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§7. Expansions Near the Origin and the Formal Group 251

If pR = 0, i.e., R has characteristic a prime p, then it can be shown that

[p](X) = c 1 X p h + c 2 X 2p h + c 3 X 3p h +· · · ,

when h ≥ 1 is an integer called the height of the formal group. When [p](X) = 0,
we set h =∞.If R = k an algebraically closed field, then the height classifies the
formal group up to isomorphism.
In Chapter 13 we will see that the height of the formal group of an ellipticcurve
in characteristic p is either 1, which is the usual case, or 2, which is called the su-
persingular case. This is related to the Hasse invariant which can be defined as the
p
coefficient of X in [p](X) above.
(7.4) Remark. A formal group F(X, Y) over R, like an ellipticcurve, has an invari-
ant differential ω(Y)=(D 1 F(0, Y)) −1 dY. Observe that the derivative D 1 F(X, Y)=
1+ higher-order terms, and hence the inverse power series is defined. The invariance
property for a differential A(X)dX means that for a variable T

A(X)dX = A(F(X, T ))D 1 F(X, T )dX.
For A(Y) = (D 1 F(0, Y)) −1 we calculate from the associative law using the chain
rule the following relations:
(D 1 F)(F(X, Y), T ) · D 1 F(X, Y) = D 1 F(X, F(Y, T )).

Setting X = 0

(D 1 F)(Y, T ) · D 1 F(0, T ) = D 1 F(0, F(Y, T ))

which gives
D 1 F(0, Y) −1 dY = D 1 F(0, F(Y, T )) −1 (D 1 F(Y, T ))dY.

(7.5) Example. The formal group E (X, Y) of an ellipticcurve E is defined over
Z[a 1 , a 2 , a 3 , a 4 , a 6 ] and the coefficients specialize to the field of definition k of E.
The invariant differential of the formal group E is just the differential ω of E with
the expansion given in (6.1).
(7.6) Definition. A formal logarithm for a formal group F(X, Y) over R is a power
series L(X) in R[[X]] with L(X) = X+ higher-order terms and F(X, Y) =
L −1 (L(X) + L(Y)).

In other words L(F(X, Y)) = L(X) + L(Y) which says that L : F → G a is
an isomorphism onto the formal additive group. Then [m] F (X) = L −1 (mL(X)) and
from

L (Y) · (D 1 F)(0, Y) = L (0) = 1,
we see that the invariant differential is given by