La matrice de logarithme en termes de chiffres $p$-adiques (The logarithm matrix in terms of $p$-adic digits)

TL;DR

This paper introduces a novel way to describe the logarithm matrix of modular forms using distributions and $p$-adic digits, extending previous work to cases where $a_p eq 0$ and multi-variable distributions.

Contribution

It generalizes the description of the logarithm matrix for modular forms to include the case $a_p eq 0$ via a new distribution matrix framework.

Findings

New description of the logarithm matrix in terms of distributions.

Extension of methods to multi-variable distributions.

Characterization of matrices by $p$-adic digits.

Abstract

Nous donnons une nouvelle description de la matrice de logarithme d'une forme modulaire en termes de distributions, g\'en\'eralisant le travail de Dion et Lei pour le cas $a_p=0$. Ce qui nous permet d'inclure le cas $a_p\ne 0$ est une nouvelle d\'efinition, celle d'une matrice de distributions, et la caract\'erisation de cette matrice par de chiffres $p$-adiques. On peut appliquer ces m\'ethodes au cas correspondant d'une distribution \`a plusieurs variables. -- We give a new description of the logarithm matrix of a modular form in terms of distributions, generalizing the work of Dion and Lei for the case $a_p=0$. What allows us to include the case $a_p\ne 0$ is a new definition, that of a distribution matrix, and the characterization of this matrix by $p$-adic digits. One can apply these methods to the corresponding case of distributions in multiple variables.