Publications and communications of Gabor Wiese [gabor.wiese@uni.lux]
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See detailUnramifiedness of Galois representations attached to weight one Hilbert modular eigenforms mod p
Dimitrov, Mladen; Wiese, Gabor UL

E-print/Working paper (2015)

The main result of this article states that the Galois representation attached to a Hilbert modular eigenform over Fp of parallel weight one and level prime to p is unramified above p. This includes the ... [more ▼]

The main result of this article states that the Galois representation attached to a Hilbert modular eigenform over Fp of parallel weight one and level prime to p is unramified above p. This includes the important case of eigenforms that do not lift to Hilbert modular forms in characteristic 0 of parallel weight one. The proof is based on the observation that parallel weight one forms in characteristic p embed into the ordinary part of parallel weight p in two different ways per place above p, namely via ‘partial’ Frobenius operators. These are defined in the article along with and based on Hecke operators Tp for p dividing p. The theorem is deduced from known local properties of the Galois representation attached to ordinary eigenforms in characteristic 0. [less ▲]

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See detailCompatible systems of symplectic Galois representations and the inverse Galois problem III. Automorphic construction of compatible systems with suitable local properties
Arias De Reyna Dominguez, Sara UL; Dieulefait, L.; Shin, S. W. et al

in Mathematische Annalen (2015), 361(3), 909-925

This article is the third and last part of a series of three articles about compatible systems of symplectic Galois representations and applications to the inverse Galois problem. This part proves the ... [more ▼]

This article is the third and last part of a series of three articles about compatible systems of symplectic Galois representations and applications to the inverse Galois problem. This part proves the following new result for the inverse Galois problem for symplectic groups. For any even positive integer n and any positive integer d, PSp_n(F_{l^d}) or PGSp_n(F_{l^d}) occurs as a Galois group over the rational numbers for a positive density set of primes l. The result is obtained by showing the existence of a regular, algebraic, self-dual, cuspidal automorphic representation of GL_n(A_Q) with local types chosen so as to obtain a compatible system of Galois representations to which the results from Part II of this series apply. [less ▲]

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See detailAutomorphic Galois representations in the inverse Galois problem
Wiese, Gabor UL

Scientific Conference (2014, June 24)

In the talk I will report on recent results on the inverse Galois problem based on compatible systems of Galois representations coming from modular and automorphic forms. The focus will be on ideas and ... [more ▼]

In the talk I will report on recent results on the inverse Galois problem based on compatible systems of Galois representations coming from modular and automorphic forms. The focus will be on ideas and strategies as well as the obstacles that are preventing us from proving much stronger theorems. In this context, the role of coefficient fields will be particularly highlighted. Many parts are joint work with Sara Arias-de-Reyna, Luis Dieulefait and Sug-Woo Shin. [less ▲]

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See detailApplying automorphic Galois representations in the inverse Galois problem
Wiese, Gabor UL

Scientific Conference (2014, March 13)

In the talk I will report on recent results on the inverse Galois problem based on compatible systems of Galois representations coming from modular and automorphic forms. The focus will be on ideas and ... [more ▼]

In the talk I will report on recent results on the inverse Galois problem based on compatible systems of Galois representations coming from modular and automorphic forms. The focus will be on ideas and strategies as well as the obstacles that are preventing us from proving much stronger theorems. In this context, the role of coefficient fields will be particularly highlighted. [less ▲]

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See detailSur les représentations galoisiennes en théorie de Galois inverse
Wiese, Gabor UL

Presentation (2014, February 10)

On présentera un travail en commun avec Sara Arias-de-Reyna, Luis Dieulefait et Sug Woo Shin où nous réalisons, pour tout n pair et tout d, le groupe PGSp_n(F_{p^d}) ou PSp_n(F_{p^d}) comme groupe de ... [more ▼]

On présentera un travail en commun avec Sara Arias-de-Reyna, Luis Dieulefait et Sug Woo Shin où nous réalisons, pour tout n pair et tout d, le groupe PGSp_n(F_{p^d}) ou PSp_n(F_{p^d}) comme groupe de Galois sur les nombres rationnels, pour p dans un ensemble de densité positive. La démonstration est basée sur des systèmes compatibles de représentations galoisiennes ayant des propriétés locales spéciales. Au début de l'exposé on esquissera la stratégie; elle est basée sur un travail en commun avec Dieulefait dans le cas de la dimension 2. On expliquera ensuite l'existence d'un corps global et minimal tel que presque toute représentation résiduelle d'un système compatible peut être définie projectivement sur le corps résiduel. En plus, on énoncera une classification simple des représentations symplectiques contenant une transvection dans leur image. Finalement, on expliquera l'existence du système compatible désiré et comment utiliser des techniques de minoration du niveau pour obtenir notre application au problème de Galois inverse. [less ▲]

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See detailOn equidistribution of signs and the Sato-Tate conjecture
Wiese, Gabor UL

Presentation (2014, January 27)

The talk concerns our joint work with Sara Arias-de-Reyna and Ilker Inam about the equidistribution of signs of coefficients of half-integral weight modular forms. The basic idea is to apply the Shimura ... [more ▼]

The talk concerns our joint work with Sara Arias-de-Reyna and Ilker Inam about the equidistribution of signs of coefficients of half-integral weight modular forms. The basic idea is to apply the Shimura lift to get into integral weight modular forms, where one can make use of the Sato-Tate theorem. The importance of error terms for the Sato-Tate theorem is stressed in this context. [less ▲]

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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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