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We consider the problem of recovering the entries of diagonal matrices $\{U_a\}_a$ for $a = 1,\ldots,t$ from multiple "incomplete" samples $\{W_a\}_a$ of the form $W_a=PU_aQ$, 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., Comment: 16 pages

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., Comment: v3: minor revision, might differ slightly from the published version

The main result of this article states that the Galois representation attached to a Hilbert modular eigenform defined over $\overline{\mathbb{F}}_{p}$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.

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}., 23 pages; v4: minor changes

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.

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. MSC (2010): 11F80 (Galois representations); 20C25 (Projective representations and multipliers), 12F12 (Inverse Galois theory).

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 Hal\'asz' Theorem. Moreover, applying a result of Serre we remove all unproved assumptions., Comment: 4 pages, main result made unconditional, minor changes due to referee's reports

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 side-results, we obtain cases of the unramified Fontaine-Mazur conjecture, and in many cases positively answer a question of Greenberg and Coleman on the splitting behaviour at p of p-adic Galois representations attached to newforms. 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., 43 pages; minor corrections and improvements following referee's comments

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.

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., Comment: 17 pages. This manuscript is extracted from an old version of our paper arXiv:1203.6552

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. A key ingredient is a classification 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., Comment: 14 pages; the proof of the classification result has been significantly shortened by appealing to results of Kantor

The behaviour of Hecke polynomials modulophas been the subject of some studies. In this paper we show that ifpis a prime, the set of integersNsuch that the Hecke polynomialsTN,χℓ,kfor all primes ℓ, all weightsk≥ 2 and all characters χ taking values in {±1} splits completely modulophas density 0, unconditionally forp= 2 and under the Cohen–Lenstra heuristics forp≥ 3. The method of proof is based on the construction of suitable dihedral modular forms.

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[subscript n](F[subscript ℓ[superscript d]]) or PGSp[subscript n](F[subscript ℓ[superscript d]]) occurs as a Galois group over the rational numbers for a positive density set of primes ℓ. The result depends on some work of Arthur’s, which is conditional, but expected to become unconditional soon. The result is obtained by showing the existence of a regular, algebraic, self-dual, cuspidal automorphic representation of GL[subscript n](A[subscript 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., National Science Foundation (U.S.) (DMS-1162250), Alfred P. Sloan Foundation. Fellowship

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., Comment: 25 pages; v2: one of the conjectures from v1 now proved; v3: restructered parts of the article; v4: minor corrections and changes

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., Comment: 16 pages; removed notational ambiguities and some misleading terminology, otherwise only minor changes

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., Comment: 26 pages; to appear in The Ramanujan Journal

Let f be a cusp form of weight k+1/2 and at most quadratic nebentype character whose Fourier coefficients a(n) are all real. We study an equidistribution conjecture of Bruinier and Kohnen for the signs of a(n). We prove this conjecture for certain subfamilies of coefficients that are accessible via the Shimura lift by using the Sato-Tate equidistribution theorem for integral weight modular forms. Firstly, an unconditional proof is given for the family {a(tp^2)}_p where t is a squarefree number and p runs through the primes. In this case, the result is in terms of natural density. To prove it for the family {a(tn^2)}_n where t is a squarefree number and n runs through all natural numbers, we assume the existence of a suitable error term for the convergence of the Sato-Tate distribution, which is weaker than one conjectured by Akiyama and Tanigawa. In this case, the results are in terms of Dedekind-Dirichlet density., 8 pages; typos corrected, final version, accepted for publication in Archiv der Mathematik

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., Comment: 18 pages, accepted for publication in Journal f\"ur die reine und angewandte Mathematik

In this article new cases of the Inverse Galois Problem are established. The main result is that for a fixed integer n, there is a positive density set of primes p such that PSL_2(F_{p^n}) occurs as the Galois group of some finite extension of the rational numbers. These groups are obtained as projective images of residual modular Galois representations. Moreover, families of modular forms are constructed such that the images of all their residual Galois representations are as large as a priori possible. Both results essentially use Khare's and Wintenberger's notion of good-dihedral primes. Particular care is taken in order to exclude nontrivial inner twists., Comment: 17 pages

We study modular Galois representations mod pm. We show that there are three progressively weaker notions of modularity for a Galois representation mod pm: We have named these "strongly", "weakly", and "dc-weakly" modular. Here, "dc" stands for "divided congruence" in the sense of Katz and Hida. These notions of modularity are relative to a fixed level M. Using results of Hida we display a level-lowering result ("stripping-of-powers of p away from the level"): A mod pm strongly modular representation of some level Npr is always dc-weakly modular of level N (here, N is a natural number not divisible by p). We also study eigenforms mod pm corresponding to the above three notions. Assuming residual irreducibility, we utilize a theorem of Carayol to show that one can attach a Galois representation mod pm to any "dc-weak" eigenform, and hence to any eigenform mod pm in any of the three senses. We show that the three notions of modularity coincide when m = 1 (as well as in other particular cases), but not in general.

The article motivates, presents and describes large computer calculations concerning the asymptotic behaviour of arithmetic properties of coefficient fields of modular forms. The observations suggest certain patterns, which deserve further study., 19 pages, 76 figures

This article starts a computational study of congruences of modular forms and modular Galois representations modulo prime powers. Algorithms are described that compute the maximum integer modulo which two monic coprime integral polynomials have a root in common in a sense that is defined. These techniques are applied to the study of congruences of modular forms and modular Galois representations modulo prime powers. Finally, some computational results with implications on the (non-)liftability of modular forms modulo prime powers and possible generalisations of level raising are presented., Comment: 26 pages

In this article we report on extensive calculations concerning the Gorenstein defect for Hecke algebras of spaces of modular forms of prime weight p at maximal ideals of residue characteristic p such that the attached mod p Galois representation is unramified at p and the Frobenius at p acts by scalars. The results lead us to the ask the question whether the Gorenstein defect and the multplicity of the attached Galois representation are always equal to 2. We review the literature on the failure of the Gorenstein property and multiplicity one, discuss in some detail a very important practical improvement of the modular symbols algorithm over finite fields and include precise statements on the relationship between the Gorenstein defect and the multiplicity of Galois representations. Appendix A: Manual of Magma package HeckeAlgebra, Appendix B: Tables of Hecke algebras., Comment: 52 pages LaTeX, 2 appendices

Let f be a non-CM newform of weight k > 1. Let L be a subfield of the coefficient field of f. We completely settle the question of the density of the set of primes p such that the p-th coefficient of f generates the field L. This density is determined by the inner twists of f. As a particular case, we obtain that in the absence of non-trivial inner twists, the density is 1 for L equal to the whole coefficient field. We also present some new data on reducibility of Hecke polynomials, which suggest questions for further investigation., 13 pages, more complete result, some corollaries added

We consider mod [math] modular Galois representations which are unramified at [math] such that the Frobenius element at [math] acts through a scalar matrix. The principal result states that the multiplicity of any such representation is bigger than [math] .

In this article we prove conditions under which a certain parabolic group cohomology space over a finite field F is a faithful module for the Hecke algebra of Katz modular forms over an algebraic closure of F. These results can e.g. be used to compute Katz modular forms of weight one with methods of linear algebra over F. This is essentially Chapter 3 of my thesis., 26 pages; small corrections and changes

We announce the creation of a database of invariant rings. This database contains a large number of invariant rings of finite groups, mostly in the modular case. It gives information on generators and structural properties of the invariant rings. The main purpose is to provide a tool for researchers in invariant theory.

In this thesis we investigate 2-dimensional, continuous, odd, residual Galois representations and their images. This manuscript consists of two parts.In the first part of this thesis we analyse a local-global problem for elliptic curves over number fields. Let E be an elliptic curve over a number field K, and let ℓ be a prime number. If E admits an ℓ-isogeny locally at a set of primes with density one then does E admit an ℓ-isogeny over K? The study of the Galois representation associated to the ℓ-torsion subgroup of E is the crucial ingredient used to solve the problem. We characterize completely the cases where the local-global principle fails, obtaining an upper bound for the possible values of ℓ for which this can happen. In the second part of this thesis, we outline an algorithm for computing the image of a residual modular 2-dimensional semi-simple Galois representation. This algorithm determines the image as a finite subgroup of GL₂(F¯ℓ), up to conjugation, as well as certain local properties of the representation and tabulate the result in a database. In this part of the thesis we show that, in almost all cases, in order to compute the image of such a representation it is sufficient to know the images of the Hecke operators up to the Sturm bound at the given level n. In addition, almost all the computations are performed in positive characteristic.In order to obtain such an algorithm, we study the local description of the representation at primes dividing the level and the characteristic: this leads to a complete description of the eigenforms in the old-space. Moreover, we investigate the conductor of the twist of a representation by characters and the coefficients of the form of minimal level and weight associated to it in order to optimize the computation of the projective image.The algorithm is designed using results of Dickson, Khare-Wintenberger and Faber on the classification, up to conjugation, of the finite subgroups of PGL₂(F¯ℓ). We characterize each possible case giving a precise description and algorithms to deal with it. In particular, we give a new approach and a construction to deal with irreducible representations with projective image isomorphic to either the symmetric group on 4 elements or the alternating group on 4 or 5 elements., Dans cette thèse, on étudie les représentations 2-dimensionnelles continues du groupe de Galois absolu d'une clôture algébrique fixée de Q sur les corps finis qui sont modulaires et leurs images. Ce manuscrit se compose de deux parties.Dans la première partie, on étudie un problème local-global pour les courbes elliptiques sur les corps de nombres. Soit E une courbe elliptique sur un corps de nombres K, et soit l un nombre premier. Si E admet une l-isogénie localement sur un ensemble de nombres premiers de densité 1 alors est-ce que E admet une l-isogénie sur K ? L'étude de la repréesentation galoisienne associéee à la l-torsion de E est l'ingrédient essentiel utilisé pour résoudre ce problème. On caractérise complètement les cas où le principe local-global n'est pas vérifié, et on obtient une borne supérieure pour les valeurs possibles de l pour lesquelles ce cas peut se produire.La deuxième partie a un but algorithmique : donner un algorithme pour calculer les images des représentations galoisiennes 2-dimensionnelles sur les corps finis attachées aux formes modulaires. L'un des résultats principaux est que l'algorithme n'utilise que des opérateurs de Hecke jusqu'à la borne de Sturm au niveau donné n dans presque tous les cas. En outre, presque tous les calculs sont effectués en caractéristique positive. On étudie la description locale de la représentation aux nombres premiers divisant le niveau et la caractéristique. En particulier, on obtient une caractérisation précise des formes propres dans l'espace des formes anciennes en caractéristique positive.On étudie aussi le conducteur de la tordue d'une représentation par un caractère et les coefficients de la forme de niveau et poids minimaux associée. L'algorithme est conçu à partir des résultats de Dickson, Khare-Wintenberger et Faber sur la classification, à conjugaison près, des sous-groupes finis de PGL₂(F¯ℓ). On caractérise chaque cas en donnant une description et des algorithmes pour le vérifier. En particulier, on donne une nouvelle approche pour les représentations irréductibles avec image projective isomorphe soit au groupe symétrique sur 4 éléments ou au groupe alterné sur 4 ou 5 éléments.