Tannakian duality for Anderson–Drinfeld motives and algebraic independence of Carlitz logarithms

We develop a theory of Tannakian Galois groups for t-motives and relate this to the theory of Frobenius semilinear difference equations. We show that the transcendence degree of the period matrix associated to a given t-motive is equal to the dimension of its Galois group. Using this result we prove that Carlitz logarithms of algebraic functions that are linearly independent over the rational function field are algebraically independent. Mathematics Subject Classification (2000) Primary: 11J93; Secondary: 11G09, 12H10, 14L17     

G-Closed fields and imbeddings of quadratic number fields

A field, K, that has no extensions with Galois group isomorphic to G is called G-closed. It is proved that a finite extension of K admits an infinite number of nonisomorphic extensions with Galois group G. A trinomial of degree n is exhibited with Galois group, the symmetric group of degree n, and with prescribed discriminant. This result is used to show that any quadratic extension of an A_n-closed field admits an extension with Galois group A_n.     

On v-adic periods of t-motives

In this paper, we prove the equality between the transcendental degree of the field generated by the v-adic periods of a t-motive M and the dimension of the Tannakian Galois group for M, where v is a “finite” place of the rational function field over a finite field. As an application, we prove the algebraic independence of certain “formal” polylogarithms.     

Torsors under finite and flat group schemes of rank <Emphasis Type="Italic">p</Emphasis> with Galois action

In this note we study the geometry of torsors under flat and finite commutative group schemes of rank p above curves in characteristic p, and above relative curves over a complete discrete valuation ring of inequal characteristic. In both cases we study the Galois action of the Galois group of the base field on these torsors. We also study the degeneration of _p -torsors, from characteritic 0 to characteristic p, and show that this degeneration is compatible with the Galois action. We then discuss the lifting of torsors under flat and commutative group schemes of rank p from positive to zero characteristics. Finally, for a proper and smooth curve X over a complete discrete valuation field, of inequal characteristic, which has good reduction, we show the existence of a canonical Galois equivariant filtration, on the first étale cohomology group of the geometric fibre of X, with values in _p .     

Cohomology of the Weil group of a p-adic field

We establish duality and vanishing results for the cohomology of the Weil group of a p-adic field. Among them is a duality theorem for finitely generated modules, which implies Tate–Nakayama Duality. We prove comparison results with Galois cohomology, which imply that the cohomology of the Weil group determines that of the Galois group. When the module is defined by an abelian variety, we use these comparison results to establish a duality theorem analogous to Tate?s duality theorem for abelian varieties over p-adic fields.     

Ramification of a finite flat group scheme over a local field

In this paper, we analyze ramification in the sense of Abbes-Saito of a finite flat group scheme over the ring of integers of a complete discrete valuation field of mixed characteristic (0,p). We deduce that its Galois representation depends only on its reduction modulo explicitly computed p-power. We also give a new proof of a theorem of Fontaine on ramification of a finite flat Galois representation, and extend it to the case where the residue field may be imperfect.     

Explicit construction of PSL(2,7) Fields over Q(t) Embeddable in SL(2,7) Fields

We employ the recent results of Mestre [C. R. Acad. Sei. Paris, t. 319, Série I, pp. 781-782] to exhibit polynomials over Q(t) with the property that their splitting fields are regular, have Galois group PSL(2,7), and are embeddable in fields with Galois group SL(2,7) over Q(t).     

Motivic cohomology of the simplicial motive of a Rost variety

We compute the motivic cohomology groups of the simplicial motive X_θ of a Rost variety for an n-symbol θ in Galois cohomology of a field. As an application we compute the kernel and cokernel of multiplication by θ in Galois cohomology. We also show that the reduced norm map on K₂ of a division algebra of square-free degree is injective.     

Projective Extensions of Fields

Every field K admits proper projective extensions, that is,Galois extensions where the Galois group is a non-trivial projectivegroup, unless K is separably closed or K is a pythagorean formallyreal field without cyclic extensions of odd degree. As a consequence,it turns out that almost all absolute Galois groups decomposeas proper semidirect products. We show that each local field has a unique maximal projectiveextension, and that the same holds for each global field ofpositive characteristic. In characteristic 0, we prove thatLeopoldt's conjecture for all totally real number fields isequivalent to the statement that, for all totally real numberfields, all projective extensions are cyclotomic. So the realizabilityof any non-procyclic projective group as Galois group over Qproduces counterexamples to the Leopoldt conjecture.     

Pro-P galois groups of function fields over local fields

For non-archimedean local field K and a prime number p we compute the finitely generated pro-p (closed) subgroups of the absolute Galois group of K(t). In addition, we characterize the finitely generated pro-p groups which occur as the maximal pro-p Galois group of algebraic extensions of K(t) containing a primitive pth root of unity.     

Polynomials with PSL(2, 7) as Galois group

We describe a technique for determining the set-transitivity of the Galois group of a polynomial over the rationals. As an application we give a short proof that the polynomial P₇(x) = x⁷ ? 154x + 99 has the simple group PSL(2, 7) of order 168 as its Galois group over the rationals. A similar method is used to prove that the associated splitting field is not that of the polynomial x⁷ ? 7x + 3 given by Trinks [9].     

Linear feedback systems and the groups of Galois and Lie

Our purpose in this paper is to delineate precisely the extent to which one can make explicit calculations involving the most basic linear feedback systems. Our results center around the Galois theory of the “root-locus” equation p(s) + kq(s) = 0 and the Lie symmetries associated with the related differential equation p(D)x + k(t)q(D)x = 0, D = d/dt. We show that the Galois theory leads to a more refined classification, but that these theories are related in a substantial way. Considerable insight into this is obtained through the study of the monodromy group associated with algebraic curve defined by p(s) + kq(s) = 0.     

A birational anabelian reconstruction theorem for curves over algebraically closed fields in Arbitrary Characteristic

The aim of Bogomolov’s programme is to prove birational anabelian conjectures for function fields K|k of varieties of dimension ≥ 2 over algebraically closed fields. The present article is concerned with the 1-dimensional case. While it is impossible to recover K|k from its absolute Galois group alone, we prove that it can be recovered from the pair (Aut(\(\bar K\)|k), Aut(\(\bar K\)|K)), consisting of the absolute Galois group of K and the larger group of field automorphisms fixing only the base field.     

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).     

Killing wild ramification

We compute the inertia group of the compositum of wildly ramified Galois covers. It is used to show that even the p-part of the inertia group of a Galois cover of ?¹ branched only at infinity can be reduced if there is a jump in the lower ramification filtration at two and a certain linear disjointness statement holds.     

Galois subfields of tame division algebras

We show that a finite-dimensional tame division algebra D over a Henselian field F has a maximal subfield Galois over F if and only if its residue division algebra \(\overline D \) has a maximal subfield Galois over the residue field \(\overline F \).     

Commutative Abelian Galois extensions of a Banach algebra

Let A be a commutative unital Banach algebra with connected maximal ideal space X. We show that the Gelfand transform induces an isomorphism between the group of commutative Galois extensions of A with given finite Abelian Galois group, and the corresponding group of extensions of C(X). This result is applied, when X is sufficiently nice, to construct a separable projective finitely generated faithful Banach A-algebra whose maximal ideal space is a given finitely fibered covering space of X.     

Groups of p-absolute Galois type that are not absolute Galois groups

Let p be a prime. We study pro-p groups of p-absolute Galois type, as defined by Lam–Liu–Sharifi–Wake–Wang. We prove that the pro-p completion of the right-angled Artin group associated to a chordal simplicial graph is of p-absolute Galois type, and moreover it satisfies a strong version of the Massey vanishing property. Also, we prove that Demushkin groups are of p-absolute Galois type, and that the free pro-p product — and, under certain conditions, the direct product — of two pro-p groups of p-absolute Galois type satisfying the Massey vanishing property, is again a pro-p group of p-absolute Galois type satisfying the Massey vanishing property. Consequently, there is a plethora of pro-p groups of p-absolute Galois type satisfying the Massey vanishing property that do not occur as absolute Galois groups.     

Construction inconditionnelle de groupes de Galois motiviquesAn unconditional construction of motivic Galois groups

We attach to any “classical” Weil cohomology theory over a field a motivic Galois group, defined up to an inner automorphism. We also study the specialisation of numerical motives and the behaviour of motivic Galois group by specialisation. To cite this article: Y. André, B. Kahn, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 989–994.     

Lifting Galois representations over arbitrary number fields

It is proved that every two-dimensional residual Galois representation of the absolute Galois group of an arbitrary number field lifts to a characteristic zero p-adic representation, if local lifting problems at places above p are unobstructed.