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

Let $k$ be a field of prime characteristic $p$.

###### Definition

Let $A$ be an abelian variety of dimension $g$. Let $l$ be a prime number.

For $n\in ℕ$ the kernel $\mathrm{ker}{l}^{n}{\mathrm{id}}_{A}\right)$ is a finite $k$-group of rank ${l}^{2ng}$. We define

$A\left(l\right):={\cup }_{n}\mathrm{ker}\left({l}^{n}{\mathrm{id}}_{A}\right)$A(l):=\cup_n ker(l^n id_A)

We have $A\left(l\right){\otimes }_{k}\overline{k}\simeq \left(ℚ\left({ℤ}_{l}\right){\right)}^{2g}$. If $l\ne p$, then $A\left(l\right)$ is an étale formal k-group?.

###### Definition

We define

${H}^{1}\left(A,l\right):={\mathrm{hom}}_{{ℤ}_{l}}\left(A\left(l\right){\otimes }_{k}\overline{k},{ℚ}_{l}/{ℤ}_{l}\right)$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 $2g$ over ${ℤ}_{l}$ and also a Galois module.

If $l=p$, then $A\left(p\right)$ is a $p$-divisible group of height $2g$. In this case we define

${H}^{1}\left(A,p\right):=M\left(A\left(p\right)\right)$H^1(A,p):=M(A(p))

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

For any prime $l$, the assignation

$A↦{H}^{1}\left(A,l\right)$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)