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

$$A(l):={\cup }_n\ker (l^n{\mathrm{id}}_A)$$A(l):=\cup_n ker(l^n id_A)

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

Definition

We define

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

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

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

as the Dieudonné module of $A(p)$. 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 $W(k)$.

For any prime $l$, the assignation

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

is a functor.