Splitting fields of X^n-X-1 and modular forms

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 ▲]

Fast computation of half-integral weight modular forms

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 ▲]

Unlikely Revelations? -- The Hidden Lattice Problem

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 ▲]

Galois families of modular forms and application to weight one

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 ▲]

The Hidden Lattice Problem

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 ▲]

Fast bases for half-integral weight modular forms

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 ▲]

Unramifiedness of Galois representations attached to weight one Hilbert modular eigenforms mod p

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 ▲]

Algèbre (notes du cours, 3ème semestre BMATH)

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.

Non-ramification de l'algèbre de Hecke en poids un

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 ▲]

Finiteness questions for Galois representations

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 ▲]