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252 12. Endomorphisms of Elliptic Curves

ω(Y) = L (Y)dY = dL(Y).

n=1 n T )(dT/T ) we have an expansion
Thus from an expansion for ω(T ) = ( ∞ a n
for the formal logarithm
∞
a n n

L(T ) = T , where a 1 = 1.
n
n=1
Hence the formal logarithm exists if and only if given the coefficients a n of the
invariant differential ω in R we can form a n /n in R.Ina Q-algebra R this is always
possible, and in general in characteristic zero it is possible in an extension ring R⊗Q
of R.
In characteristic p > 0 we can always form a n /n for n < p, and thus the
coefficient a p plays a basic role especially for our applications in Chapter 13 to
elliptic curves. we will use:
(7.7) Proposition. The coefficient a p in the expansion of the invariant differential of
p
a formal group F(X, Y) in characteristic p is the coefficent of X in the expansion
of [p] F (X).