In the present paper, we give necessary and sufficient conditions for a birational Galois section of a projective smooth curve over either the field of rational numbers or an imaginary quadratic field to be geometric. As a consequence, we prove that, over such a small number field, to prove the birational section conjecture for projective smooth curves, it suffices to verify that, roughly speak...

We describe some of the basic ideas of Galois theory for commutative S-algebras originally formulated by John Rognes. We restrict attention to the case of finite Galois groups. We describe the general framework developed by Rognes. Central rôles are played by the notion of strong duality and a trace or norm mapping constructed by Greenlees and May in the context of generalized Tate cohomology. ...

We prove Serre’s conjecture for the case of Galois representations of Serre’s weight 2 and level 1. We do this by combining the potential modularity results of Taylor and lowering the level for Hilbert modular forms with a Galois descent argument, properties of universal deformation rings, and the non-existence of p-adic Barsotti-Tate conductor 1 Galois representations proved in [Di3].

In this paper, the changes of representations of a group are used in order to describe its action as algebraic Galois group of an univariate polynomial on the roots of factors of any Lagrange resolvent. By this way, the Galois group of resolvent factors are pre-determinated. In follows, different applications are exposed; in particular, some classical results of algebraic Galois theory.

1. Table of symbols 2 2. Transcendence for Drinfeld modules 2 2.1. Wade’s results 2 2.2. Drinfeld modules 3 2.3. The Weierstraß-Drinfeld correspondence 3 2.4. Carlitz 5 2.5. Yu’s work 6 3. t-Modules 7 3.1. Definitions 7 3.2. Yu’s sub-t-module theorem 8 3.3. Yu’s version of Baker’s theorem 8 3.4. Proof of Baker-Yu 8 3.5. Quasi-periodic functions 9 3.6. Derivatives and linear independence 12 3.7....

In the year 1923 Emil Artin introduced his new L-functions belonging to Galois characters. But his theory was still incomplete in two respects. First, the theory depended on the validity of the General Reciprocity Law which Artin was unable at that time to prove in full generality. Secondly, in the explicit definition of L-functions the ramified primes could not be taken into account; hence tha...

Let K be a complete discrete valuation field with finite residue field F of order q. We call such a field a local field. The geometric Frobenius FrF is the inverse of the map a → a in the absolute Galois group GF = Gal(F̄ /F ). The Weil group WK is defined as the inverse image of the subgroup 〈FrF 〉 ⊂ GF by the canonical map GK = Gal(K̄/K) → GF . For an element σ ∈ WK , let n(σ) denote the intege...

In the mid-1960s Borevič and Faddeev initiated the study of the Galois module structure of groups of pth-power classes of cyclic extensions K/F of pth-power degree. They determined the structure of these modules in the case when F is a local field. In this paper we determine these Galois modules for all base fields F . In 1947 Šafarevič initiated the study of Galois groups of maximal pextension...

What we propose here is to reduce the size of Galois lattices still conserving their formal structure and exhaustivity. For that purpose we use a preliminary partition of the instance set, representing the association of a “type” to each instance. By redefining the notion of extent of a term in order to cope, to a certain degree (denoted as α), with this partition, we define a particular family...

This article continues investigation of ways to obtain pseudo-random sequences over Galois field via generating LRS over Galois ring and complication it. Previous work is available at http://eprint.iacr.org/2016/212 In this work we provide low rank estimations for sequences generated by two different designs based on coordinate sequences of linear recurrent sequences (LRS) of maximal period (MP...