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Transcendance de périodes: état des connaissances

Michel Waldschmidt
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Abstract

Les nombres réels ou complexes forment un ensemble ayant la puissance du continu. Parmi eux, ceux qui sont intéressants, qui apparaissent naturellement, qui méritent notre attention, forment un ensemble dénombrable. Dans cet état d'esprit nous nous intéressons aux périodes au sens de Kontsevich et Zagier. Nous faisons le point sur l'état de nos connaissances concernant la nature arithmétique de ces nombres: décider si une période est un nombre rationnel, algébrique irrationnel ou au contraire transcendant est l'objet de quelques théorèmes et de beaucoup de conjectures. Nous précisons aussi ce qui est connu sur l'approximation diophantienne de tels nombres, par des nombres rationnels ou algébriques.

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Michel Waldschmidt. 2005-02-28. Transcendance de périodes: état des connaissances. https://arxiv.org/abs/math/0502582

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