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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

Solving equations over finite fields is an important problem from both theoretical and practice points of view. The problem of solving explicitly the equation P a ( X ) = 0 over the finite field F Q , where P a ( X ) : = X q + 1 + X + a , Q = p n , q = p k , a ∈ F Q ⁎ and p is a prime, arises in many different contexts including finite geometry, the inverse Galois problem [1] , the construction of difference sets with Singer parameters [9] , determining cross-correlation between m-sequences [10] and to construct error correcting codes [5] , cryptographic APN functions [6] , [7] , designs [21] , as well as to speed up the index calculus method for computing discrete logarithms on finite fields [11] , [12] and on algebraic curves [18] . In fact, the research on this specific problem has a long history of more than a half-century from the year 1967 when Berlekamp, Rumsey and Solomon [2] firstly considered a very particular case with k = 1 and p = 2 . In this article, we discuss the equation P a ( X ) = 0 without any restriction on p and gcd ⁡ ( n , k ) . In a very recent paper [15] , the authors have left open a problem that could definitely solve this equation. More specifically, for the cases of one or two F Q -zeros, explicit expressions for these rational zeros in terms of a were provided, but for the case of p gcd ⁡ ( n , k ) + 1 F Q − zeros it was remained open to compute explicitly the zeros. This paper solves the remained problem, thus now the equation X p k + 1 + X + a = 0 over F p n is completely solved for any prime p, any integers n and k.

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!

We discuss some results on the divisibility of the class numbers of pairs of quadratic fields. We also discuss a problem arises from the inverse Galois problem.

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

We introduce splitting coverings to enhance the well known analogy between field extensions and covering spaces. Semi-topological Galois groups are defined for Weierstrass polynomials and a Galois correspondence is proved. Combining results from braid groups, we solve the topological inverse Galois problem. As an application, symmetric and cyclic groups are realized over ℚ.

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.