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Demazure, lectures on p-divisible groups, V.2, points of finite order and endomorphisms

Let kk be a field of prime characteristic pp.

Definition

Let AA be an abelian variety of dimension gg. Let ll be a prime number.

For nn\in \mathbb{N} the kernel kerl nid A)ker l^n id_A) is a finite kk-group of rank l 2ngl^{2 n g}. We define

A(l):= nker(l nid A)A(l):=\cup_n ker(l^n id_A)

We have A(l) kk¯(( l)) 2gA(l)\otimes_k \overline k\simeq (\mathbb{Q}(\mathbb{Z}_l))^{2g}. If lpl\neq p, then A(l)A(l) is an étale formal k-group?.

Definition

We define

H 1(A,l):=hom l(A(l) kk¯, l/ l)H^1(A,l):=hom_{\mathbb{Z}_l}(A(l)\otimes_k \overline k, \mathbb{Q}_l /\mathbb{Z}_l)

This is a free module of rank 2g2g over l\mathbb{Z}_l and also a Galois module.

If l=pl=p, then A(p)A(p) is a pp-divisible group of height 2g2g. In this case we define

H 1(A,p):=M(A(p))H^1(A,p):=M(A(p))

as the Dieudonné module of A(p)A(p). It is an F-lattice? (defined in Demazure, lectures on p-divisible groups, IV.1, isogenies) over kk, and in particular a free module of rank 2g2g over W(k)W(k).

For any prime ll, the assignation

AH 1(A,l)A\mapsto H^1(A,l)

is a functor.

Created on May 28, 2012 00:08:34 by Stephan Alexander Spahn (79.227.178.105)