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The geometry of Hida families I: \(\Lambda\)-adic de Rham cohomology. (English) Zbl 1441.11098
Summary: We construct the \(\Lambda\)-adic de Rham analogue of Hida’s ordinary \(\Lambda\)-adic étale cohomology and of Ohta’s \(\Lambda\)-adic Hodge cohomology, and by exploiting the geometry of integral models of modular curves over the cyclotomic extension of \(\mathbb{Q}_p\), we give a purely geometric proof of the expected finiteness, control, and \(\Lambda\)-adic duality theorems. Following M. Ohta [J. Reine Angew. Math. 463, 49–98 (1995; Zbl 0827.11025)], we then prove that our \(\Lambda\)-adic module of differentials is canonically isomorphic to the space of ordinary \(\Lambda\)-adic cuspforms. In the sequel [the author, Compos. Math. 154, No. 4, 719–760 (2018; Zbl 1441.11097)] to this paper, we construct the crystalline counterpart to Hida’s ordinary \(\Lambda\)-adic étale cohomology, and employ integral \(p\)-adic Hodge theory to prove \(\Lambda\)-adic comparison isomorphisms between all of these cohomologies. As applications of our work in this paper and [the author, loc. cit.], we will be able to provide a “cohomological” construction of the family of \((\varphi,\Gamma)\)-modules attached to Hida’s ordinary \(\Lambda\)-adic étale cohomology by J. Dee [J. Algebra 235, No. 2, 636–664 (2001; Zbl 0984.11062)], as well as a new and purely geometric proof of Hida’s finiteness and control theorems. We are also able to prove refinements of the main theorems in B. Mazur and A. Wiles [Compos. Math. 59, 231–264 (1986; Zbl 0654.12008)] and Ohta [loc. cit.].

11F33 Congruences for modular and \(p\)-adic modular forms
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11G18 Arithmetic aspects of modular and Shimura varieties
11R23 Iwasawa theory
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