Fix a prime number p [ges ] 5 and a positive integer N prime to p. We consider the projective limits of p-adic étale cohomology groups of the modular curves X1(Npr) and Y1(Npr) (r [ges ] 1), which are denoted by ESp(N) ${\biib Z}$p and GES p(N)${\biib Z}$p, respectively. Let e* ′ be the projector to the direct sum of the ωi-eigenspaces of the ordinary part, for i [nequiv ] 0, −1 mod p−1. Our main result states that e* ′ GESp (N)${\biib Z}$p has a good p-adic Hodge structure, which can be described in terms of λ-adic modular forms, extending the previously known result for e*′ ESp (N)${\biib Z}$p. We then apply the method of Harder and Pink to the Galois representation on e*′ ESp(N) ${\biib Z}$p to construct large unramified abelian p-extensions over cyclotomic ${\bib Z}$p-extensions of abelian number fields.

Let X be an m dimensional smooth projective variety with a Kähler metric. We construct a metrized line bundle $\cL$ with a rational section s over the product $\cC$p(X)× $\cC $q(X) of Chow varieties $\cC$p(X), $\cC$q(X) such that $\[{1\over (m-1)!}\log|s(A,B)|^2=\langle A, B\rangle \]$ for disjoint A, B. That gives an answer to a part of Barry Mazur‘s proposal in a private communication to Bruno Horris about the Archimedean height pairing 〈 A, B〉 on a smooth projective variety X.

Cet article constitue une partie importante de ma thèse. Je l‘ai écrit sous la direction avisée de Henri Carayol, à qui je dois un très grand merci car ce sont ses conseils et son optimisme qui m‘ont permis d‘en venir à bout. Je tiens également à remercier Laurent Clozel pour toutes ses remarques, utiles et pertinentes.