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...

In this note we study Morita contexts and Galois extensions for corings. For a coring C over a (not necessarily commutative) ground ring A we give equivalent conditions for M to satisfy the weak. resp. the strong structure theorem. We also characterize the so called cleft C-Galois extensions over commutative rings. Our approach is similar to that of Y. Doi and A. Masuoka in their work on (cleft...

The Langlands philosophy contemplates the relation between auto-morphic representations and Galois representations. A particularly interesting case is that of the non-selfdual automorphic representations of GL 3. Clozel conjectured that the L-functions of certain of these are equal to L-functions of Galois representations. Here we announce that we found an example of such an automorphic represe...

We mainly investigate the structures of skew cyclic and skew quasicyclic codes of arbitrary length over Galois rings. Similar to [5], our results show that the skew cyclic codes are equivalent to either cyclic and quasi-cyclic codes over Galois rings. Moreover, we give a necessary and sufficient condition for skew cyclic codes over Galois rings to be free. A sufficient condition for 1-generator...

We study the action of the absolute Galois group Gal(Q̄/Q) on the algebraic fundamental group of P1C\{0, 1,∞}. We define a class of functions called Belyi-extending maps, and using them we are able to (1) give new Galois invariants of Grothendieck’s dessins d’enfants and (2) give explicit relations on the fσ in terms of which the Galois action on π̂1(P 1 C \ {0, 1,∞}) is known.

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...

We describe an algorithm for computing the Galois automorphisms of a Galois extension which generalizes the algorithm of Acciaro and Klüners to the non-Abelian case. This is much faster in practice than algorithms based on LLL or factorization.

We show that the Galois group of any Schubert problem involving lines in projective space contains the alternating group. This constitutes the largest family of enumerative problems whose Galois groups have been largely determined. Using a criterion of Vakil and a special position argument due to Schubert, our result follows from a particular inequality among Kostka numbers of two-rowed tableau...

A dynamical model is constructed of the northern summertime global circulation, maintained by empirically derived forcing, based on the same dynamical code that has recently been used to study African easterly waves (AEWs) as convectively triggered perturbations (Thorncroft et al.; Leroux and Hall). In the configuration used here, the model faithfully simulates the observed mean distributions o...

Let N be a bijective measure space. D. Sasaki’s classification of positive definite, globally universal domains was a milestone in homological Galois theory. We show that |O| ≤ K. The groundbreaking work of H. Thomas on contra-arithmetic planes was a major advance. Recent developments in analytic group theory [36, 36] have raised the question of whether Fréchet’s conjecture is false in the cont...