TY - JOUR

T1 - The geometry of Hida families I

T2 - Λ -adic de Rham cohomology

AU - Cais, Bryden

N1 - Funding Information:
During the writing of this paper, the author was partially supported by an NSA Young Investigator Grant (H98230-12-1-0238) and an NSF RTG (DMS-0838218).

PY - 2018/10/1

Y1 - 2018/10/1

N2 - 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 (Cais, Compos Math, to appear) 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 (Cais, Compos Math, to appear), 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 Dee (J Algebra 235(2), 636–664, 2001), 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 Mazur and Wiles (Compos Math 59(2):231–264, 1986) and Ohta (J Reine Angew Math 463:49–98, 1995).

AB - 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 (Cais, Compos Math, to appear) 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 (Cais, Compos Math, to appear), 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 Dee (J Algebra 235(2), 636–664, 2001), 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 Mazur and Wiles (Compos Math 59(2):231–264, 1986) and Ohta (J Reine Angew Math 463:49–98, 1995).

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U2 - 10.1007/s00208-017-1608-1

DO - 10.1007/s00208-017-1608-1

M3 - Article

AN - SCOPUS:85039036417

VL - 372

SP - 781

EP - 844

JO - Mathematische Annalen

JF - Mathematische Annalen

SN - 0025-5831

IS - 1-2

ER -