Publications and communications of Gabor Wiese [gabor.wiese@uni.lux]
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See detailOn Galois Representations of Weight One
Wiese, Gabor UL

in Documenta Mathematica (2014), 19

A two-dimensional Galois representation into the Hecke algebra of Katz modular forms of weight one over a finite field of characteristic p is constructed and is shown to be unramified at p in most cases.

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See detailApplying modular Galois representations to the Inverse Galois Problem
Wiese, Gabor UL

in Oberwolfach Reports (2014), 11(1), 305-309

For many finite groups, the Inverse Galois Problem can be approached through modular/automorphic Galois representations. This is a report explaining the basic strategy, ideas and methods behind some ... [more ▼]

For many finite groups, the Inverse Galois Problem can be approached through modular/automorphic Galois representations. This is a report explaining the basic strategy, ideas and methods behind some recent results. It focusses mostly on the 2-dimensional case and underlines in particular the importance of understanding coefficient fields. [less ▲]

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See detailComputations with Modular Forms: Proceedings of a Summer School and Conference, Heidelberg, August/September 2011
Wiese, Gabor UL; Böckle, Gebhard

Book published by Springer (2014)

This volume contains original research articles, survey articles and lecture notes related to the Computations with Modular Forms 2011 Summer School and Conference, held at the University of Heidelberg. A ... [more ▼]

This volume contains original research articles, survey articles and lecture notes related to the Computations with Modular Forms 2011 Summer School and Conference, held at the University of Heidelberg. A key theme of the Conference and Summer School was the interplay between theory, algorithms and experiment. The 14 papers offer readers both, instructional courses on the latest algorithms for computing modular and automorphic forms, as well as original research articles reporting on the latest developments in the field. The three Summer School lectures provide an introduction to modern algorithms together with some theoretical background for computations of and with modular forms, including computing cohomology of arithmetic groups, algebraic automorphic forms, and overconvergent modular symbols. The 11 Conference papers cover a wide range of themes related to computations with modular forms, including lattice methods for algebraic modular forms on classical groups, a generalization of the Maeda conjecture, an efficient algorithm for special values of p-adic Rankin triple product L-functions, arithmetic aspects and experimental data of Bianchi groups, a theoretical study of the real Jacobian of modular curves, results on computing weight one modular forms, and more. [less ▲]

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See detailImages of Galois representations and the inverse Galois problem
Wiese, Gabor UL

Presentation (2013, October 10)

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See detailOn symplectic Galois representations and the inverse Galois problem
Wiese, Gabor UL

Scientific Conference (2013, September 27)

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See detailOn Serre weights and discriminants
Wiese, Gabor UL

Scientific Conference (2013, August 28)

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See detailBeschleunigung durch Abstraktion
Wiese, Gabor UL

Conference given outside the academic context (2013)

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See detailQuestions on Galois Representations Modulo Prime Powers
Wiese, Gabor UL

Presentation (2013, April 23)

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

Presentation (2013, April 19)

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See detailFour lectures on Modular Galois Representations and Applications
Wiese, Gabor UL

Presentation (2013, April)

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See detailModular forms and the inverse Galois problem
Wiese, Gabor UL

Scientific Conference (2013, February 02)

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See detailModular forms and the inverse Galois problem
Wiese, Gabor UL

Presentation (2013, January 17)

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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 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 detailEquidistribution of signs for modular eigenforms of half integral weight
Inam, Ilker; Wiese, Gabor UL

in Archiv der Mathematik [=ADM] (2013), 101(4), 331--339

Let f be a cusp form of weight k+1/2 and at most quadratic nebentype character whose Fourier coefficients a(n) are all real. We study an equidistribution conjecture of Bruinier and Kohnen for the signs of ... [more ▼]

Let f be a cusp form of weight k+1/2 and at most quadratic nebentype character whose Fourier coefficients a(n) are all real. We study an equidistribution conjecture of Bruinier and Kohnen for the signs of a(n). We prove this conjecture for certain subfamilies of coefficients that are accessible via the Shimura lift by using the Sato-Tate equidistribution theorem for integral weight modular forms. Firstly, an unconditional proof is given for the family {a(tp^2)}_p where t is a squarefree number and p runs through the primes. In this case, the result is in terms of natural density. To prove it for the family {a(tn^2)}_n where t is a squarefree number and n runs through all natural numbers, we assume the existence of a suitable error term for the convergence of the Sato-Tate distribution, which is weaker than one conjectured by Akiyama and Tanigawa. In this case, the results are in terms of Dedekind-Dirichlet density. [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 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 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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