Publications and communications of Gabor Wiese [gabor.wiese@uni.lux]
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See detailSur les représentations galoisiennes des formes modulaires de Hilbert de poids un en caractéristique p
Wiese, Gabor UL

Presentation (2017, April 11)

Dans cet exposé, je vais donner les idées principales de mon travail avec Mladen Dimitrov dans lequel nous démontrons que la représentation galoisienne associée à toute forme modulaire de Hilbert de poids ... [more ▼]

Dans cet exposé, je vais donner les idées principales de mon travail avec Mladen Dimitrov dans lequel nous démontrons que la représentation galoisienne associée à toute forme modulaire de Hilbert de poids un en caractéristique p et de niveau premier à p est non-ramifiée en p. Ce résultat s'applique notamment à des formes qui ne se relèvent pas en caractéristique zéro, et peut être vu comme la précision d'un aspect du poids dans la généralisation de la conjecture de modularité de Serre aux formes de Hilbert. [less ▲]

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See detailSur les représentations galoisiennes des formes modulaires de Hilbert de poids un en caractéristique p
Wiese, Gabor UL

Presentation (2017, January 19)

Dans cet exposé, je vais donner les idées principales de mon travail avec Mladen Dimitrov dans lequel nous démontrons que la représentation galoisienne associée à toute forme modulaire de Hilbert de poids ... [more ▼]

Dans cet exposé, je vais donner les idées principales de mon travail avec Mladen Dimitrov dans lequel nous démontrons que la représentation galoisienne associée à toute forme modulaire de Hilbert de poids un en caractéristique p et de niveau premier à p est non-ramifiée en p. Ce résultat s'applique notamment à des formes qui ne se relèvent pas en caractéristique zéro, et peut être vu comme la précision d'un aspect du poids dans la généralisation de la conjecture de modularité de Serre aux formes de Hilbert. [less ▲]

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See detailCompatible systems of symplectic Galois representations and the inverse Galois problem I. Images of projective representations
Arias De Reyna Dominguez, Sara UL; Dieulefait, Luis; Wiese, Gabor UL

in Transactions of the American Mathematical Society (2017), 369

This article is the first part of a series of three articles about compatible systems of symplectic Galois representations and applications to the inverse Galois problem. In this first part, we determine ... [more ▼]

This article is the first part of a series of three articles about compatible systems of symplectic Galois representations and applications to the inverse Galois problem. In this first part, we determine the smallest field over which the projectivisation of a given symplectic group representation satisfying some natural conditions can be defined. The answer only depends on inner twists. We apply this to the residual representations of a compatible system of symplectic Galois representations satisfying some mild hypothesis and obtain precise information on their projective images for almost all members of the system, under the assumption of huge residual images, by which we mean that a symplectic group of full dimension over the prime field is contained up to conjugation. Finally, we obtain an application to the inverse Galois problem. [less ▲]

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See detailOn Galois representations of mod p Hilbert eigenforms of weight one
Wiese, Gabor UL

Scientific Conference (2016, August 16)

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See detailClassification of subgroups of symplectic groups over finite fields containing a transvection
Arias De Reyna Dominguez, Sara UL; Dieulefait, Luis; Wiese, Gabor UL

in Demonstratio Mathematica (2016), 49(2), 129-148

In this note we give a self-contained proof of the following classification (up to conjugation) of subgroups of the general symplectic group of dimension n over a finite field of characteristic l, for l ... [more ▼]

In this note we give a self-contained proof of the following classification (up to conjugation) of subgroups of the general symplectic group of dimension n over a finite field of characteristic l, for l at least 5, which can be derived from work of Kantor: G is either reducible, symplectically imprimitive or it contains Sp(n, l). This result is for instance useful for proving "big image" results for symplectic Galois representations. [less ▲]

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See detailCompatible systems of symplectic Galois representations and the inverse Galois problem II. Transvections and huge image
Arias De Reyna Dominguez, Sara UL; Dieulefait, Luis; Wiese, Gabor UL

in Pacific Journal of Mathematics (2016), 281(1), 1-16

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

This article is the second part of a series of three articles about compatible systems of symplectic Galois representations and applications to the inverse Galois problem. This part is concerned with symplectic Galois representations having a huge residual image, by which we mean that a symplectic group of full dimension over the prime field is contained up to conjugation. We prove a classification result on those subgroups of a general symplectic group over a finite field that contain a nontrivial transvection. Translating this group theoretic result into the language of symplectic representations whose image contains a nontrivial transvection, these fall into three very simply describable classes: the reducible ones, the induced ones and those with huge image. Using the idea of an (n,p)-group of Khare, Larsen and Savin we give simple conditions under which a symplectic Galois representation with coefficients in a finite field has a huge image. Finally, we combine this classification result with the main result of the first part to obtain a strenghtened application to the inverse Galois problem. [less ▲]

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See detailHilbertian fields and Galois representations
Bary-Soroker, Lior; Fehm, Arno; Wiese, Gabor UL

in Journal für die Reine und Angewandte Mathematik (2016), 712

We prove a new Hilbertianity criterion for fields in towers whose steps are Galois with Galois group either abelian or a product of finite simple groups. We then apply this criterion to fields arising ... [more ▼]

We prove a new Hilbertianity criterion for fields in towers whose steps are Galois with Galois group either abelian or a product of finite simple groups. We then apply this criterion to fields arising from Galois representations. In particular we settle a conjecture of Jarden on abelian varieties. [less ▲]

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See detailA Short Note on the Bruinier-Kohnen Sign Equidistribution Conjecture and Halasz' Theorem
Inam, lker; Wiese, Gabor UL

in International Journal of Number Theory (2016), 12(2), 357-360

In this note, we improve earlier results towards the Bruinier-Kohnen sign equidistribution conjecture for half-integral weight modular eigenforms in terms of natural density by using a consequence of ... [more ▼]

In this note, we improve earlier results towards the Bruinier-Kohnen sign equidistribution conjecture for half-integral weight modular eigenforms in terms of natural density by using a consequence of Halász' Theorem. Moreover, applying a result of Serre we remove all unproved assumptions. [less ▲]

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See detailOn certain finiteness questions in the arithmetic of modular forms
Kiming, Ian; Rustom, Nadim; Wiese, Gabor UL

in Journal of the London Mathematical Society (2016), 94(2), 479-502

We investigate certain finiteness questions that arise naturally when studying approximations modulo prime powers of p-adic Galois representations coming from modular forms. We link these finiteness ... [more ▼]

We investigate certain finiteness questions that arise naturally when studying approximations modulo prime powers of p-adic Galois representations coming from modular forms. We link these finiteness statements with a question by K. Buzzard concerning p-adic coefficient fields of Hecke eigenforms. Specifically, we conjecture that, for fixed N, m, and prime p with p not dividing N, there is only a finite number of reductions modulo p^m of normalized eigenforms on \Gamma_1(N). We consider various variants of our basic finiteness conjecture, prove a weak version of it, and give some numerical evidence. [less ▲]

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See detailTopics on modular Galois representations modulo prime powers
Tsaknias, Panagiotis; Wiese, Gabor UL

E-print/Working paper (2016)

This article surveys modularity, level raising and level lowering questions for two-dimensional representations modulo prime powers of the absolute Galois group of the rational numbers. It contributes ... [more ▼]

This article surveys modularity, level raising and level lowering questions for two-dimensional representations modulo prime powers of the absolute Galois group of the rational numbers. It contributes some new results and describes algorithms and a database of modular forms orbits and higher congruences. [less ▲]

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See detailUnramifiedness of Galois representations attached to weight one Hilbert modular eigenforms mod p
Wiese, Gabor UL

Scientific Conference (2015, November 13)

The talk will summarise the main ideas underlying the recent joint work with Mladen Dimitrov, proving that the existence of Hecke operators T_P, for P dividing p, implies that the Galois representation ... [more ▼]

The talk will summarise the main ideas underlying the recent joint work with Mladen Dimitrov, proving that the existence of Hecke operators T_P, for P dividing p, implies that the Galois representation attached to a mod p Hilbert modular eigenform of parallel weight one and prime-to-p level is unramified above p. This applies, in particular, to non-liftable mod p eigenforms, and can be seen as a refinement of the weight aspect in generalisations of Serre's Modularity Conjecture to Hilbert modular forms. [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 detailOn conjectures of Sato-Tate and Bruinier-Kohnen
Arias De Reyna Dominguez, Sara UL; Inam, Ilker; Wiese, Gabor UL

in Ramanujan Journal, The (2015), 36(3), 455-481

This article covers three topics. (1) It establishes links between the density of certain subsets of the set of primes and related subsets of the set of natural numbers. (2) It extends previous results on ... [more ▼]

This article covers three topics. (1) It establishes links between the density of certain subsets of the set of primes and related subsets of the set of natural numbers. (2) It extends previous results on a conjecture of Bruinier and Kohnen in three ways: the CM-case is included; under the assumption of the same error term as in previous work one obtains the result in terms of natural density instead of Dedekind-Dirichlet density; the latter type of density can already be achieved by an error term like in the prime number theorem. (3) It also provides a complete proof of Sato-Tate equidistribution for CM modular forms with an error term similar to that in the prime number theorem. [less ▲]

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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 detailAlgèbre 1 (BASI filière mathématiques, 2015)
Wiese, Gabor UL

Learning material (2015)

Course notes with exercises from the lecture Algèbre 1, taught in the BASI track mathematics at the University of Luxembourg in 2015.

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See detailCommutative Algebra (Master in Mathematics, 2015)
Wiese, Gabor UL

Learning material (2015)

Lecture notes with exercise sheets from the lecture Commutative Algebra held in winter term 2015 in the Master in Mathematics at the University of Luxembourg.

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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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