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Galois families of modular forms and application to weight one ; ; Wiese, Gabor in Israel Journal of Mathematics (in press) We introduce Galois families of modular forms. They are a new kind of family coming from Galois representations of the absolute Galois groups of rational function fields over the rational field. We ... [more ▼] We introduce Galois families of modular forms. They are a new kind of family coming from Galois representations of the absolute Galois groups of rational function fields over the rational field. We exhibit some examples and provide an infinite Galois family of non-liftable weight one Katz modular eigenforms over an algebraic closure of F_p for p in {3,5,7,11}. [less ▲] Detailed reference viewed: 65 (1 UL)Simultaneous Diagonalization of Incomplete Matrices and Applications Coron, Jean-Sébastien ; Notarnicola, Luca ; Wiese, Gabor in Proceedings of the Fourteenth Algorithmic Number Theory Symposium (ANTS-XIV), edited by Steven Galbraith, Open Book Series 4, Mathematical Sciences Publishers, Berkeley, 2020 (2020), 4 We consider the problem of recovering the entries of diagonal matrices {U_a}_a for a = 1, . . . , t from multiple “incomplete” samples {W_a}_a of the form W_a = P U_a Q, where P and Q are unknown matrices ... [more ▼] We consider the problem of recovering the entries of diagonal matrices {U_a}_a for a = 1, . . . , t from multiple “incomplete” samples {W_a}_a of the form W_a = P U_a Q, where P and Q are unknown matrices of low rank. We devise practical algorithms for this problem depending on the ranks of P and Q. This problem finds its motivation in cryptanalysis: we show how to significantly improve previous algorithms for solving the approximate common divisor problem and breaking CLT13 cryptographic multilinear maps. [less ▲] Detailed reference viewed: 107 (15 UL)Fast computation of half-integral weight modular forms ; Wiese, Gabor E-print/Working paper (2020) For many studies, it is essential to have a large number of Fourier coefficients of modular forms at one's disposal, for example Sato-Tate like problems where one needs many Fourier coefficients to obtain ... [more ▼] For many studies, it is essential to have a large number of Fourier coefficients of modular forms at one's disposal, for example Sato-Tate like problems where one needs many Fourier coefficients to obtain significant statistical results. In this article, we exhibit bases for the space of modular forms of any half-integral weight and level 4, which have the property that many coefficients can be computed (relatively) quickly on a computer. [less ▲] Detailed reference viewed: 13 (0 UL)Number Theory for Cryptography (Lecture Notes) Wiese, Gabor Learning material (2020) In these lectures (8 hours taught in November 2020), we mention some topics from (algebraic) number theory as well as some related concepts from (algebraic) geometry that can be useful in cryptography. We ... [more ▼] In these lectures (8 hours taught in November 2020), we mention some topics from (algebraic) number theory as well as some related concepts from (algebraic) geometry that can be useful in cryptography. We cannot go deeply into any of the topics and most results will be presented without any proofs. One of the things that one encounters are `ideal lattices'. In the examples I saw, this was nothing but (an ideal in) an order in a number field, which is one of the concepts that we present here in its mathematical context (i.e. embedded in a conceptual setting). It has been noted long ago (already in the 19th century) that number fields and function fields of curves have many properties in common. Accordingly, we shall also present some basic topics on affine plane curves and their function fields. This leads us to mention elliptic curves, however, only in an affine version (instead of the better projective one); we cannot go deeply into that topic at all. The material presented here is classical and very well known. [less ▲] Detailed reference viewed: 19 (7 UL)Dihedral Universal Deformations Deo, Shaunak ; Wiese, Gabor in Research in Number Theory (2020), 6 This article deals with universal deformations of dihedral representations with a particular focus on the question when the universal deformation is dihedral. Results are obtained in three settings: (1 ... [more ▼] This article deals with universal deformations of dihedral representations with a particular focus on the question when the universal deformation is dihedral. Results are obtained in three settings: (1) representation theory, (2) algebraic number theory, (3) modularity. As to (1), we prove that the universal deformation is dihedral if all infinitesimal deformations are dihedral. Concerning (2) in the setting of Galois representations of number fields, we give sufficient conditions to ensure that the universal deformation relatively unramified outside a finite set of primes is dihedral, and discuss in how far these conditions are necessary. As a side-result, we obtain cases of the unramified Fontaine-Mazur conjecture. As to (3), we prove a modularity theorem of the form `R=T' for parallel weight one Hilbert modular forms for cases when the minimal universal deformation is dihedral. [less ▲] Detailed reference viewed: 127 (0 UL)Algèbre (notes du cours, 3ème semestre BMATH) Wiese, Gabor Learning material (2020) These are the lecture notes for the lecture `Algèbre' in the 3rd semester of the Bachelor in Mathematics at the University of Luxembourg. Last update: winter term 2020. Detailed reference viewed: 74 (4 UL)Commutative Algebra (lecture notes, Master in Mathematics, Master in Secondary Education) Wiese, Gabor Learning material (2020) These are the lecture notes for the course Commutative Algebra in the Master in Mathematics and the Master in Secondary Education at the University of Luxembourg. Last update: winter term 2020. Detailed reference viewed: 131 (14 UL)On the distribution of coefficients of half-integral weight modular forms and the Bruinier-Kohnen Conjecture ; ; et al E-print/Working paper (2020) This work represents a systematic computational study of the distribution of the Fourier coefficients of cuspidal Hecke eigenforms of level Gamma_0(4) and half-integral weights. Based on substantial ... [more ▼] This work represents a systematic computational study of the distribution of the Fourier coefficients of cuspidal Hecke eigenforms of level Gamma_0(4) and half-integral weights. Based on substantial calculations, the question is raised whether the distribution of normalised Fourier coefficients with bounded indices can be approximated by a generalised Gaussian distribution. Moreover, it is argued that the apparent symmetry around zero of the data lends strong evidence to the Bruinier-Kohnen Conjecture on the equidistribution of signs and even suggests the strengthening that signs and absolute values are distributed independently. [less ▲] Detailed reference viewed: 13 (0 UL)Non-ramification de l'algèbre de Hecke en poids un Wiese, Gabor Presentation (2019, May 09) Soit S l'espace des formes modulaires de Hilbert paraboliques en poids parallèle un géométriquement défini sur un corps fini et soit T l'algèbre de Hecke qui agit fidèlement sur S. Nous esquissons une ... [more ▼] Soit S l'espace des formes modulaires de Hilbert paraboliques en poids parallèle un géométriquement défini sur un corps fini et soit T l'algèbre de Hecke qui agit fidèlement sur S. Nous esquissons une démonstration que la représentation galoisienne à valeurs dans T est non-ramifiée au dessus de p. Ceci peut être considéré comme un premier pas vers un théorème de type R=T liant les formes modulaires de poids un et les représentations non-ramifiées au-dessus de p. Travail commun avec Shaunak Deo et Mladen Dimitrov. [less ▲] Detailed reference viewed: 32 (2 UL)Dihedral Universal Deformations Wiese, Gabor Presentation (2019, February 19) Detailed reference viewed: 36 (0 UL)Finiteness questions for Galois representations Wiese, Gabor Presentation (2019, January 29) Let p be a prime number. Due to classical work of Shimura and Deligne, to any "newform" (a modular form that is an eigenfunction for the Hecke operators and assumed of level one in the talk) one attaches ... [more ▼] Let p be a prime number. Due to classical work of Shimura and Deligne, to any "newform" (a modular form that is an eigenfunction for the Hecke operators and assumed of level one in the talk) one attaches a p-adic Galois representation. Since there are infinitely many newforms, there are infinitely many attached p-adic Galois representations. However, if one reduces them modulo p, there are only finitely many (up to isomorphism). It is tempting to ask what happens "in between", 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 and explain partial results, including a relation to a strong question by Kevin Buzzard. The talk is based on joint work with Ian Kiming and Nadim Rustom. [less ▲] Detailed reference viewed: 41 (0 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: 40 (5 UL)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: 116 (3 UL)Unramifiedness of Galois representations attached to weight one Hilbert modular eigenforms mod p ; Wiese, Gabor in Journal of the Institute of Mathematics of Jussieu (2018) The main result of this article states that the Galois representation attached to a Hilbert modular eigenform defined over F_p^bar of parallel weight 1 and level prime to p is unramified above p. This ... [more ▼] The main result of this article states that the Galois representation attached to a Hilbert modular eigenform defined over F_p^bar of parallel weight 1 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 1. The proof is based on the observation that parallel weight 1 forms in characteristic p embed into the ordinary part of parallel weight p forms in two different ways per prime dividing p, namely via `partial' Frobenius operators. MSC: 11F80 (primary); 11F41, 11F33 Keywords: Hilbert modular forms modulo p, weight one, Galois representations [less ▲] Detailed reference viewed: 85 (5 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: 160 (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: 164 (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: 104 (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: 48 (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: 48 (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: 74 (5 UL) |
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