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Topics on modular Galois representations modulo prime powers ; Wiese, Gabor in Böckle, Gebhard; Decker, Wolfram; Malle, Gunter (Eds.) Algorithmic and Experimental Methods in Algebra, Geometry, and Number Theory (in press) 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 ▲] Detailed reference viewed: 17 (2 UL)On the arithmetic of modular forms Wiese, Gabor Presentation (2017, June 28) In this overview talk, we will illustrate the arithmetic significance of the coefficients of modular forms, from classical examples to the modern point of view of Galois representations. From that on, one ... [more ▼] In this overview talk, we will illustrate the arithmetic significance of the coefficients of modular forms, from classical examples to the modern point of view of Galois representations. From that on, one is naturally lead to questions on the distribution of the coefficients in various senses. We will briefly touch on some of them, report on recent theorems and state some open questions. When investigating arithmetical properties of coefficients through Galois representations, their ramification properties are an important object to study. One notices that forms of -so called- weight one play a special role that will be explained. The importance of Galois representations coming from modular forms is that one expects all Galois representations satisfying reasonable "geometric" assumption to arise in such a way. As a special case, we will explain the theorem of Khare and Wintenberger, formerly called Serre's modularity conjecture, and conjectural generalisations. Whereas Galois representations are usually very hard to calculate directly on a computer, modular forms are pretty simple to compute. We will briefly touch on this and what kind of information one can get. If time allows, we will also touch on an application to the so-called inverse Galois problem. [less ▲] Detailed reference viewed: 5 (0 UL)On the arithmetic of modular forms Wiese, Gabor Scientific Conference (2017, June 15) In this short overview talk, we will stress the arithmetic significance of the coefficients of modular forms. This naturally leads to questions on the distribution of the coefficients in various senses ... [more ▼] In this short overview talk, we will stress the arithmetic significance of the coefficients of modular forms. This naturally leads to questions on the distribution of the coefficients in various senses. We will briefly touch on some of them and state some open questions. The arithmetic information in the coefficients of a Hecke eigenform is summarised in the attached Galois representation. When studying its ramification properties, one notices that forms of weight one play a special role that we will explain. [less ▲] Detailed reference viewed: 7 (2 UL)On Galois representations of weight one Wiese, Gabor Scientific Conference (2017, June 08) Modular forms of weight one play a special role, especially those that are geometrically defined over a finite field of characteristic p. For instance, in general they cannot be obtained as reductions ... [more ▼] Modular forms of weight one play a special role, especially those that are geometrically defined over a finite field of characteristic p. For instance, in general they cannot be obtained as reductions from weight one forms in characteristic zero. Another property is that if the level is prime-to p, then the attached mod p Galois representation is unramified at p. It is known that this property characterises weight one forms (if p>2). In this talk, I will present the approach chosen in joint work with Mladen Dimitrov to prove the unramifiedness above p in the case of Hilbert modular forms of parallel weight one over finite fields of characteristic p and level prime-to p. The approach is based on Hecke theory and exhibits an interesting behaviour of the Galois representation into an appropriate higher weight integral Hecke algebra. [less ▲] Detailed reference viewed: 6 (0 UL)On certain finiteness questions in the arithmetic of Galois representations Wiese, Gabor Presentation (2017, June 07) Let p be a fixed prime number. It has been known for a long time that there are only finitely many Galois extensions K/Q with Galois group a finite irreducible subgroup of GL_2(F_p^bar) that are imaginary ... [more ▼] Let p be a fixed prime number. It has been known for a long time that there are only finitely many Galois extensions K/Q with Galois group a finite irreducible subgroup of GL_2(F_p^bar) that are imaginary and unramified outside p. In contrast, there are infinitely many such with Galois group inside GL_2(Z_p^bar), even if one restricts to ones coming from modular forms (this restriction is believed to be local at p). It is tempting to ask what happens "in between" F_p^bar and Z_p^bar, i.e. whether there is still finiteness modulo fixed prime powers. In the talk, I will motivate and explain a conjecture made with Ian Kiming and Nadim Rustom stating that the set of such Galois extensions `modulo p^m' (a proper definition will be given in the talk) coming from modular forms is finite. I will present partial results and a relation of the finiteness conjecture to a strong question by Kevin Buzzard. The talk is based on joint work with Ian Kiming and Nadim Rustom. [less ▲] Detailed reference viewed: 5 (1 UL)Sur les représentations galoisiennes des formes modulaires de Hilbert de poids un en caractéristique p Wiese, Gabor 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 ▲] Detailed reference viewed: 37 (1 UL)Sur les représentations galoisiennes des formes modulaires de Hilbert de poids un en caractéristique p Wiese, Gabor 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 ▲] Detailed reference viewed: 15 (1 UL)Linear Algebra 2 Wiese, Gabor ; Notarnicola, Luca ; Notarnicola, Massimo Learning material (2017) Detailed reference viewed: 26 (1 UL)Compatible systems of symplectic Galois representations and the inverse Galois problem I. Images of projective representations Arias De Reyna Dominguez, Sara ; ; Wiese, Gabor 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 ▲] Detailed reference viewed: 92 (19 UL)On Galois representations of mod p Hilbert eigenforms of weight one Wiese, Gabor Scientific Conference (2016, August 16) Detailed reference viewed: 34 (6 UL)A Short Note on the Bruinier-Kohnen Sign Equidistribution Conjecture and Halasz' Theorem ; Wiese, Gabor 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 ▲] Detailed reference viewed: 45 (9 UL)Compatible systems of symplectic Galois representations and the inverse Galois problem II. Transvections and huge image Arias De Reyna Dominguez, Sara ; ; Wiese, Gabor 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 ▲] Detailed reference viewed: 37 (3 UL)On certain finiteness questions in the arithmetic of modular forms ; ; Wiese, Gabor 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 ▲] Detailed reference viewed: 50 (6 UL)Hilbertian fields and Galois representations ; ; Wiese, Gabor 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 ▲] Detailed reference viewed: 101 (8 UL)Classification of subgroups of symplectic groups over finite fields containing a transvection Arias De Reyna Dominguez, Sara ; ; Wiese, Gabor 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 ▲] Detailed reference viewed: 69 (6 UL)Unramifiedness of Galois representations attached to weight one Hilbert modular eigenforms mod p Wiese, Gabor 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 ▲] Detailed reference viewed: 18 (1 UL)Algèbre 1 (BASI filière mathématiques, 2015) Wiese, Gabor 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. Detailed reference viewed: 33 (2 UL)Compatible 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 ; ; 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 ▲] Detailed reference viewed: 62 (12 UL)On conjectures of Sato-Tate and Bruinier-Kohnen Arias De Reyna Dominguez, Sara ; ; Wiese, Gabor 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 ▲] Detailed reference viewed: 93 (10 UL) |
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