# formal group laws of Abelian varieties in positive characteristic

Let $G$ be an algebraic group defined over an (algebraically closed) field $k$. Then one can obtain a formal group law by completing the multiplication map $m: G \times G \to G$ at the unit of $G$.

The formal group law of $\mathbb{G}_a$ thus obtained will be given by the series $F(x,y)=x+y$ (up to isomorphism of formal group laws), as one can easily see.

In characteristic 0, over $\mathbb{C}$ at least, Abelian varietes are just quotients of finite products $\mathbb{G}_a$ by a lattice, so they have the same formal group law.

What can one say about the formal group laws of Abelian varieties (now understood as just projective algebraic groups) over fields other than $\mathbb C$? How can one see that it is isomorphic to $F(x,y)=x+y$ (or $n$-dimensional versien thereof)? What is it in characteric $p$?

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For any (smooth, finite-dimensional) commutative formal group $G$ over a field of char. 0, the theory of logarithms provides an isomorphism to a power of the formal additive group. But in positive characteristic $p$ the formal group of an abelian variety "is" the identity component of its $p$-divisible group, so this has a rich theory of moduli in dimension $> 1$. In dimension 1 over a separably closed field, a $p$-divisible group with finite height is determined up to isomorphism by its height (e.g., height 1 for an ordinary elliptic curve, height 2 for a supersingular one). – user76758 Feb 3 '14 at 16:10
The theory of $p$-divisible groups associated to abelian varieties is very classical. You may want to start with Tate's lecture in the Driebergen conference on local fields (1966), or if you read french, with Serre's talk at the Bourbaki seminar, vol. 10 (1966-68). – abx Feb 3 '14 at 16:52
It should be noted that identifying which $p$-divisible groups arise from abelian varieties is a rather subtle question (e.g., over finite fields of size $q$ there is a Weil-number restriction on the $q$-Frobenius, though there are further subtleties since a simple abelian variety may have non-isosimple $p$-divisible group), so it is quite remarkable that for many questions about abelian varieties in char. $p$ (such as for deformation theory) one can nonetheless reduce oneself to results that are valid more generally for $p$-divisible groups. – user76758 Feb 3 '14 at 22:06