In this paper, we are interested in the tame version of the Fontaine-Mazur conjecture. After recalling the role of étale cohomology in the context of this conjecture, we establish a relationship between it and the computation of the cohomological dimension of the pro-p-groups GS that appear. We then look at this conjecture by viewing the pro-pproup GS as a quotient of a Galois group with rami c...

0.2. Structure of the proof. 1. As in [dJ], we choose a projection X 99K P of relative dimension 1, and apply semistable reduction to obtain a model X ′ → B over a suitable Galois base change B → P , with Galois group G. 2. We apply induction on the dimension to B , therefore we may assume that B is smooth, and that the discriminant locus of X ′ → B is a G-strict divisor of normal crossings. 3....

Which invariants of a Galois p-extension of local number fields L/K (residue field of char p, and Galois group G) determine the structure of the ideals in L as modules over the group ring Zp[G], Zp the p-adic integers? We consider this question within the context of elementary abelian extensions, though we also briefly consider cyclic extensions. For elementary abelian groups G, we propose and ...

Let p be prime. Let L/K be a finite, totally ramified, purely inseparable extension of local fields, [L : K] = p, n ≥ 2. It is known that L/K is Hopf Galois for numerous Hopf algebras H, each of which can act on the extension in numerous ways. For a certain collection of such H we construct “Hopf Galois scaffolds” which allow us to obtain a Hopf analogue to the Normal Basis Theorem for L/K. The...

Differential Galois theory has known an outburst of activity in the last decade. To pinpoint what triggered this renewal is probably a matter of personal taste; all the same, let me start the present review by a tentative list, restricted on purpose to “non-obviously differential” occurrences of the theory (and also, as in the book under review, to the Galois theory of linear differential equat...

We present an efficient erasure decoding algorithm for generalized Reed Solomon codes constructed by utilizing the structure of the inverse of the VanderMonde matrices. Given an [n, k] generalized Reed Solomon code, decoding for r erasures, where r = n − k, requires rk + n Galois field multiplications and r(k − 1) Galois field additions after setting up the decoding structures for an erasure pa...

Galois corings with a group-like element [4] provide a neat framework to understand the analogies between several theories like the Faithfully Flat Descent for (noncommutative) ring extensions [26], Hopf-Galois algebra extensions [27], or noncommutative Galois algebra extensions [23, 15]. A Galois coring is isomorphic in a canonical way to the Sweedler’s canonical coring A ⊗B A associated to a ...

Let p > 2 be prime, and let n,m ∈ N be given. For cyclic extensions E/F of degree p that contain a primitive pth root of unity, we show that the associated Fp[Gal(E/F )]-modules H(GE , μp) have a sparse decomposition. When E/F is additionally a subextension of a cyclic, degree p extension E/F , we give a more refined Fp[Gal(E/F )]-decomposition of H (GE , μp).

Let F be a field containing a primitive pth root of unity, and let U be an open normal subgroup of index p of the absolute Galois group GF of F . Using the Bloch-Kato Conjecture we determine the structure of the cohomology group H(U, Fp) as an Fp[GF /U ]-module for all n ∈ N. Previously this structure was known only for n = 1, and until recently the structure even of H(U, Fp) was determined onl...