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See detailSplitting fields of X^n-X-1 and modular forms
Wiese, Gabor UL

Presentation (2022, July 21)

In his article `On a theorem of Jordan', Serre considered the family of polynomials f_n(X) = X^n-X-1 and the counting function of the number of roots of f_n over the finite field F_p, seen as function in ... [more ▼]

In his article `On a theorem of Jordan', Serre considered the family of polynomials f_n(X) = X^n-X-1 and the counting function of the number of roots of f_n over the finite field F_p, seen as function in p. He explicitly showed the `modularity' of this function for n=3,4. In this talk, I report on joint work with Alfio Fabio La Rosa and Chandrashekhar Khare, in which we treat the case n=5 in several different ways. [less ▲]

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See detailUnlikely Revelations? -- The Hidden Lattice Problem
Wiese, Gabor UL

Presentation (2022)

In this talk, which is based on joint work with Luca Notarnicola, I will present the Hidden Lattice Problem (HLP), which is the task of recovering a "small" lattice from the knowledge of only one or a few ... [more ▼]

In this talk, which is based on joint work with Luca Notarnicola, I will present the Hidden Lattice Problem (HLP), which is the task of recovering a "small" lattice from the knowledge of only one or a few of its vectors. This problem can be traced back at least to the work on the Hidden Subset Sum Problem by Nguyen and Stern, who also came up with the "orthogonal lattice attack" for solving this kind of problem. The main novelty that I am going to discuss and illustrate is an alternative algorithm for the HLP. [less ▲]

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See detailCommutative Algebra (lecture notes, Master in Mathematics, Master in Secondary Education)
Wiese, Gabor UL

Learning material (2022)

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 2022.

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See detailA Conjecture of Coleman on the Eisenstein Family
Advocaat, Bryan UL; Kiming, Ian; Wiese, Gabor UL

E-print/Working paper (2022)

We prove for primes $p\ge 5$ a conjecture of Coleman on the analytic continuation of the family of modular functions $\frac{\Es_\k}{V(\Es_\k)}$ derived from the family of Eisenstein series $\Es_\k$. The ... [more ▼]

We prove for primes $p\ge 5$ a conjecture of Coleman on the analytic continuation of the family of modular functions $\frac{\Es_\k}{V(\Es_\k)}$ derived from the family of Eisenstein series $\Es_\k$. The precise, quantitative formulation of the conjecture involved a certain on $p$ depending constant. We show by an example that the conjecture with the constant that Coleman conjectured cannot hold in general for all primes. On the other hand, the constant that we give is also shown not to be optimal in all cases. The conjecture is motivated by its connection to certain central statements in works by Buzzard and Kilford, and by Roe, concerning the ``halo'' conjecture for the primes $2$ and $3$, respectively. We show how our results generalize those statements and comment on possible future developments. [less ▲]

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See detailSplitting fields of X^n-X-1 (particularly for n=5), prime decomposition and modular forms
Khare, Chandrashekhar; La Rosa, Alfio Fabio UL; Wiese, Gabor UL

E-print/Working paper (2022)

We study the splitting fields of the family of polynomials $f_n(X)= X^n-X-1$. This family of polynomials has been much studied in the literature and has some remarkable properties. Serre related the ... [more ▼]

We study the splitting fields of the family of polynomials $f_n(X)= X^n-X-1$. This family of polynomials has been much studied in the literature and has some remarkable properties. Serre related the function on primes $N_p(f_n)$, for a fixed $n \leq 4$ and $p$ a varying prime, which counts the number of roots of $f_n(X)$ in $\mathbb F_p$ to coefficients of modular forms. We study the case $n=5$, and relate $N_p(f_5)$ to mod $5$ modular forms over $\mathbb Q$, and to characteristic 0, parallel weight 1 Hilbert modular forms over $\mathbb Q(\sqrt{19 \cdot 151})$. [less ▲]

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See detailFast computation of half-integral weight modular forms
Inam, Ilker; Wiese, Gabor UL

in Rocky Mountain Journal of Mathematics (2022), 52(4), 1395-1401

To study statistical properties of modular forms, including for instance Sato-Tate like problems, it is essential to be able to compute a large number of Fourier coefficients. In this article, we show ... [more ▼]

To study statistical properties of modular forms, including for instance Sato-Tate like problems, it is essential to be able to compute a large number of Fourier coefficients. In this article, we show that this can be achieved in level 4 for a large range of half-integral weights by making use of one of three explicit bases, the elements of which can be calculated via fast power series operations. [less ▲]

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See detailGalois Families of Modular Forms
Wiese, Gabor UL

Presentation (2021, April 16)

Following a joint work with Sara Arias-de-Reyna and François Legrand, we present a new kind of families of modular forms. They come from representations of the absolute Galois group of rational function ... [more ▼]

Following a joint work with Sara Arias-de-Reyna and François Legrand, we present a new kind of families of modular forms. They come from representations of the absolute Galois group of rational function fields over Q. As a motivation and illustration, we discuss in some details one example: an infinite Galois family of Katz modular forms of weight one in characteristic 7, all members of which are non-liftable. This may be surprising because non-liftability is a feature that one might expect to occur only occasionally. [less ▲]

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See detailFast bases for half-integral weight modular forms
Wiese, Gabor UL

Scientific Conference (2021)

In this joint work with Ilker Inam, we exploit classical results of Kohnen and Cohen to give explicit bases of the spaces of half-integral weight modular forms of level Gamma_0(4) in any weight, which can ... [more ▼]

In this joint work with Ilker Inam, we exploit classical results of Kohnen and Cohen to give explicit bases of the spaces of half-integral weight modular forms of level Gamma_0(4) in any weight, which can be compared to the Miller bases for integral weight modular forms of level 1. They are very simple and can be computed quickly by performing power series multiplications. In further work with Inam et al., we apply them to a computational study of the distribution of signs of such modular forms. [less ▲]

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See detailAlgèbre linéaire 2 (BMATH 2021)
Wiese, Gabor UL

Learning material (2021)

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See detailThe Hidden Lattice Problem
Notarnicola, Luca UL; Wiese, Gabor UL

E-print/Working paper (2021)

We consider the problem of revealing a small hidden lattice from the knowledge of a low-rank sublattice modulo a given sufficiently large integer -- the {\em Hidden Lattice Problem}. A central motivation ... [more ▼]

We consider the problem of revealing a small hidden lattice from the knowledge of a low-rank sublattice modulo a given sufficiently large integer -- the {\em Hidden Lattice Problem}. A central motivation of study for this problem is the Hidden Subset Sum Problem, whose hardness is essentially determined by that of the hidden lattice problem. We describe and compare two algorithms for the hidden lattice problem: we first adapt the algorithm by Nguyen and Stern for the hidden subset sum problem, based on orthogonal lattices, and propose a new variant, which we explain to be related by duality in lattice theory. Following heuristic, rigorous and practical analyses, we find that our new algorithm brings some advantages as well as a competitive alternative for algorithms for problems with cryptographic interest, such as Approximate Common Divisor Problems, and the Hidden Subset Sum Problem. Finally, we study variations of the problem and highlight its relevance to cryptanalysis. [less ▲]

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See detailGalois families of modular forms and application to weight one
Arias-de-Reyna, Sara; Legrand, François; Wiese, Gabor UL

in Israel Journal of Mathematics (2021), 244

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 ▲]

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See detailOn the distribution of coefficients of half-integral weight modular forms and the Bruinier-Kohnen Conjecture
Inam, Ilker; Demirkol Özkaya, Zeynep; Tercan, Elif et al

in Turkish Journal of Mathematics (2021), 45(6), 2427-2440

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 ▲]

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See detailSimultaneous Diagonalization of Incomplete Matrices and Applications
Coron, Jean-Sébastien UL; Notarnicola, Luca UL; Wiese, Gabor UL

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, December)

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 ▲]

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

in Journal of the Institute of Mathematics of Jussieu (2020), 19(2), 281-306

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 ▲]

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See detailAlgèbre (notes du cours, 3ème semestre BMATH)
Wiese, Gabor UL

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.

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See detailNumber Theory for Cryptography (Lecture Notes)
Wiese, Gabor UL

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 ▲]

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See detailDihedral Universal Deformations
Deo, Shaunak UL; Wiese, Gabor UL

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 ▲]

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See detailNon-ramification de l'algèbre de Hecke en poids un
Wiese, Gabor UL

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 ▲]

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See detailDihedral Universal Deformations
Wiese, Gabor UL

Presentation (2019, February 19)

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See detailFiniteness questions for Galois representations
Wiese, Gabor UL

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 ▲]

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