Construction of some wreath products as Galois groups of normal real extensions of the rationals

Let K be a real algebraic number field. Suppose that G occurs as a Galois group of a normal real extension field of K. Using elementary methods, we show that certain types of split extensions of an elementary abelian 2-group by G also occur as Galois groups of normal real extensions of K. Among other examples, we show that Sylow 2-subgroups of the symmetric and alternating groups of degree 2ⁿ, as well as the Weyl groups of type B_n and D_n, occur as Galois groups of real extensions of the rationals.     

Some Galois groups over number fields

Some conditions are stated which imply that certain finite groups are Galois groups over some number fields and related fields.     

Frobenius Galois groups over quadratic fields

There exists a quadratic fieldQ(√D) over which every Frobenius group is realizable as a Galois group.     

Galois groups over complete valued fields

We propose an elementary algebraic approach to the patching of Galois groups. We prove that every finite group is regularly realizable over the field of rational functions in one variable over a complete discrete valued field. Partially supported by NSF grant DMS 9306479.     

Computation of Galois groups over function fields

Symmetric function theory provides a basis for computing Galois groups which is largely independent of the coefficient ring. An exact algorithm has been implemented over in Maple for degree up to 8. A table of polynomials realizing each transitive permutation group of degree 8 as a Galois group over the rationals is included.

     

Abhyankar's conjecture on Galois groups over curves

Sans résuméOblatum 5-I-1993En hommage à Armand Borel avec notre admiration     

The Galois group of a radical extension of the rationals

The Galois group of the splitting field of an irreducible binomialx ^2e −a overQ is computed explicitly as a full subgroup of the holomorph of the cyclic group of order 2 ^e . The general casex ⁿ −a is also effectively computed.     

Galois cohomology of the classical groups over imperfect fields

Elaborating on techniques of Bayer-Fluckiger and Parimala, we prove the following strong version of Serre’s Conjecture II for classical groups: let G be a simply connected absolutely simple group of outer type A_n or of type B_n, C_n or D_n (non trialitarian) defined over an arbitrary field F. If the separable dimension of F is at most 2 for every torsion prime of G, then every G-torsor is trivial.     

On search over rationals

Papadimitriou and Reiss have independently shown how a rational x = p/q, p, q positive integers with p,q?N can be found in O(logN) queries of the form ‘is x?y?’ We examine some of the algorithmic implications of their findings.     

Galois groups of some iterated polynomials over cyclotomic extensions

Let \(\varphi _p(z)=(z-1)^p+2-\zeta _p\), where \(\zeta _p\in \bar{\mathbb {Q}}\) is a primitive pth root of unity. Building on previous work, we show that the nth iterate \(\varphi _p^n(z)\) has Galois group \([C_p]^n\), an iterated wreath product of cyclic groups, whenever p is not a Wieferich prime.     

The free topological group over the rationals

In this paper we investigate the topological structure of the Graev free topological group over the rationals. We show that this free group fails to be a k-space and fails to carry the weak topology generated by its subspaces of words of length less than or equal to n. As tools in this investigation we establish some properties of net convergence in free groups and also some properties of certain canonical maps which are closely related to the topological structure of free groups.     

Computing modular Galois representations

We compute modular Galois representations associated with a newform $f$ , and study the related problem of computing the coefficients of $f$ modulo a small prime $\ell $ . To this end, we design a practical variant of the complex approximations method presented in Edixhoven and Couveignes (Ann. of Math. Stud., vol. 176, Princeton University Press, Princeton, 2011). Its efficiency stems from several new ingredients. For instance, we use fast exponentiation in the modular jacobian instead of analytic continuation, which greatly reduces the need to compute abelian integrals, since most of the computation handles divisors. Also, we introduce an efficient way to compute arithmetically well-behaved functions on jacobians, a method to expand cuspforms in quasi-linear time, and a trick making the computation of the image of a Frobenius element by a modular Galois representation more effective. We illustrate our method on the newforms $\Delta $ and $E_4 \cdot \Delta $ , and manage to compute for the first time the associated faithful representations modulo $\ell $ and the values modulo $\ell $ of Ramanujan’s $\tau $ function at huge primes for $\ell \in \{ 11,13,17,19,29\}$ . In particular, we get rid of the sign ambiguity stemming from the use of a projective representation as in Bosman (On the computation of Galois representations associated to level one modular forms. arxiv.org/abs/0710.1237, 2007). As a consequence, we can compute the values of $\tau (p)~\mathrm{mod}~2^{11} \times 3^6 \times 5^3 \times 7 \times 11 \times 13 \times 17 \times 19 \times 23 \times 29 \times 691 \approx 2.8 \times 10^{19}$ for huge primes $p$ . The representations we computed lie in the jacobian of modular curves of genus up to $22$ .     

Covering groups of almost simple groups as Galois groups over ℚab(t)

In this paper we construct Galois extensions with the rigidity method and apply a criterion [15] for solving central embedding problems over ?^ab(t) to realize regularly the covering groups of most of the classical groups and the sporadic groups as Galois groups over ?^ab(t).     

Time-frequency analysis on the adeles over the rationals

We show that the construction of Gabor frames in $L^2(\mathbb{R})$ with generators in ${\mathbf{S}}_0(\mathbb{R})$ and with respect to time-frequency shifts from a rectangular lattice $\alpha \mathbb{Z}\times \beta \mathbb{Z}$ is equivalent to the construction of certain Gabor frames for $L^2$ over the adeles over the rationals and the group $\mathbb{R}\times {\mathbb{Q}}_p$. Furthermore, we detail the connection between the construction of Gabor frames on the adeles and on $\mathbb{R}\times {\mathbb{Q}}_p$ with the construction of certain Heisenberg modules.     

Motives with exceptional Galois groups and the inverse Galois problem

We construct motivic ?-adic representations of $\textup {Gal}(\overline {\mathbb{Q}}/\mathbb{Q})$ into exceptional groups of type E ₇,E ₈ and G ₂ whose image is Zariski dense. This answers a question of Serre. The construction is uniform for these groups and is inspired by the Langlands correspondence for function fields. As an application, we solve new cases of the inverse Galois problem: the finite simple groups $E_{8}(\mathbb{F}_{\ell})$ are Galois groups over $\mathbb{Q}$ for large enough primes ?.     

Galois cohomology of the classical groups over fields of cohomological dimension≦2

Oblatum 8-XII-1994 7-III-1995     

Approximating absolute Galois groups

In this paper we identify a class of profinite groups (totally torsion free groups) that includes all separable Galois groups of fields containing an algebraically closed subfield, and demonstrate that it can be realized as an inverse limit of torsion free virtually finitely generated abelian (tfvfga) profinite groups. We show by examples that the condition is quite restrictive. In particular, semidirect products of torsion free abelian groups are rarely totally torsion free. The result is of importance for K-theoretic applications, since descent problems for tfvfga groups are relatively manageable.