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The geometry of Hida families. II: \(\Lambda\)-adic \((\varphi,\Gamma)\)-modules and \(\Lambda\)-adic Hodge theory. (English) Zbl 1441.11097
Summary: We construct the \(\Lambda\)-adic crystalline and Dieudonné analogues of Hida’s ordinary \(\Lambda\)-adic étale cohomology, and employ integral \(p\)-adic Hodge theory to prove \(\Lambda\)-adic comparison isomorphisms between these cohomologies and the \(\Lambda\)-adic de Rham cohomology studied in [the author, Math. Ann. 372, No. 1–2, 781–844 (2018; Zbl 1441.11098)] as well as Hida’s \(\Lambda\)-adic étale cohomology. As applications of our work, we 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)], and we give a new and purely geometric proof of Hida’s finiteness and control theorems. We also prove suitable \(\Lambda\)-adic duality theorems for each of the cohomologies we construct.

MSC:
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
14F30 \(p\)-adic cohomology, crystalline cohomology
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