Galois problem and are able to realise general symplectic groups as Galois groups of tame extensions of $\mathbb{Q}$., Comment: v2: Expanded to include application to tame inverse Galois problem. To appear in Manuscripta Mathematica

We consider the moduli of elliptic curves with G-structures, where G is a finite 2-generated group. When G is abelian, a G-structure is the same as a classical congruence level structure. There is a natural action of $$\text {SL}_2(\mathbb {Z})$$ on these level structures. If $$\Gamma $$ is a stabilizer of this action, then the quotient of the upper half plane by $$\Gamma $$ parametrizes isomorphism classes of elliptic curves equipped with G-structures. When G is sufficiently nonabelian, the stabilizers $$\Gamma $$ are noncongruence. Using this, we obtain arithmetic models of noncongruence modular curves as moduli spaces of elliptic curves equipped with nonabelian G-structures. As applications we describe a link to the Inverse Galois Problem, and show how our moduli interpretations explains the bad primes for the Unbounded Denominators Conjecture, and allows us to translate the conjecture into the language of geometry and Galois theory.

Let $m$ be an integer such that $m \geq 7$ and $m \equiv 0,1,7 \mod 8$. We construct strictly compatible systems of representations of $\Gamma_{\mathbb Q} \to \mathrm{Spin}_m(\overline{\mathbb Q}_l) \xrightarrow{\mathrm{spin}} \mathrm{GL}_N(\overline{\mathbb Q}_l)$ that is potentially automorphic and motivic. As an application, we prove instances of the inverse Galois problem for the $\mathbb F_p$--points of the spin groups. For odd $m$, we compare our examples with the work of A. Kret and S. W. Shin, which studies automorphic Galois representations valued in $\mathrm{Spin}_m$., Comment: 21 pages. Comments are welcome!

Fix a positive integer g and a squarefree integer m. We prove the existence of a genus g curve $$C/{\mathbb {Q}}$$ C / Q such that the mod m representation of its Jacobian is tame. The method is to analyse the period matrices of hyperelliptic Mumford curves, which could be of independent interest. As an application, we study the tame version of the inverse Galois problem for symplectic matrix groups over finite fields.

This article is the first part of a series of three articles about compatible systems of symplectic Galois representations and applications to the inverse Galois problem. In this first part, we determine the smallest field over which the projectivisation of a given symplectic group representation satisfying some natural conditions can be defined. The answer only depends on inner twists. We apply this to the residual representations of a compatible system of symplectic Galois representations satisfying some mild hypothesis and obtain precise information on their projective images for almost all members of the system, under the assumption of huge residual images, by which we mean that a symplectic group of full dimension over the prime field is contained up to conjugation. Finally, we obtain an application to the inverse Galois problem. MSC (2010): 11F80 (Galois representations); 20C25 (Projective representations and multipliers), 12F12 (Inverse Galois theory).

Let $m$ be an integer greater than three and $\ell$ be an odd prime. In this paper, we prove that at least one of the following groups: $\mbox{P}\Omega^\pm_{2m}(\mathbb{F}_{\ell^s})$, $\mbox{PSO}^\pm_{2m}(\mathbb{F}_{\ell^s})$, $\mbox{PO}_{2m}^\pm(\mathbb{F}_{\ell^s})$ or $\mbox{PGO}^\pm_{2m}(\mathbb{F}_{\ell^s})$ is a Galois group of $\mathbb{Q}$ for infinitely many integers $s > 0$. This is achieved by making use of a slight modification of a group theory result of Khare, Larsen and Savin, and previous results of the author on the images of the Galois representations attached to cuspidal automorphic representations of $\mbox{GL}_{2m}(\mathbb{A}_\mathbb{Q})$.., Comment: Revised version - referees' comments added. The final version is to appear in Proc. Amer. Math. Soc

Let $S$ be a smooth cubic surface over a finite field $\mathbb F_q$. It is known that $\#S(\mathbb F_q) = 1 + aq + q^2$ for some $a \in \{-2,-1,0,1,2,3,4,5,7\}$. Serre has asked which values of a can arise for a given $q$. Building on special cases treated by Swinnerton-Dyer, we give a complete answer to this question. We also answer the analogous question for other del Pezzo surfaces, and consider the inverse Galois problem for del Pezzo surfaces over finite fields. Finally we give a corrected version of Manin's and Swinnerton-Dyer's tables on cubic surfaces over finite fields., 25 pages. Fixed various typos and improved exposition

The Galois representations associated to weight 1 eigenforms over $$\bar{\mathbb {F}}_{p}$$ are remarkable in that they are unramified at $$p$$ , but the effective computation of these modular forms presents challenges. One complication in this setting is that a weight 1 cusp form over $$\bar{\mathbb {F}}_{p}$$ need not arise from reducing a weight 1 cusp form over $$\bar{\mathbb {Q}}$$ . In this article we propose a unified Hecke stability method for computing spaces of weight 1 modular forms of a given level in all characteristics simultaneously. Our main theorems outline conditions under which a finite-dimensional Hecke module of ratios of modular forms must consist of genuine modular forms. We conclude with some applications of the Hecke stability method that are motivated by the refined inverse Galois problem.

In this article, we consider the inverse Galois problem for parameterized differential equations over k ( ( t ) ) ( x ) with k any field of characteristic zero and use the method of patching over fields due to Harbater and Hartmann. We show that every connected semisimple k ( ( t ) ) -split linear algebraic group is a parameterized differential Galois group over k ( ( t ) ) ( x ) .

We study the moduli space M ~ \widetilde {\mathscr {M}} of marked cubic surfaces. By classical work of A. B. Coble, this has a compactification M ~ \widetilde {M} , which is linearly acted upon by the group W ( E 6 ) W(E_6) . M ~ \widetilde {M} is given as the intersection of 30 cubics in P 9 \mathbf {P}^9 . For the morphism M ~ → P ( 1 , 2 , 3 , 4 , 5 ) \widetilde {\mathscr {M}} \to \mathbf {P}(1,2,3,4,5) forgetting the marking, followed by Clebsch’s invariant map, we give explicit formulas, i.e., Clebsch’s invariants are expressed in terms of Coble’s irrational invariants. As an application, we give an affirmative answer to the inverse Galois problem for cubic surfaces over Q \mathbb {Q} .

We describe a method for counting the number of extensions of $\mathbb{Q}_p$ with a given Galois group $G$, founded upon the description of the absolute Galois group of $\mathbb{Q}_p$ due to Jannsen and Wingberg. Because this description is only known for odd $p$, our results do not apply to $\mathbb{Q}_2$. We report on the results of counting such extensions for $G$ of order up to $2000$ (except those divisible by $512$), for $p=3,5,7,11,13$. In particular, we highlight a relatively short list of minimal $G$ that do not arise as Galois groups. Motivated by this list, we prove two theorems about the inverse Galois problem for $\mathbb{Q}_p$: one giving a necessary condition for $G$ to be realizable over $\mathbb{Q}_p$ and the other giving a sufficient condition., Comment: Presented at ANTS 13 (2018)

We completely solve the inverse Galois problem for del Pezzo surfaces of degree $2$ and $3$ over all finite fields., Comment: 2nd version, 6 pages, to appear in Math. Res. Lett

Let $F$ be a number field with ring of integers $O_F$ and let $G$ be a finite group. We describe an approach to the study of the set of realisable classes in the locally free class group $Cl(O_FG)$ of $O_FG$ that involves applying the work of the second-named author in the context of relative algebraic $K$ theory. When $G$ is of odd order, we show (subject to certain conditions) that the set of realisable classes is a subgroup of $Cl(O_FG)$. This may be viewed as being a partial analogue of a classical theorem of Shafarevich on the inverse Galois problem for soluble groups in the setting of Galois module theory., Final version. To appear in Algebra and Number Theory

We describe a construction of plane quartics with prescribed Galois operation on the 28 bitangents, in the particular case of a Galois invariant Cayley octad. As an application, we solve the inverse Galois problem for degree two del Pezzo surfaces in the corresponding particular case., Comment: arXiv admin note: substantial text overlap with arXiv:1708.00071

Given a $G$-Galois branched cover of the projective line over a number field $K$, we study whether there exists a closed point of $\mathbb{P}^1_K$ with a connected fiber such that the $G$-Galois field extension induced by specialization "arithmetically descends" to $\mathbb{Q}$ (i.e., there exists a $G$-Galois field extension of $\mathbb{Q}$ whose compositum with the residue field of the point is equal to the specialization). We prove that the answer is frequently positive (whenever $G$ is regularly realizable over $\mathbb{Q}$) if one first allows a base change to a finite extension of $K$. If one does not allow base change, we prove that the answer is positive when $G$ is cyclic. Furthermore, we provide an explicit example of a Galois branched cover of $\mathbb{P}^1_K$ with no $K$-rational points of arithmetic descent., Comment: We have generalized Theorem 4.1 (in the current numbering) to apply to all cyclic groups. The proof is somewhat dissimilar to the proof of the corresponding statement in the previous version (Theorem 5.1 in v2). Subsequently, some of the numbering has changed

We discuss the inverse Galois problem with normal basis, concerning Kummer theories for algebraic tori, in the framework of group schemes. The unit group scheme of a group algebra plays an important role in this article, as was pointed out by Serre~[8]. We develop our argument not only over a field but also over a ring, considering integral models of Kummer theories for algebraic tori.

The inverse Galois problem is concerned with finding a Galois extension of a field K with given Galois group. In this paper we consider the particular case where the base field is K = F p ( t ) . We give a conjectural formula for the minimal number of primes, both finite and infinite, ramified in G-extensions of K, and give theoretical and computational proofs for many cases of this conjecture.

We prove the existence of certain rationally rigid triples in${E}_{8}(p)$for good primes$p$(i.e. $p>5$) thereby showing that these groups occur as Galois groups over the field of rational numbers. We show that these triples arise from rigid triples in the algebraic group and prove that they generate an interesting subgroup in characteristic zero. As a byproduct of the proof, we derive a remarkable symmetry between the character table of a finite reductive group and that of its dual group. We also give a short list of possible overgroups of regular unipotent elements in simple exceptional groups.

Let k be any field, G be a finite group acting on the rational function field k ( x g : g ∈ G ) by h ⋅ x g = x h g for any h , g ∈ G . Define k ( G ) = k ( x g : g ∈ G ) G . Noether’s problem asks whether k ( G ) is rational (= purely transcendental) over k . A weaker notion, retract rationality introduced by Saltman, is also very useful for the study of Noether’s problem. We prove that, if G is a Frobenius group with abelian Frobenius kernel, then k ( G ) is retract k -rational for any field k satisfying some mild conditions. As an application, we show that, for any algebraic number field k , for any Frobenius group G with Frobenius complement isomorphic to S L 2 ( F 5 ) , there is a Galois extension field K over k whose Galois group is isomorphic to G , i.e. the inverse Galois problem is valid for the pair ( G , k ) . The same result is true for any non-solvable Frobenius group if k ( ζ 8 ) is a cyclic extension of k .

There are several variants of the inverse Galois problem which involve restrictions on ramification. In this paper we give sufficient conditions that a given finite group $G$ occurs infinitely often as a Galois group over the rationals $\mathbb Q$ with all nontrivial inertia groups of order $2$. Notably any such realization of $G$ can be translated up to a quadratic field over which the corresponding realization of $G$ is unramified. The sufficient conditions are imposed on a parametric polynomial with Galois group $G$--if such a polynomial is available--and the infinitely many realizations come from infinitely many specializations of the parameter in the polynomial. This will be applied to the three finite simple groups $A_5$, $PSL_2(7)$ and $PSL_3(3)$. Finally, the applications to $A_5$ and $PSL_3(3)$ are used to prove the existence of infinitely many optimally intersective realizations of these groups over the rational numbers (proved earlier for $PSL_2(7)$ by the first author)., 12 pages. Link to arXiv added to first reference

Let $U$ be an affine smooth curve defined over an algebraically closed field of positive characteristic. The Abhyankar Conjecture (proved by Raynaud and Harbater in 1994) describes the set of finite quotients of Grothendieck's \'etale fundamental group of $U$. In this paper, we consider a purely inseparable analogue of this problem, formulated in terms of Nori's profinite fundamental group scheme, and give a partial answer to it., Comment: 2nd version. Added Remark 3.7(4). Filled a gap in Lemma 4.2. Corrected some typos

In this paper we prove that for each integer of the form $n=4\varpi$ (where $\varpi$ is a prime between $17$ and $73$) at least one of the following groups: $P\Omega^+_n(\mathbb{F}_{\ell^s})$, $PSO^+_n(\mathbb{F}_{\ell^s})$, $PO_n^+(\mathbb{F}_{\ell^s})$ or $PGO^+_n(\mathbb{F}_{\ell^s})$ is a Galois group of $\mathbb{Q}$ for almost all primes $\ell$ and infinitely many integers $s > 0$. This is achieved by making use of the classification of small degree representations of finite simple groups of Lie type in defining characteristic of F. L\"ubeck and a previous result of the author on the image of the Galois representations attached to RAESDC automorphic representations of $GL_n(\mathbb{A}_\mathbb{Q})$., Comment: This paper replaces "L\"ubeck's classification of Chevalley groups representations and the inverse Galois problem for some orthogonal groups" arXiv:1712.05528v2. Revised version - referees' comments added. The final version is to appear in J. Number Theory

We consider the inverse Galois problem over function fields of positive characteristic p, for example, the inverse Galois problem over the projective line. We describe a method to construct certain Galois covers of the projective line and other curves, which are ordinary in the sense that their Jacobian has maximal p-torsion. We do this by constructing Galois covers of ordinary semi-stable curves, and then deforming them into smooth Galois covers., Comment: correction of a few small errors, improvements made in presentation, comments are still welcome!

A field K admits proper projective extensions, i.e. Galois extensions where the Galois group is a nontrivial projective group, unless K is separably closed or K is a pythagorean formally real field without cyclic extensions of odd degree. As a consequence, it turns out that almost all absolute Galois groups decompose as proper semidirect products. We show that each local field has a unique maximal projective extension, and that the same holds for each global field of positive characteristic. In characteristic 0, we prove that Leopoldt’s conjecture for all totally real number fields is equivalent to the statement that, for all totally real number fields, all projective extensions are cyclotomic. So the realizability of any non-procyclic projective group as Galois group over Q produces counterexamples to the Leopoldt conjecture, while the non-realizability may produce counterexamples to the classical inverse Galois problem.

In this paper, we generalize the Hurwitz space which is defined by Fried and Völklein by replacing constant Teichmüller level structures with non-constant Teichmüller level structures defined by finite étale group schemes. As an application, we give some examples of projective general symplectic groups over finite fields which occur as quotients of the absolute Galois group of the field of rational numbers $\mathbb Q$.

This article is the second part of a series of three articles about compatible systems of symplectic Galois representations and applications to the inverse Galois problem. This part is concerned with symplectic Galois representations having a huge residual image, by which we mean that a symplectic group of full dimension over the prime field is contained up to conjugation. A key ingredient is a classification of symplectic representations whose image contains a nontrivial transvection: these fall into three very simply describable classes, the reducible ones, the induced ones and those with huge image. Using the idea of an (n,p)-group of Khare, Larsen and Savin we give simple conditions under which a symplectic Galois representation with coefficients in a finite field has a huge image. Finally, we combine this classification result with the main result of the first part to obtain a strenghtened application to the inverse Galois problem., Comment: 14 pages; the proof of the classification result has been significantly shortened by appealing to results of Kantor

The minimal ramification problem may be considered as a quantitative version of the inverse Galois problem. For a nontrivial finite group $G$, let $m(G)$ be the minimal integer $m$ for which there exists a $G$-Galois extension $N/\mathbb{Q}$ that is ramified at exactly $m$ primes (including the infinite one). So, the problem is to compute or to bound $m(G)$.In this paper, we bound the ramification of extensions $N/\mathbb{Q}$ obtained as a specialization of a branched covering $\unicode[STIX]{x1D719}:C\rightarrow \mathbb{P}_{\mathbb{Q}}^{1}$. This leads to novel upper bounds on $m(G)$, for finite groups $G$ that are realizable as the Galois group of a branched covering. Some instances of our general results are: $$\begin{eqnarray}1\leqslant m(S_{k})\leqslant 4\quad \text{and}\quad n\leqslant m(S_{k}^{n})\leqslant n+4,\end{eqnarray}$$ for all $n,k>0$. Here $S_{k}$ denotes the symmetric group on $k$ letters, and $S_{k}^{n}$ is the direct product of $n$ copies of $S_{k}$. We also get the correct asymptotic of $m(G^{n})$, as $n\rightarrow \infty$ for a certain class of groups $G$.Our methods are based on sieve theory results, in particular on the Green–Tao–Ziegler theorem on prime values of linear forms in two variables, on the theory of specialization in arithmetic geometry, and on finite group theory.

Let R be a domain, complete with respect to a norm which defines a non-discrete topology on R. We prove that the quotient field of R is ample, generalizing a theorem of Pop. We then consider the case where R is a ring of arithmetic power series which are holomorphic on the closed disc of radius 0

We will give an explicit polynomial over ℚ with 3 parameters of degree 40 as a result of the inverse Galois problem. Its Galois group over ℚ (resp. ℚ(√-3)) is isomorphic to PGSp4(3) (resp. PSp4(3)) and it is a regular PSp4(3)-polynomial over ℚ(p√−3). To construct the polynomial and prove its properties above we use some results of Siegel modular forms and permutation group theory.

Cet article donne une application du principe de fonctorialite de Langlands au probleme classique suivant: quels groupes finis, en particulier quels groupes simples, apparaissent comme groupes de Galois sur ℚ ? Soit l une nombre premier et t un entier positif. Nous montrons que les groupes finis simples de type de Lie B n (l k ) = 3DSO 2n+1 (F l k) der lorsque l ≡ 3,5 (mod 8) et G 2 (l k ) sont des groupes de Galois sur ℚ pour un entier k divisant t. En particulier, pour chacun de ces deux types de Lie et pour un entier l fixe, nous construisons une infinite de groupes de Galois, mais nous n'avons pas de controle precis sur k.

For a finite set S of primes of a number field K and for σ1, . . . , σe ∈ Gal(K) we denote the field of totally S-adic numbers by Ktot,S and the fixed field of σ1, . . . , σe in Ktot,S by Ktot,S(σ). We prove that for almost all σ ∈ Gal(K) the absolute Galois group of Ktot,S(σ) is the free product of Fe and a free product of local factors over S. MR Classification: 12E30 Directory: \Jarden\Diary\HJPd 10 May, 2009 * Research supported by the Minkowski Center for Geometry at Tel Aviv University, established by the Minerva Foundation. Introduction The Inverse Galois Problem asks whether every finite group is realizable over Q. Although this has been shown to be true for many finite groups, including the symmetric and alternating groups (Hilbert), we are still very far from the solution of the problem. One could ask, more generally, what is the structure of the absolute Galois group of Q. Here we do not even have a plausible conjecture. However, we do know the structure of the absolute Galois group of certain distinguished algebraic extensions of Q, or, more generally, of a countable Hilbertian field K. We fix a separable closure Ks and an algebraic closure K of K and let Gal(K) = Gal(Ks/K) be the absolute Galois group of K. Our goal is to explore the absolute Galois groups of large algebraic extensions of K having interesting diophantine or arithmetical properties. Our study is motivated by two earlier results. By the free generators theorem Gal(Ks(σ)) is, for almost all σ ∈ Gal(K), the free profinite group Fe on e generators (Jarden [FrJ, Thm. 18.5.6]). On the other hand, if K is a global field and S1 is a finite set of primes of K, then the absolute Galois group Gal(Ktot,S1) of the maximal S1-adic extension of K is a free product of local groups (Pop [Pop4, Thm. 3]). In this work we simultaneously generalize both results and prove that Gal(Ks(σ) ∩ Ktot,S1) is, for almost all σ ∈ Gal(K), the free product of Fe and a free product of local groups. Here is a detailed account of our result. The main theorem. For each e-tuple σ = (σ1, . . . , σe) ∈ Gal(K) we denote the fixed field in Ks (resp. K) of σ1, . . . , σe by Ks(σ) (or K(σ) if char(K) = 0). We know that for almost all σ ∈ Gal(K) the field Ks(σ) is PAC [FrJ, Thm. 18.6.1] and Gal(Ks(σ)) ∼= Fe [FrJ, Thm. 18.5.6]. Here “almost all” is meant in the sense of the Haar measure of Gal(K) and we say that a field M is PAC if every absolutely irreducible variety V defined over M has an M -rational point. The PAC property of the field Ks(σ) implies that if w is a nontrivial valuation of Ks(σ), then the Henselian closure of Ks(σ) at w is Ks [FrJ, Cor. 11.5.5]. To bring valuations into the game we consider a finite set S1 of absolute values