We now want to generalize this from elliptic curves to abelian varieties. Given an abelian variety $\mathcal{A}$, we consider its torsion points $\mathcal{A}[m]$ as a finite flat group scheme over $\mathbb{Z}$ of rank $m^{2g}$.
Finite flat group schemes
Definition 1. An affine group scheme $G$ over a ring $R$ is a group object in the category of affine $R$-schemes. That is we have a multiplication map $m \colon G \times G \to G$ and $e \colon \operatorname{Spec} R \to G$ and $i \colon G \to G$ satisfying the usual axioms.
Then if we write $G = \operatorname{Spec} A$ we get a map $c \colon A \to A \otimes_R A$ and $e \colon A \to R$ and $i \colon A \to A$ satisfying the dual axioms.
Definition 2. A (commutative) Hopf algebra is an $R$-algebra $A$ with these structures satisfying the usual axioms.
In this case, we write $I = \ker(e \colon A \to R)$ for the augmentation ideal.
Example 3. For $\mathbb{G}_ m = \operatorname{Spec} R[T^{\pm 1}]$ we have the maps defined as $$ c \colon T \mapsto T \otimes T, \quad e \colon T \mapsto 1, \quad i \colon T \mapsto T^{-1}. $$ For $\mu_n = \operatorname{Spec} R[T]/(T^n - 1)$ we have the same formula. This has rank $n$.
Example 4. Let $G$ be a group scheme that is finite free of rank $2$. Then writing $A \cong R \oplus R$, we see that $I$ is generated by $$ I = R (e(0,1), e(1,0)) \subseteq R^{\oplus 2} \cong A. $$ Because this is an ideal, writing $I = R \alpha$ we get $A \cong R[x] / (x^2 + ax)$. Using all the axioms, we can work out that the comultiplication must be of the form $$ x \mapsto x \otimes 1 + 1 \otimes x + b x \otimes x $$ where $ab = 2$.
For $A$ a finite locally free Hopf algebra over a ring $R$, we can define a Hopf algebra structure on $A^\vee = \Hom_R(A, R)$. For example, the usual multiplication $$ m \colon A \otimes A \to A $$ corresponds to $$ c^\vee \colon A^\vee \to A^\vee \otimes_R A^\vee. $$ This can be checked to be a Hopf algebra.
Definition 5. For $G$ a commutative finite locally free group scheme, we define its Cartier dual $G^\vee$ by looking at its dual (commutative and cocommutative) Hopf algebra.
We also have the functor of points description $$ G^\vee(R^\prime) = \operatorname{Hom}_ {R^\prime}(G_{R^\prime}, \mathbb{G}_ {m,R^\prime}) = \operatorname{Hom}_ {R^\prime}^\mathrm{Hopf}(R^\prime[T^{\pm 1}], A \otimes_R R^\prime). $$
Example 6. Consider this group scheme $G_{a,b}$ that is finite locally free of rank $2$ and take its dual. The points are given by sending $T \mapsto 1 - ex$ where $e^2 + be = 0$. Then this can be written as $R[y] / (y^2 + by)$ where the group law is $$ y \mapsto y \otimes 1 + 1 \otimes y + a y \otimes y. $$ That is, we get $G_{a,b}^\vee = G_{b,a}$.
Deligne’s theorem
Theorem 7 (Deligne). If $G = \operatorname{Spec} A$ is a commutative finite locally free group scheme over a ring $R$ of rank $m$, then $mG = 0$.
This is a bit complicated. First, base changing along the structure map $R \to A$, we may reduce to showing that $m$ kills $G(R)$. Given any finite locally free $R$-algebra $S$, we may define a norm map $$ N \colon S \to R; \quad s \mapsto \det(s \colon S \to S) \in R. $$ Then we can look at the norm map $N \colon A \to R$ and base change it to $$ \tilde{N} \colon A^\vee \otimes A \to A^\vee. $$ Now given $u \in G(R)$, we may vew it as an element of $A^\vee$. We then compute $$ \tilde{N}(\mathrm{id}_ A) = \tilde{N}(T_u (\mathrm{id}_ A)) = N(u) \tilde{N}(\mathrm{id}_ A) = u^m \tilde{N}(\mathrm{id}_ A). $$ This shows that $u^m = 1$.
Proposition 8. For $A$ an (commutative co-commutative) $R$-Hopf algebra, we have $\Omega_{A/R}^1 \cong A \otimes_R I/I^2$.
Proof.
The Kahler differentials can be considered as the normal bundle to $\Delta \colon G \to G \times G$. But under an automorphism of $G \times G$ this is the same as $$ G \to G \times G; \quad g \mapsto (g,e). $$ The normal bundle here is just $A \otimes_R I/I^2$.
Proposition 9. If $m$ annihilates $G$, then $m \Omega_{A/R}^1 = 0$.
Proof.
It suffices to show that $m^\ast \colon e^\ast \Omega_{A/R}^1 \to e^\ast \Omega_{A/R}^1$ is given by multiplication by $m$. For that, it suffices to show that $$ c^\ast \colon I/I^2 \oplus I/I^2 \cong e^\ast \Omega_{A \otimes_R A/R}^1 \to e^\ast \Omega_{A/R}^1 \cong I/I^2 $$ is addition. This is essentially saying that $$ c(x) \equiv x \otimes 1 + 1 \otimes x \pmod{I^2} $$ for $x \in I$.
Corollary 10 (Cartier). In characteristic zero, every finite locally free group scheme is étale.