Ordinary p-adic étale Cohomology Groups Attached to Towers of Elliptic Modular Curves

Abstract

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.