Une version relative de la conjecture des périodes de Kontsevich-Zagier

Abstract

Nous partons d’une série $F=\sum_{r\gg -\infty} f_r \cdot\varpi^r$ où $\varpi$ est l’indéterminée et les coefficients $f_r=f_r(z_1,\dots, z_n)$ sont des fonctions holomorphes définies sur un voisinage ouvert du polydisque fermé $\bar{\mathbb{D}}^n=\{(z_1,\dots,z_n);\, |z_i|\leq 1\}$. En intégrant les coefficients de cette série sur le $n$-cube réel $[0,1]^n$, on obtient la série de Laurent $\int_{[0,1]^n}F$. Lorsque $F$ est algébrique nous dirons que $\int_{[0,1]^n}F$ est une série de périodes. Dans cet article, nous cherchons à déterminer les séries algébriques $F$ telles que $\int_{[0,1]^n}F$ est nulle. En principle, ceci fournit des informations sur les propriétés de transcendance des séries de périodes. Notre résultat principal rappelle la conjecture des périodes de Kontsevich-Zagier sous une forme remaniée.

We start with a series $F=\sum_{r\gg -\infty} f_r \cdot\varpi^r$ with indeterminate $\varpi$ and where the coefficients $f_r=f_r(z_1,\dots,z_n)$ are holomorphic functions defined on an open neighborhood of the closed polydisc $\bar{\mathbb{D}}^n\!=\!\{(z_1,\dots,z_n);\, |z_i|\!\leq\! 1\}$. Integrating the coefficients of this series on the $n$-dimensional real cube $[0,1]^n$ yields a Laurent series $\int_{[0,1]^n}F$. When $F$ is algebraic we say that $\int_{[0,1]^n}F$ is a series of periods. In this article, our goal is to determine the algebraic series $F$ such that $\int_{[0,1]^n}F$ is zero. In principle, this gives informations on the transcendence properties of series of periods. Our main result is reminiscent to the Kontsevich-Zagier conjecture on periods in a modified form.

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Authors

Joseph Ayoub

Institut für Mathematik Universität Zürich, Zürich, Switzerland
CNRS LAGA Université Paris 13, Villetaneuse France