For a finite Galois extension of fields L/k with Galois group G, we study a functor from the G-equivariant stable homotopy category to the stable motivic homotopy category over k induced by the classical Galois correspondence. We show that after completing at a prime and η (the motivic Hopf map) this results in a full and faithful embedding whenever k is real closed and L = k[i]. It is a full a...

In previous work [7], the authors constructed and studied a lift of the Galois correspondence to stable homotopy categories. In particular, if L/k is a nite Galois extension of elds with Galois group G, there is a functor c∗ L/k : SHG → SHk from the G-equivariant stable homotopy category to the stable motivic homotopy category over k such that c∗ L/k (G/H+) = Spec(L)+. The main theorem of [7] s...

The Jacobson-Bourbaki Theorem for division rings was formulated in terms of corings by Sweedler in [14]. Finiteness conditions hypotheses are not required in this new approach. In this paper we extend Sweedler’s result to simple artinian rings using a particular class of corings, comatrix corings. A Jacobson-Bourbaki like correspondence for simple artinian rings is then obtained by duality.

In the late 1960s, Robert Langlands proposed a new and far-reaching connection between the representation theory of Lie groups over real and -adic fields, and the structure of the Galois groups of these fields [24]. Even though this local Langlands correspondence remains largely conjectural, the relation that it predicts between representation theory and number theory has profoundly changed our...

The aim of this paper is to use the “amplification technique” to obtain estimates on the dimension of spaces of automorphic forms associated to Galois representations; these bounds improve nontrivially on the work of Duke ([D]). A cuspidal representation π of GL2(AQ) is associated to a 2-dimensional Galois representation ρ : Gal(Q/Q) → GL2(C) if, for each place v, the local representation πv is...

This thesis is concerned with the definition and the study of properties of homotopic Hopf-Galois extensions in the category Ch 0 k of chain complexes over a field k, equipped with its projective model structure. Given a differential graded k-Hopf algebra H of finite type, we define a homotopic H-Hopf-Galois extension to be a morphism ' : B ! A of augmented H-comodule dg-k-algebras, where B is ...

Differential Galois theory, the theory of strongly normal extensions, has unfortunately languished. This may be due to its reliance on Kolchin’s elegant, but not widely adopted, axiomatization of the theory of algebraic groups. This paper attempts to revive the theory using a differential scheme in place of those axioms. We also avoid using a universal differential field, instead relying on a c...

We show that the p-adic Galois representations attached to Hilbert modular forms of motivic weight are potentially semistable at all places above p and are compatible with the local Langlands correspondence at these places, proving this for those forms not covered by the previous works of T. Saito and of D. Blasius and J. Rogawski. 2000 Mathematics Subject Classification: 11F80, 11F41

We analyze the notions of indiscernibility and indeterminacy in the light of the Galois theory of field extensions and the generalization to K -algebras proposed by Grothendieck. Grothendieck’s reformulation of Galois theory permits to recast the Galois correspondence between symmetry groups and invariants as a Galois– Grothendieck duality between G-spaces and the minimal observable algebras th...

For a prime number p > 2, we give a direct proof of Breuil’s classification of finite flat group schemes killed by p over the valuation ring of a p-adic field with perfect residue field. As application we establish a correspondence between finite flat group schemes and Faltings’s strict modules which respects associated Galois modules via the Fontaine-Wintenberger field-of-norms functor