TY - JOUR

T1 - The geometry of Hida families II

T2 - Λ-adic (p, Γ)-modules and Λ-adic Hodge theory

AU - Cais, Bryden

N1 - Funding Information:
During the writing of this paper, the author was partially supported by an NSA Young Investigator (H98230-12-1-0238) and an NSF RTG grant (DMS-0838218). This journal is ©c Foundation Compositio Mathematica 2018.

PY - 2018/4/1

Y1 - 2018/4/1

N2 - We construct the -adic crystalline and Dieudonné analogues of Hida's ordinary -adic étale cohomology, and employ integral -adic Hodge theory to prove -adic comparison isomorphisms between these cohomologies and the -adic de Rham cohomology studied in Cais [The geometry of Hida families I: -adic de Rham cohomology, Math. Ann. (2017), doi:10.1007/s00208-017-1608-1] as well as Hida's -adic étale cohomology. As applications of our work, we provide a 'cohomological' construction of the family of -modules attached to Hida's ordinary -adic étale cohomology by Dee [ - modules for families of Galois representations, J. Algebra 235 (2001), 636-664], and we give a new and purely geometric proof of Hida's finiteness and control theorems. We also prove suitable -adic duality theorems for each of the cohomologies we construct.

AB - We construct the -adic crystalline and Dieudonné analogues of Hida's ordinary -adic étale cohomology, and employ integral -adic Hodge theory to prove -adic comparison isomorphisms between these cohomologies and the -adic de Rham cohomology studied in Cais [The geometry of Hida families I: -adic de Rham cohomology, Math. Ann. (2017), doi:10.1007/s00208-017-1608-1] as well as Hida's -adic étale cohomology. As applications of our work, we provide a 'cohomological' construction of the family of -modules attached to Hida's ordinary -adic étale cohomology by Dee [ - modules for families of Galois representations, J. Algebra 235 (2001), 636-664], and we give a new and purely geometric proof of Hida's finiteness and control theorems. We also prove suitable -adic duality theorems for each of the cohomologies we construct.

KW - Hida families

KW - crystalline cohomology

KW - de Rham cohomology

KW - integral p-adic Hodge theory

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U2 - 10.1112/S0010437X17007680

DO - 10.1112/S0010437X17007680

M3 - Article

AN - SCOPUS:85054162402

VL - 154

SP - 719

EP - 760

JO - Compositio Mathematica

JF - Compositio Mathematica

SN - 0010-437X

IS - 4

ER -