Frobenius modules are difference modules with respect to a Frobenius operator. Here we show that over non-archimedean complete differential fields Frobenius modules define differential modules with the same Picard-Vessiot ring and the same Galois group schemes up to extension by constants. Moreover, these Frobenius modules are classified by unramified Galois representations over the base field. This leads among others to the solution of the inverse differential Galois problem for $p$-adic differential...

This paper contains an application of Langlands’ functoriality principle to the following classical problem: which finite groups, in particular which simple groups appear as Galois groups over $\mathbb{Q}$? Let $\ell$ be a prime and $t$ a positive integer. We show that that the finite simple groups of Lie type $B_n\left({\ell }^k\right)=3DS{O}_{2n+1}{\left({𝔽}_{{\ell }^k}\right)}^{der}$ if $\ell \equiv 3,5(mod8)$ and $G_2\left({\ell }^k\right)$ appear as Galois groups over $\mathbb{Q}$, for some $k$ divisible by $t$. In particular, for each of the two Lie types and fixed $\ell$ we construct infinitely many Galois groups but we do not have a precise control...

Our main result combines three topics: it contains a Grunwald-Wang type conclusion, a version of Hilbert’s irreducibility theorem and a $p$-adic form à la Harbater, but with good reduction, of the Regular Inverse Galois Problem. As a consequence we obtain a statement that questions the RIGP over $\mathbb{Q}$. The general strategy is to study and exploit the good reduction of certain twisted models of the covers and of the associated moduli spaces.

This paper considers some refined versions of the Inverse Galois Problem. We study the local or global behavior of rational specializations of some finite Galois covers of ${\mathbb{P}}_{\mathbb{Q}}^1$.

This article examines the realizability of groups of order 64 as Galois groups over arbitrary fields. Specifically, we provide necessary and sufficient conditions for the realizability of 134 of the 200 noncyclic groups of order 64 that are not direct products of smaller groups.

Let p be an odd prime and k an arbitrary field of characteristic not p. We determine the obstructions for the realizability as Galois groups over k of all groups of orders p 5 and p 6 that have an abelian quotient obtained by factoring out central subgroups of order p or p 2. These obstructions are decomposed as products of p-cyclic algebras, provided that k contains certain roots of unity.

2000 Mathematics Subject Classification: 12F12.We find the obstructions to realizability of groups of order 32 as Galois groups over arbitrary field of characteristic not 2. We discuss explicit extensions and automatic realizations as well.This work is partially supported by project of Shumen University

We reformulate more explicitly the results of Momose, Ribet and Papier concerning the images of the Galois representations attached to newforms without complex multiplication, restricted to the case of weight $2$ and trivial nebentypus. We compute two examples of these newforms, with a single inner twist, and we prove that for every inert prime greater than $3$ the image is as large as possible. As a consequence, we prove that the groups $\text{PGL}(2,{𝔽}_{{\ell }^2})$ for every prime $(\ell \equiv 3,5(\text{mod}8),\ell \>3)$, and $\text{PGL}(2,{𝔽}_{{\ell }^5})$ for every prime $\ell \neg \equiv 0\pm 1\left(\text{mod}11\right);\ell \>3)$, are Galois groups...

Let ${\mathbb{Z}}_d\wr$m$bethegeneralizedalternatinggroup.Weprovethatalldoublecoversof$ℤd ≀ m$occurasGaloisgroupsoveranyalgebraicnumberfield.WefurtherrealizesomeofthesedoublecoversastheGaloisgroupsofregularextensionsof\mathbb{Q}\left(T\right).Ifdisoddandm>7,theneverycentralextensionof$ℤd ≀ m$occursastheGaloisgroupofaregularextensionof\mathbb{Q}\left(T\right).Wefurtherimprovesomeofourearlierresultsconcerningdoublecoversofthegeneralizedsymmetricgroup$ℤd ≀ m$.$

In [Michailov I.M., On Galois cohomology and realizability of 2-groups as Galois groups, Cent. Eur. J. Math., 2011, 9(2), 403–419] we calculated the obstructions to the realizability as Galois groups of 14 non-abelian groups of order 2n, n ≥ 4, having a cyclic subgroup of order 2n−2, over fields containing a primitive 2n−3th root of unity. In the present paper we obtain necessary and sufficient conditions for the realizability of the remaining 8 groups that are not direct products of smaller groups....