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Courbes intégrales : transcendance et géométrie

Abstract : This thesis is devoted to the study of some questions motivated by Nesterenko's theorem on the algebraic independence of values of Eisenstein series E₂, E₄, E₆. It is divided in two parts.In the first part, comprising the first two chapiters, we generalize the algebraic differential equations satisfied by Eisenstein series that lie in the heart of Nesterenko's method, the Ramanujan equations. These generalizations, called 'higher Ramanujan equations', are obtained geometrically from vector fields naturally defined on certain moduli spaces of abelian varieties. In order to justify the interest of the higher Ramanujan equations in Transcendence Theory, we also show that values of a remarkable particular solution of these equations are related to 'periods' of abelian varieties.In the second part (third chapter), we study Nesterenko's method per se. We establish a geometric statement, containing the theorem of Nesterenko, on the transcendence of values of holomorphic maps from a disk to a quasi-projective variety over Q¯ defined as integral curves of some vector field. These maps are required to satisfy some integrality property, besides a growth condition and a strong form of Zariski-density that are natural for integral curves of algebraic vector fields.
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Tiago Jardim da Fonseca. Courbes intégrales : transcendance et géométrie. Théorie des nombres [math.NT]. Université Paris Saclay (COmUE), 2017. Français. ⟨NNT : 2017SACLS515⟩. ⟨tel-01685449⟩

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