Automatic realization of Hopf Galois structures

We consider Hopf Galois structures on a separable field extension [Formula: see text] of degree [Formula: see text], for [Formula: see text] an odd prime number, [Formula: see text]. For [Formula: see text], we prove that [Formula: see text] has at most one abelian type of Hopf Galois structures. For a nonabelian group [Formula: see text] of order [Formula: see text], with commutator subgroup of order [Formula: see text], we prove that if [Formula: see text] has a Hopf Galois structure of type [Formula: see text], then it has a Hopf Galois structure of type [Formula: see text], where [Formula: see text] is an abelian group of order [Formula: see text] and having the same number of elements of order [Formula: see text] as [Formula: see text], for [Formula: see text].

Admissible Galois Structures on the categories dual to some varieties of universal algebras

The subject of the paper is suggested by G. Janelidze and motivated by his earlier result giving a positive answer to the question posed by S. MacLane whether the Galois theory of homogeneous linear ordinary differential equations over a differential field (which is Kolchin–Ritt theory and an algebraic version of Picard–Vessiot theory) can be obtained as a particular case of G. Janelidze’s Galois theory in categories. One ground category in the Galois structure involved in this theory is dual to the category of commutative rings with unit, and another one is dual to the category of commutative differential rings with unit. In the present paper, we apply the general categorical construction, the particular case of which gives this Galois structure, by replacing “commutative rings with unit” by algebras from any variety \mathscr{V} of universal algebras satisfying the amalgamation property and a certain condition (of the syntactical nature) for elements of amalgamated free products which was introduced earlier, and replacing “commutative differential rings with unit” by \mathscr{V}-algebras equipped with additional unary operations which satisfy some special identities to construct a new Galois structure. It is proved that this Galois structure is admissible. Moreover, normal extensions with respect to it are characterized in the case where \mathscr{V} is any of the following varieties: abelian groups, loops and quasigroups.

Troisi\`eme groupe de cohomologie non ramifi\'ee des torseurs universels sur les surfaces rationnelles

Let $k$ a field of characteristic zero. Let $X$ be a smooth, projective, geometrically rational $k$-surface. Let $\mathcal{T}$ be a universal torsor over $X$ with a $k$-point et $\mathcal{T}^c$ a smooth compactification of $\mathcal{T}$. There is an open question: is $\mathcal{T}^c$ $k$-birationally equivalent to a projective space? We know that the unramified cohomology groups of degree 1 and 2 of $\mathcal{T}$ and $\mathcal{T}^c$ are reduced to their constant part. For the analogue of the third cohomology groups, we give a sufficient condition using the Galois structure of the geometrical Picard group of $X$. This enables us to show that $H^{3}_{nr}(\mathcal{T}^{c},\mathbb{Q}/\mathbb{Z}(2))/H^3(k,\mathbb{Q}/\mathbb{Z}(2))$ vanishes if $X$ is a generalised Ch\^atelet surface and that this group is reduced to its $2$-primary part if $X$ is a del Pezzo surface of degree at least 2. Comment: 27 page, in French

ON THE GALOIS STRUCTURE OF ARITHMETIC COHOMOLOGY I: COMPACTLY SUPPORTED -ADIC COHOMOLOGY

We investigate the Galois structures of $p$-adic cohomology groups of general $p$-adic representations over finite extensions of number fields. We show, in particular, that as the field extensions vary over natural families the Galois modules formed by these cohomology groups always decompose as the direct sum of a projective module and a complementary module of bounded $p$-rank. We use this result to derive new (upper and lower) bounds on the changes in ranks of Selmer groups over extensions of number fields and descriptions of the explicit Galois structures of natural arithmetic modules.

On Mordell–Weil groups and congruences between derivatives of twisted Hasse–Weil L-functions

LetAbe an abelian variety defined over a number fieldkand letFbe a finite Galois extension ofk. Letpbe a prime number. Then under certain not-too-stringent conditions onAandFwe compute explicitly the algebraic part of thep-component of the equivariant Tamagawa number of the pair(h^{1}(A_{/F})(1),\mathbb{Z}[{\rm Gal}(F/k)]). By comparing the result of this computation with the theorem of Gross and Zagier we are able to give the first verification of thep-component of the equivariant Tamagawa number conjecture for an abelian variety in the technically most demanding case in which the relevant Mordell–Weil group has strictly positive rank and the relevant field extension is both non-abelian and of degree divisible byp. More generally, our approach leads us to the formulation of certain precise families of conjecturalp-adic congruences between the values ats=1of derivatives of the Hasse–WeilL-functions associated to twists ofA, normalised by a product of explicit equivariant regulators and periods, and to explicit predictions concerning the Galois structure of Tate–Shafarevich groups. In several interesting cases we provide theoretical and numerical evidence in support of these more general predictions.