2014/07/22 by Bryden Cais, Cais, Bryden
Mathematics · #11F33 #11F67 #11G18 #11R23 #Advanced Algebra and Geometry #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11F33 #msc:11F67 #msc:11G18 #msc:11R23
paper · pdf · doi:10.48550/arxiv.1407.5707
This article is a revised version of part of arXiv:1209.0046
openalex publication_date 2014/07/22 · arxiv created 2016/06/08 · arxiv updated 2016/06/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We construct the Λ-adic de Rham analogue of Hida's ordinary Λ-adic étale cohomology and of Ohta's Λ-adic Hodge cohomology, and by exploiting the geometry of integral models of modular curves over the cyclotomic extension of Qp, we give a purely geometric proof of the expected finiteness, control, and Λ-adic duality theorems. Following Ohta, we then prove that our Λ-adic module of differentials is canonically isomorphic to the space of ordinary Λ-adic cuspforms. In the sequel to this paper, we construct the crystalline counterpart to Hida's ordinary Λ-adic étale cohomology, and employ integral p-adic Hodge theory to prove Λ-adic comparison isomorphisms between all of these cohomologies. As applications of our work in this paper and the sequel, we will be able to provide a "cohomological" construction of the family of (φ,Γ)-modules attached to Hida's ordinary Λ-adic étale cohomology by the work of Dee, as well as a new and purely geometric proof of Hida's finitenes and control theorems. We are also able to prove refinements of theorems of Mazur-Wiles and of Ohta.