Part I. The origins of the Langlands Program 9 1. The Langlands correspondence over number fields 9 1.1. Galois group 9 1.2. Abelian class field theory 10 1.3. Frobenius automorphisms 13 1.4. Rigidifying ACFT 14 1.5. Non-abelian generalization? 15 1.6. Automorphic representations of GL2(AQ) and modular forms 18 1.7. Elliptic curves and Galois representations 22 2. From number fields to function...

Using the middle convolution functor MCχ introduced by N. Katz, we prove the existence of rigid local systems whose monodromy is dense in the simple algebraic group G2. We derive the existence of motives for motivated cycles which have a motivic Galois group of type G2. Granting Grothendieck’s standard conjectures, the existence of motives with motivic Galois group of type G2 can be deduced, gi...

In this note we study a certain formal group law over a complete discrete valuation ring F[[un−1]] of characteristic p > 0 which is of height n over the closed point and of height n− 1 over the generic point. By adjoining all coefficients of an isomorphism between the formal group law on the generic point and the Honda group law Hn−1 of height n − 1, we get a Galois extension of the quotient fi...

This paper is concerned with difference equations on elliptic curves. We establish some general properties of the difference Galois groups of equations of order two, and give applications to the calculation of some difference Galois groups. For instance, our results combined with a result from transcendence theory due to Schneider allow us to identify a large class of discrete Lamé equations wi...

Proof. The discriminant of x + 1 is D = 256 = 2. We have x + 1 ≡ (x + 1) (mod 2). Let p be an odd prime (so p D), and suppose the irreducible factors of x + 1 have degrees n1, n2, . . . , nk. By Corollary 41, the Galois group of x + 1 contains an element with cycle structure (n1, n2, . . . , nk). Since the Galois group of x +1 over Q is the Klein 4-group, in which every element has order dividi...

1. Let A be a complete field with respect to a discreet valuation,2 and suppose that the residue class field of K is finite and has characteristic p. A group which is finite and whose order is not divisible by p is said to be ^-regular. A normal extension of A whose Galois group is ^-regular will be called a ^-regular extension of K. The object of this paper is to characterize those groups whic...

We present short elementary proofs of the well-known Ruffini-Abel-Galois theorems on unsolvability of algebraic equations in radicals. This proof is obtained from existing expositions by stripping away material not required for the proof (but presumably required elsewhere). In particular, we do not use the terms ‘Galois group’ and even ‘group’. However, our presentation is a good way to learn (...

In this article we prove the following: A topos with a point is connected atomic if and only if it is the classifying topos of a localic group, and this group can be taken to be the locale of automorphisms of the point. We explain and give the necessary definitions to understand this statement. The hard direction in this equivalence was first proved in print in [4], Theorem 1, Section 3, Chapte...

Let k be a perfect field and k̄ an algebraic closure of k. Let f be a separable univariate polynomial of k[X] with degree n, and Ω be an ordered set of the n distinct roots of f in k̄. In Valibouze (1999) the notion of ideal of Ω-relations invariant by a subset L of the symmetric group of degree n is introduced. It generalizes the notion of ideal of relations and the notion of ideal of symmetric ...

GivenS|R a finite Galois extension of finite chain rings andB anS-linear code we define twoGalois operators, the closure operator and the interior operator. We proof that a linear code is Galois invariant if and only if the row standard form of its generator matrix has all entries in the fixed ring by the Galois group and show a Galois correspondence in the class of S-linear codes. As applicati...