Galois Embedding Research Articles

About One Galois Embedding Problem

The Galois embedding problem of an extension with elementary Abelian 2-group into an extension with the Galois group isomorphic to the group of unitriangular matrices over the 2-element field is considered. It is proved that the solvability of the maximal accompanying problem with central kernel of period 2 is sufficient for the solvability of the original problem.

Arithmetic properties encoded in the Galois module structure of [formula omitted

The power classes of a field are well-known for their ability to parameterize elementary p-abelian Galois extensions. These classical objects have recently been reexamined through the lens of their Galois module structure. Module decompositions have been computed in several cases, providing deep new insight into absolute Galois groups. The surprising result in each case is that there are far fewer isomorphism types of indecomposables than one would expect generically, with summands predominately free over associated quotient rings. Though non-free summands are the exception both in their form and prevalence, they play the critical role in controlling arithmetic conditions in the field which allow the rest of the decomposition to be so simple.Suppose m,n∈N and p is prime. In a recent paper, a surprising and elegant decomposition for pmth power classes has been computed when the underlying Galois group is a cyclic group of order pn. As with previous module decompositions, at most one non-free summand appears. Outside of a particular special case when p=2, the structure of this exceptional summand was determined by a vector a∈{−∞,0,…,n}m and a natural number d. In this paper we give field-theoretic interpretations for a and d, showing they are related to the solvability of a family of Galois embedding problems and the cyclotomic character associated to K/F.

Twists of non-hyperelliptic curves

In this paper we present a method for computing the set of twists of a non-singular projective curve defined over an arbitrary (perfect) field k . The method is based on a correspondence between twists and solutions to a Galois embedding problem. When in addition, this curve is non-hyperelliptic we show how to compute equations for the twists. If k=\mathbb{F}_q the method then becomes an algorithm, since in this case, it is known how to solve the Galois embedding problems that appear. As an example we compute the set of twists of the non-hyperelliptic genus 6 curve x^7-y^3-1=0 when we consider it defined over a number field such that [k(\zeta_{21}):k]=12 . For each twist equations are exhibited.

A note on Galois embeddings of abelian varieties

In this note we show that if an abelian variety possesses a Galois embedding into some projective space, then it must be isogenous to the self product of an elliptic curve. We prove moreover that the self product of an elliptic curve always has infinitely many Galois embeddings.

Smooth quotients of bi-elliptic surfaces

We consider the quotient X of bi-elliptic surface by a finite automorphism group. If X is smooth, then it is a bi-elliptic surface or ruled surface with irregularity one. As a corollary any bi-elliptic surface cannot be Galois covering of projective plane, hence does not have any Galois embedding.

$p$-groups have unbounded realization multiplicity

In this paper we interpret the solutions to a particular Galois embedding problem over an extension K/F satisfying Gal(K/F) ≃ Z/p n Z in terms of certain Galois submodules within the parameterizing space of elementary p-abelian extensions of K; here p is a prime. Combined with some basic facts about the module structure of this param- eterizing space, this allows us to exhibit a class of p-groups whose realization multiplicity is unbounded.

On the Galois embedding problem associated to the universal central extension of the alternating group of degree 6

We obtain an explicit expression for the obstruction to the solvability of the Galois embedding problem associated to the universal central extension of the alternating group of degree 6.

A Note on Minimal Galois Embeddings of Abelian Surfaces

We show that the least number N such that an abelian surface has a Galois embedding in \mathbb{P}^N is seven and then we give examples of such surfaces.

On the embedding problem for number fields in the case of an elementary Abellan kernel

The Galois embedding problem is considered in the case of number fields and an elementary Abelian kernel. New cases are discovered, in which the consistency condition is sufficient for the existence of a solution of the embedding problem. In particular, it is true when the order of the kernel is the cube of a prime integer.

Galois embedding problems with cyclic quotient of orderp

LetK/F be a cyclic field extension of odd prime degree. We consider Galois embedding problems involving Galois groups with common quotient Gal(K/F) such that corresponding normal subgroups are indecomposable $$\mathbb{F}_p \left[ {Gal\left( {K/F} \right)} \right] - modules$$ . For these embedding problems we prove conditions on solvability, formulas for explicit construction, and results on automatic realizability.

Galois embedding problems and symmetric powers

We consider a Galois embedding problem given by a finite projective group and its special linear lifting. We give an equivalent condition to the solvability of such an embedding problem and a method of construction of its solutions.

Resolution of a family of Galois embedding problems with cyclic kernel

In this paper we compute the obstruction and the solutions of cyclic embedding problems given by (E): \quad 0 \rightarrow \mathbb{Z}/n\mathbb{Z} \rightarrow E \rightarrow \Gamma=\mathbb{Z}/n\mathbb{Z} \times \stackrel{m)}{\cdots} \times \mathbb{Z}/n\mathbb{Z} \rightarrow 0 , with \mathbb{Z}/n\mathbb{Z} trivial \Gamma -modulo, finding adequate representations of \Gamma in the automorphisms group of a generalized Clifford algebra.

The Valentiner group as Galois group

We obtain the complete set of solutions to the Galois embedding problem given by the Valentiner group as a triple cover of the alternating group A 6 A_6 .

Galois realizability of non-split group extensions of C2 by ( C2) r×( C4) s×( D4) t

The focus of this paper is Galois embedding problems associated with extensions of C 2 by groups of the form ( C 2) r ×( C 4) s ×( D 4) t over arbitrary fields of characteristic not 2. As an application of our general solution, we obtain obstructions to the realizability of 13 groups of order 32 for which obstructions were not previously known.

Octahedral Galois Representations Arising From Q-Curves of Degree 2

AbstractGenerically, one can attach to a Q-curve C octahedral representations ρ: Gal() → GL2() coming from the Galois action on the 3-torsion of those abelian varieties of GL2-type whose building block is C. When C is defined over a quadratic field and has an isogeny of degree 2 to its Galois conjugate, there exist such representations ρ having image into GL2(F9). Going the other way, we can ask which mod 3 octahedral representations ρ of Gal() arise from Q-curves in the above sense. We characterize those arising from quadratic Q-curves of degree 2. The approach makes use of Galois embedding techniques in GL2(F9), and the characterization can be given in terms of a quartic polynomial defining the S4-extension of Q corresponding to the projective representation .

QUADRATIC CORESTRICTION, -EMBEDDING PROBLEMS, AND EXPLICIT CONSTRUCTION

ABSTRACT Let be a quadratic extension of fields of characteristic not 2. We give a natural correspondence between cup-products of square classes of and certain Galois embedding problems over . Under this correspondence, the obstruction to the embedding problem associated to is the corestriction from to of the quaternion algebra over given by . We reduce the explicit construction problem for such embedding problems to the construction of an -algebra isomorphism from an algebra expressed as a tensor product of quaternion algebras to a matrix ring. We apply these results to determine the obstructions to, and a method for explicit construction for, all -embedding problems extending or , where the subdirect products are taken over a common factor group .

QUADRATIC DESCENT FOR QUATERNION ALGEBRAS

Given a Galois embedding problem H → G = Gal(L/F) with kernel of order two and F Hilbertian, we consider how obstructions OK to subgroup embedding problems H 0 → G 0 = Gal(L/K) for [K : F] = 2 descend to the obstruction OF to the original embedding problem, up to Br2(K/F). In particular, to such an obstruction OK we associate a tower of Z/2Z-embedding problems and prove that the contribution of OK to OF is given by the obstruction to the last embedding problem in the tower. We show that such an association in fact holds generally for central, Brauer Z/pZ-problems. When OK is the class of a quaternion algebra, we give an explicit representation of OF up to Br2(K/F). As a consequence, we represent in terms of quaternion algebras over F the obstructions to Z/2Z × Z/2Z-embedding problems not previously determined over F.

On the antisymmetry of Galois embeddings

A Galois connection is a ‘natural’ way of relating two partially-ordered spaces (for example, semantic models of different expressive power). The most common form of Galois connection, which reflects one model being finer than the other, is that of a Galois embedding. This paper shows a published claim, ‘ two partially-ordered spaces related by a Galois embedding in each direction are isomorphic’, to be false in general but true in the finite case. The proof we give provides an application of a ‘disjoint from’ calculus.

Explicit Solutions of Galois Embedding Problems by Means of Generalized Clifford Algebras

In this paper we study Galois embedding problems given by central extensions with cyclic kernel. We find a new expression for the obstruction to the solvability of these embedding problems in terms of Galois symbols. We also give a method to construct the solutions when these problems are solvable. We find a solution from a coordinate of the norm of an adequate element in a generalized Clifford algebra.

Relative Embedding Problems

We consider Galois embedding problems G ↠ H ≅ Gal ⁡ ( X / Z ) G\twoheadrightarrow H\cong \operatorname {Gal}(X/Z) such that a Galois embedding problem G ↠ Gal ⁡ ( Y / Z ) G\twoheadrightarrow \operatorname {Gal}(Y/Z) is solvable, where Y / Z Y/Z is a Galois subextension of X / Z X/Z . For such embedding problems with abelian kernel, we prove a reduction theorem, first in the general case of commutative k k -algebras, then in the more specialized field case. We demonstrate with examples of dihedral embedding problems that the reduced embedding problem is frequently of smaller order. We then apply these results to the theory of obstructions to central embedding problems, considering a notion of quotients of central embedding problems, and classify the infinite towers of metacyclic p p -groups to which the reduction theorem applies.