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See detailWinter School on Galois Theory, Volume 2
Wiese, Gabor UL; Arias De Reyna Dominguez, Sara UL; Bary-Soroker, Lior

Book published by University of Luxembourg / Campus Kirchberg (2013)

Galois Theory plays a key role in many mathematical disciplines, such as number theory, algebra, topology, and geometry. This special volume of the Luxembourg based peer-reviewed mathematics journal ... [more ▼]

Galois Theory plays a key role in many mathematical disciplines, such as number theory, algebra, topology, and geometry. This special volume of the Luxembourg based peer-reviewed mathematics journal "Travaux mathématiques" unites four instructional texts that have grown out of lectures delivered at the Winter School on Galois Theory held at the University of Luxembourg in February 2012. It also includes one research article. Gebhard Böckle's contribution is a quite comprehensive survey on Galois representations. It focusses on the key ideas, and the long list of recommended references enables the reader to pursue himself/herself any of the mentioned topics in greater depth. Michael Schein's notes sketch the proof due to Khare and Wintenberger of one of the major theorems in arithmetic algebraic geometry in recent years, namely Serre's Modularity Conjecture. Moshe Jarden's contribution is based on his book on algebraic patching. It develops the method of algebraic patching from scratch and gives applications in contemporary Galois theory. David Harbater's text is complementary to Jarden's notes, and describes recent applications of patching in other aspects of algebra, for example: differential algebra, local-global principles, quadratic forms, and more. The focus is on the big picture and on providing the reader with intuition. The research article by Wulf-Dieter Geyer and Moshe Jarden concerns model completeness of valued PAC fields. [less ▲]

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See detailWinter School on Galois Theory, Volume 1
Wiese, Gabor UL; Arias De Reyna Dominguez, Sara UL; Bary-Soroker, Lior

Book published by University of Luxembourg / Campus Kirchberg (2013)

Galois Theory plays a key role in many mathematical disciplines, such as number theory, algebra, topology, and geometry. This special volume of the Luxembourg based peer-reviewed mathematics journal ... [more ▼]

Galois Theory plays a key role in many mathematical disciplines, such as number theory, algebra, topology, and geometry. This special volume of the Luxembourg based peer-reviewed mathematics journal "Travaux mathématiques" unites two instructional texts that have grown out of lectures delivered at the Winter School on Galois Theory held at the University of Luxembourg in February 2012. The contribution by Wulf-Dieter Geyer is about "Field Theory". It can be considered as a textbook in its own right. It manages to start at the level that any student possesses after any introductory algebra course and nevertheless to lead the reader to very advanced field theory at the frontier of current research, and to cover a wealth of material. Many examples are contained, which nicely enlighten the presented concepts, very often providing counterexamples that show why certain hypotheses are necessary. One also finds a chapter on the history of field theory as well as other historical remarks throughout the text. The second contribution addresses "Profinite Groups". It is written by Luis Ribes, who is the author of two standard books on this subject. Being necessarily much shorter than the two books, it has the feature of presenting an overview stressing the main concepts and the links with Galois Theory. Since for those proofs which are not included precise references are given, the notes, due to their conciseness and nevertheless great amount of material, constitute an excellent starting point for any Master or PhD student willing to learn this subject. [less ▲]

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See detailCommutative Algebra (Master in Mathematics, 2013)
Wiese, Gabor UL

Learning material (2013)

Lecture notes from the 1st year of Master in Mathematics at the University of Luxembourg. Rings of integers of number fields and coordinate rings of plane curves are the central examples around which the ... [more ▼]

Lecture notes from the 1st year of Master in Mathematics at the University of Luxembourg. Rings of integers of number fields and coordinate rings of plane curves are the central examples around which the theory is developed: integrality, noetherian rings, localisation, Noether normalisation, Hilbert's Nullstellensatz, etc. [less ▲]

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See detailAlgèbre 1 (BASI filière mathématiques, 2013)
Wiese, Gabor UL; David, Agnès UL

Learning material (2013)

Lecture notes written in French from the Algebra 1 lecture in the 1st term of the Bachelor programme BASI branch Mathematics at the University of Luxembourg. The lecture starts with preliminaries on logic ... [more ▼]

Lecture notes written in French from the Algebra 1 lecture in the 1st term of the Bachelor programme BASI branch Mathematics at the University of Luxembourg. The lecture starts with preliminaries on logic, sets and functions, it builds the natural numbers (almost) from the Peano axioms, then constructs the integers and the rationals. Groups and rings are introduced in that context. The most basic definitions and results from abstract linear algebra are also given. The course finishes with some basic group theory. [less ▲]

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See detailAn Application of Maeda's Conjecture to the Inverse Galois Problem
Wiese, Gabor UL

in Mathematical Research Letters (2013), 20(5), 985-993

It is shown that Maeda's conjecture on eigenforms of level 1 implies that for every positive even d and every p in a density-one set of primes, the simple group PSL_2(F_{p^d}) occurs as the Galois group ... [more ▼]

It is shown that Maeda's conjecture on eigenforms of level 1 implies that for every positive even d and every p in a density-one set of primes, the simple group PSL_2(F_{p^d}) occurs as the Galois group of a number field ramifying only at p. [less ▲]

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See detailOn modular Galois representations modulo prime powers
Chen, Imin; Kiming, Ian; Wiese, Gabor UL

in International Journal of Number Theory (2013), 9(1), 91--113

We study modular Galois representations mod p^m. We show that there are three progressively weaker notions of modularity for a Galois representation mod p^m: we have named these `strongly', `weakly', and ... [more ▼]

We study modular Galois representations mod p^m. We show that there are three progressively weaker notions of modularity for a Galois representation mod p^m: we have named these `strongly', `weakly', and `dc-weakly' modular. Here, `dc' stands for `divided congruence' in the sense of Katz and Hida. These notions of modularity are relative to a fixed level M. Using results of Hida we display a `stripping-of-powers of p away from the level' type of result: A mod p^m strongly modular representation of some level Np^r is always dc-weakly modular of level N (here, N is a natural number not divisible by p). We also study eigenforms mod p^m corresponding to the above three notions. Assuming residual irreducibility, we utilize a theorem of Carayol to show that one can attach a Galois representation mod p^m to any `dc-weak' eigenform, and hence to any eigenform mod p^m in any of the three senses. We show that the three notions of modularity coincide when m=1 (as well as in other, particular cases), but not in general. [less ▲]

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See detailModulformen und das inverse Galois-Problem
Wiese, Gabor UL

Scientific Conference (2012, September 19)

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See detailOn modular Galois representations modulo prime powers
Wiese, Gabor UL

Presentation (2012, May 30)

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See detailSymplectic Galois representations and applications to the inverse Galois problem
Wiese, Gabor UL

Presentation (2012, May 30)

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See detailModulare Galois-Darstellungen und Computeralgebra
Wiese, Gabor UL

Scientific Conference (2012, May 12)

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See detailFour lectures on modular forms and Galois representations
Wiese, Gabor UL

Presentation (2012, February)

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See detailSymplectic Galois representations and applications to the inverse Galois problem
Wiese, Gabor UL

Presentation (2012, January 19)

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See detailAlgèbre 3 (théorie des corps et théorie de Galois)
Wiese, Gabor UL

Learning material (2012)

Lecture notes written in French from the Algebra 3 lecture in the 3rd term of the Bachelor programme BASI branch Mathematics (old version) at the University of Luxembourg. The lecture covers field theory ... [more ▼]

Lecture notes written in French from the Algebra 3 lecture in the 3rd term of the Bachelor programme BASI branch Mathematics (old version) at the University of Luxembourg. The lecture covers field theory and Galois theory and includes a treatment of the solvability of equations by radicals and a treatment of classical construction problems with ruler and compass. [less ▲]

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See detailLectures on Modular Galois Representations Modulo Prime Powers
Wiese, Gabor UL

Presentation (2011, December)

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See detailA computational study of the asymptotic behaviour of coefficient fields of modular forms
Mohyla, Marcel; Wiese, Gabor UL

in Actes de la Conférence ``Théorie des Nombres et Applications'' (2011)

The article motivates, presents and describes large computer calculations concerning the asymptotic behaviour of arithmetic properties of coefficient fields of modular forms. The observations suggest ... [more ▼]

The article motivates, presents and describes large computer calculations concerning the asymptotic behaviour of arithmetic properties of coefficient fields of modular forms. The observations suggest certain patterns, which deserve further study. [less ▲]

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See detailOn modular forms and the inverse Galois problem
Dieulefait, Luis; Wiese, Gabor UL

in Transactions of the American Mathematical Society (2011), 363(9), 4569--4584

In this article new cases of the Inverse Galois Problem are established. The main result is that for a fixed integer n, there is a positive density set of primes p such that PSL_2(F_{p^n}) occurs as the ... [more ▼]

In this article new cases of the Inverse Galois Problem are established. The main result is that for a fixed integer n, there is a positive density set of primes p such that PSL_2(F_{p^n}) occurs as the Galois group of some finite extension of the rational numbers. These groups are obtained as projective images of residual modular Galois representations. Moreover, families of modular forms are constructed such that the images of all their residual Galois representations are as large as a priori possible. Both results essentially use Khare's and Wintenberger's notion of good-dihedral primes. Particular care is taken in order to exclude nontrivial inner twists. [less ▲]

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See detailOn mod $p$ representations which are defined over $\Bbb F_p$: II
Kilford, L. J. P.; Wiese, Gabor UL

in Glasgow Mathematical Journal (2010), 52(2), 391--400

The behaviour of Hecke polynomials modulo p has been the subject of some study. In this note we show that, if p is a prime, the set of integers N such that the Hecke polynomials T^{N,\chi}_{l,k} for all ... [more ▼]

The behaviour of Hecke polynomials modulo p has been the subject of some study. In this note we show that, if p is a prime, the set of integers N such that the Hecke polynomials T^{N,\chi}_{l,k} for all primes l, all weights k>1 and all characters \chi taking values in {+1,-1} splits completely modulo p has density 0, unconditionally for p=2 and under the Cohen-Lenstra heuristics for odd p. The method of proof is based on the construction of suitable dihedral modular forms. [less ▲]

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See detailComputing congruences of modular forms and Galois representations modulo prime powers
Taixés i Ventosa, Xavier; Wiese, Gabor UL

in Arithmetic, geometry, cryptography and coding theory 2009 (2010)

This article starts a computational study of congruences of modular forms and modular Galois representations modulo prime powers. Algorithms are described that compute the maximum integer modulo which two ... [more ▼]

This article starts a computational study of congruences of modular forms and modular Galois representations modulo prime powers. Algorithms are described that compute the maximum integer modulo which two monic coprime integral polynomials have a root in common in a sense that is defined. These techniques are applied to the study of congruences of modular forms and modular Galois representations modulo prime powers. Finally, some computational results with implications on the (non-)liftability of modular forms modulo prime powers and possible generalisations of level raising are presented. [less ▲]

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See detailDie Serresche Modularitätsvermutung und Computer-Algebra
Wiese, Gabor UL

in Computeralgebra-Rundbrief (2010), 47

In diesem Artikel für Nichtspezialisten wird die kürzlich von Khare, Wintenberger und Kisin bewiesene Serresche Modularitätsvermutung vorgestellt und ihre Bedeutung in der Computer-Algebra erläutert.

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See detailOn modular symbols and the cohomology of Hecke triangle surfaces
Wiese, Gabor UL

in International Journal of Number Theory (2009), 5(1), 89--108

The aim of this article is to give a concise algebraic treatment of the modular symbols formalism, generalised from modular curves to Hecke triangle surfaces. A sketch is included of how the modular ... [more ▼]

The aim of this article is to give a concise algebraic treatment of the modular symbols formalism, generalised from modular curves to Hecke triangle surfaces. A sketch is included of how the modular symbols formalism gives rise to the standard algorithms for the computation of holomorphic modular forms. Precise and explicit connections are established to the cohomology of Hecke triangle surfaces and group cohomology. In all the note a general commutative ring is used as coefficient ring in view of applications to the computation of modular forms over rings different from the complex numbers. [less ▲]

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