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Jean-Christophe Feauveau

Publications and source records attributed to Jean-Christophe Feauveau.

4 recordsLinked to original sources
arXiv2018

Structure and bases of modular space sequences $(M_{2k}(Γ_0(N)))_{k\in \mathbb{N}^*}$ and $(S_{2k}(Γ_0(N)))_{k\in \mathbb{N}^*}$. Part II: a modular butterfly hunt

In the first part of this article, which contains three of them, we have identified the notion of level $N$ strong modular unit. It enabled us to structure the modular forms family $(M_{2k}(Γ_0(N)))_{k\in \mathbb{N}^*}$ and to propose the explicit bases for these spaces. It is in this perspective that we wrote this second part where the structure and explicit bases are proposed when $1\leq N \leq 10$.

math.NT↗
arXiv2018

Structure and bases of modular space sequences $(M_{2k}(Γ_0(N)))_{k\in \mathbb{N}^*}$ and $(S_{2k}(Γ_0(N)))_{k\in \mathbb{N}^*}$. Part III: Cuspidal spaces

Based on the notion of strong modular form developed in Part I, we propose to structure the family of cuspidal modular form spaces $(S_{2k}(Γ_0(N)))_{k\in \mathbb{N}^*}$ and to determine bases for each of these spaces, once known bases for the first values of $k$. We then apply these theoretical results to explicitly determine bases for space families $(S_{2k}(Γ_0(N)))_{k\in \mathbb{N}^*}$ when $1\leq N \leq 10$.

math.NT↗
arXiv2018

Structure and bases of modular space sequences $(M_{2k}(Γ_0(N)))_{k\in \mathbb{N}^*}$ and $(S_{2k}(Γ_0(N)))_{k\in \mathbb{N}^*}$. Part I : Strong modular units

The modular discriminant $Δ$ is known to structure the sequence of modular forms $(M_{2k}(SL_2(\mathbb{Z})))_{k\in \; \mathbb{N}^*}$ at level $1$.\\ For all positive integer $N$, we define a strong modular unit $Δ_N$ at level $N$ which enables one to structure the sequence $(M_{2k}(Γ_0(N)))_{k\in \; \mathbb{N}^*}$ in an identical way. We will apply this result to the bases search for each of the spaces $(M_{2k}(Γ_0(N)))_{k\in \; \mathbb{N}^*}$.\\ This article is the first in a series of three. In the second part we will propose explicit bases of $(M_{2k}(Γ_0(N)))_{k\in \; \mathbb{N}^*}$ for $1\leq N \leq 10$. Finally, in a third part, we will apply the results obtained in the first two parts to $(S_{2k}(Γ_0(N)))_{k\in \; \mathbb{N}^*}$.

math.NT↗
arXiv2017

Elliptic functions revisited

Elliptic functions are largely studied and standardized mathematical objects. The two usual approaches are due to Jacobi and Weierstrass. From a contour integral which allowed us to unify many summation formulae (Euler-MacLaurin, Poisson, Voronoï or Circle formulae), we will find the entirety of the elliptic functions, proposed either in the shape of Jacobi or Weierstrass. But with one translation which appears in their natural form. What could seem a defect will lead us to a renormalisation of the elliptic functions making it possible to determine, in a rather simple way, a Fourier series representation and a factorization of these functions.

math.CV↗