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Computational Arithmetic of Modular Forms Wiese, Gabor in Buyukasik, Engin; Inam, Ilker (Eds.) Notes from the International Autumn School on Computational Number Theory: Izmir Institute of Technology 2017 (2019) These course notes are about computing modular forms and some of their arithmetic properties. Their aim is to explain and prove the modular symbols algorithm in as elementary and as explicit terms as ... [more ▼] These course notes are about computing modular forms and some of their arithmetic properties. Their aim is to explain and prove the modular symbols algorithm in as elementary and as explicit terms as possible, and to enable the devoted student to implement it over any ring (such that a sufficient linear algebra theory is available in the chosen computer algebra system). The chosen approach is based on group cohomology and along the way the needed tools from homological algebra are provided. [less ▲] Detailed reference viewed: 178 (3 UL)Unramifiedness of weight one Hilbert Hecke algebras ; ; Wiese, Gabor E-print/Working paper (2019) We prove that the Galois pseudo-representation valued in the mod p^n parallel weight 1 Hecke algebra for GL(2) over a totally real number field F is unramified at a place above p if p-1 does not divide ... [more ▼] We prove that the Galois pseudo-representation valued in the mod p^n parallel weight 1 Hecke algebra for GL(2) over a totally real number field F is unramified at a place above p if p-1 does not divide the ramification index at that place. A novel geometric ingredient is the construction and study, in the case when p ramifies in F, of generalised Theta-operators using Reduzzi-Xiao's generalised Hasse invariants, including especially an injectivity criterion in terms of minimal weights. [less ▲] Detailed reference viewed: 73 (5 UL)MATHEMATICAL CAREERS In Luxembourg and beyond Palmirotta, Guendalina ; Notarnicola, Luca ; Wiese, Gabor Diverse speeches and writings (2019) Detailed reference viewed: 68 (1 UL)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 (2018) 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: 217 (9 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: 207 (1 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: 136 (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: 76 (1 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: 86 (3 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: 88 (5 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: 55 (3 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: 225 (24 UL)Linear Algebra 2 Wiese, Gabor ; Notarnicola, Luca ; Notarnicola, Massimo Learning material (2017) Detailed reference viewed: 212 (31 UL)On Galois representations of mod p Hilbert eigenforms of weight one Wiese, Gabor Scientific Conference (2016, August 16) Detailed reference viewed: 83 (9 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: 163 (7 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: 143 (4 UL)Algebraic Number Theory (Lecture Notes) ; Wiese, Gabor Learning material (2016) These are lecture notes for the course Algebraic Number Theory taught in the Master in Mathematics and Master in Secondary Education at the University of Luxembourg. Detailed reference viewed: 57 (5 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: 192 (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: 89 (10 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: 212 (9 UL) |
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