Les $E$-fonctions et $G$-fonctions de Siegel

Tanguy Rivoal

doi:10.5802/xups.2019-03

Les

E

-fonctions et

G

-fonctions de Siegel

Tanguy Rivoal ¹

¹ Institut Fourier, CNRS et Université Grenoble Alpes, CS 40700, 38058 Grenoble cedex 9, France

Journées mathématiques X-UPS (2019), pp. 197-298.

Résumé

Siegel a introduit en 1929 les $E$ - et $G$ -fonctions, deux classes de séries entières à coefficients algébriques qui sont solutions d’équations différentielles à coefficients polynomiaux dans le but de généraliser les théorèmes classiques d’Hermite, Lindemann et Weierstrass concernant la nature arithmétique des valeurs des fonctions exponentielle et logarithme aux points algébriques, que ces deux classes généralisent respectivement.

Une première partie introduit la théorie des approximants de Padé « explicites » et la combine avec des méthodes « inexplicites » pour aboutir au célèbre résultat (1956) de Siegel et Shidlovskii sur la nature arithmétique des valeurs de $E$ -fonctions. On indique ensuite comment Chudnovsky a complété en 1984 le programme de Siegel sur la nature diophantienne des $G$ -fonctions. Une deuxième partie aborde la nature des équations différentielles satisfaites par les $G$ -fonctions (travaux d’André, Chudnovsky et Katz) puis, par ricochet, par les $E$ -fonctions. On conclut par de nouvelles applications arithmétiques que l’on en déduit et qui complètent le théorème de Siegel-Shidlovskii (travaux d’André, Beukers, Adamczewski-Rivoal).

Publié le : 2024-08-01

DOI : 10.5802/xups.2019-03

Affiliations des auteurs :

Tanguy Rivoal ¹

¹ Institut Fourier, CNRS et Université Grenoble Alpes, CS 40700, 38058 Grenoble cedex 9, France

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Tanguy Rivoal. Les $E$-fonctions et $G$-fonctions de Siegel. Journées mathématiques X-UPS (2019), pp. 197-298. doi : 10.5802/xups.2019-03. https://proceedings.centre-mersenne.org/articles/10.5802/xups.2019-03/

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