After introducing the concepts of singular Jacobi forms, we will define quasi-Jacobi forms and study their algebraic structure. We will focus in particular on their stability under various derivations and construct sequences of bidifferential operators with the aim of finding analogs of the well-known Rankin-Cohen brackets or transvectants on algebras of modular forms. This is a joint work with François Dumas and François Martin from the University of Clermont Auvergne.
21 nov. 2024
Kloosterman sums are fundamental objects of analytic number theory, appearing, for example, as Fourier coefficients of Poincaré series or in the circle method. A detailed understanding of the properties of these sums has become an area of interest in itself. These sums are parameterized by an integer: the modulus. They are constructed by successive additions of complex numbers of norm 1, and can be considered as a sequence of partial sums. Plotted in the complex plane, these partial sums lead to some very intriguing figures, known as [Kloosterman paths](https://blogs.ethz.ch/kowalski/the-kloostermania-page/). Kowalski & Sawin have developed a probabilistic interpretation of the distribution of these paths in the case of the prime modulus. I shall present an interpretation in the case where the modules are powers of prime numbers. This work was carried out in collaboration with Guillaume Ricotta on the one hand, and Guillaume Ricotta and Igor Shparlinski on the other.
23 janv. 2024
We construct formal deformations on the algebras of quasimodular and weak Jacobi forms.
13 juil. 2018
Nous exhibons des liens entre combinatoire et théorie des nombres dans plusieurs cadres : moments de valeurs spéciales de fonctions L, surfaces à petits carreaux et études des quatre rangs des corps quadratiques.
07 mai 2014
We give a general view of the links between random matrix theory and L-functions and study the symmetry type of symmetric power L-functions of modular forms.
27 févr. 2008
We introduce the necessary tools to understand and solve couting problems in the geometry of tanslation surfaces with quasimodular forms.
09 mai 2007
Grâce à un modèle probabiliste, on évalue les moments complexes des valeurs des fonctions L de puissances symétriques de formes modulaires au bord de la bande critique et on en déduit des résultats de nature statistique sur leurs grandes valeurs.
01 nov. 2004
Études des moments des valeurs au bord de la bande critique des fonctions L de carré symétrique de formes modulaires.
04 nov. 2003
Présentation de la théorie des formes quasimodulaires.
18 sept. 2003