Let~$V$ be a complete dvr with residue field~$k$ of characteristic~$p>0$ and fraction field~$K$ of characteristic zero. Let~$\gerS$ ba a formal scheme over~$V$ and let~$\gerX\to \gerS$ be a locally projective formal abelian scheme. In this paper we prove that, under suitable natural conditions on the Hasse--Witt matrix of~$\gerX\tensor_V V/pV$, the kernel of the Frobenius morphism on~$\gerX_k$ can be canonically lifted to a finite and flat subgroup scheme of~$\gerX$ over an admissible blow up of~$\gerS$, called the ``canonical subgroup of~$\gerX$". This is done by a careful study of torsors under group schemes of order~$p$ over~$\gerX$. We also present a filtration on~$\H^1(\gerX,\mu_p)$ in the spirit of the Hodge--Tate decomposition.

The canonical subgroup for families of abelian varieties / F. Andreatta, C. Gasbarri. - In: COMPOSITIO MATHEMATICA. - ISSN 0010-437X. - 143:3(2007), pp. 566-602.

The canonical subgroup for families of abelian varieties

F. Andreatta
Primo
;
2007

Abstract

Let~$V$ be a complete dvr with residue field~$k$ of characteristic~$p>0$ and fraction field~$K$ of characteristic zero. Let~$\gerS$ ba a formal scheme over~$V$ and let~$\gerX\to \gerS$ be a locally projective formal abelian scheme. In this paper we prove that, under suitable natural conditions on the Hasse--Witt matrix of~$\gerX\tensor_V V/pV$, the kernel of the Frobenius morphism on~$\gerX_k$ can be canonically lifted to a finite and flat subgroup scheme of~$\gerX$ over an admissible blow up of~$\gerS$, called the ``canonical subgroup of~$\gerX$". This is done by a careful study of torsors under group schemes of order~$p$ over~$\gerX$. We also present a filtration on~$\H^1(\gerX,\mu_p)$ in the spirit of the Hodge--Tate decomposition.
Abelian scheme; Bloch-Kato filtration; Canonical subgroup; Frobenius endomorphism; Torsors
Settore MAT/03 - Geometria
2007
Article (author)
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2434/141687
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