A homomorphism from a formal group law F of dimension m to a formal group law G of dimension n is a collection f of n power series in m variables, such that A homomorphism with an inverse is called an isomorphism, and is called a strict isomorphism if in addition f(x) = x + terms of higher degree.
Two formal group laws with an isomorphism between them are essentially the same; they differ only by a "change of coordinates".
Over the rational numbers, there is an isomorphism from the additive formal group law to the multiplicative one, given by exp(x) − 1.
Over general commutative rings R there is no such homomorphism as defining it requires non-integral rational numbers, and the additive and multiplicative formal groups are usually not isomorphic.
Any n-dimensional formal group law gives an n-dimensional Lie algebra over the ring R, defined in terms of the quadratic part F2 of the formal group law.
[3] Over fields of non-zero characteristic, formal group laws are not equivalent to Lie algebras.
So in some sense formal group laws are the "right" substitute for Lie algebras in characteristic p > 0.
[4] In other words, there is a strict isomorphism f from the additive formal group to F, called the logarithm of F, so that Examples: If R does not contain the rationals, a map f can be constructed by extension of scalars to R ⊗ Q, but this will send everything to zero if R has positive characteristic.
Formal group laws over a ring R are often constructed by writing down their logarithm as a power series with coefficients in R ⊗ Q, and then proving that the coefficients of the corresponding formal group over R ⊗ Q actually lie in R. When working in positive characteristic, one typically replaces R with a mixed characteristic ring that has a surjection to R, such as the ring W(R) of Witt vectors, and reduces to R at the end.
When F is one-dimensional, one can write its logarithm in terms of the invariant differential ω(t).
Conversely, given a Hopf algebra whose coalgebra structure is given above, we can recover a formal group law F from it.
So 1-dimensional formal group laws are essentially the same as Hopf algebras whose coalgebra structure is given above.
Given an n-dimensional formal group law F over R and a commutative R-algebra S, we can form a group F(S) whose underlying set is Nn where N is the set of nilpotent elements of S. The product is given by using F to multiply elements of Nn; the point is that all the formal power series now converge because they are being applied to nilpotent elements, so there are only a finite number of nonzero terms.
The group-valued functor of F can also be described using the formal group ring H of F. For simplicity we will assume that F is 1-dimensional; the general case is similar.
Suppose that f is a homomorphism between one-dimensional formal group laws over a field of characteristic p > 0.
We let be for indeterminates and we define the universal ring R to be the commutative ring generated by the elements ci,j, with the relations that are forced by the associativity and commutativity laws for formal group laws.
Formal groups and formal group laws can also be defined over arbitrary schemes, rather than just over commutative rings or fields, and families can be classified by maps from the base to a parametrizing object.
The moduli space of formal group laws is a disjoint union of infinite-dimensional affine spaces, whose components are parametrized by dimension, and whose points are parametrized by admissible coefficients of the power series F. The corresponding moduli stack of smooth formal groups is a quotient of this space by a canonical action of the infinite-dimensional groupoid of coordinate changes.
This difference gives formal groups a rich geometric theory in positive and mixed characteristic, with connections to the Steenrod algebra, p-divisible groups, Dieudonné theory, and Galois representations.
For supersingular elliptic curves, this control is complete, and this is quite different from the characteristic zero situation where the formal group has no deformations.
A formal group is sometimes defined as a cocommutative Hopf algebra (usually with some extra conditions added, such as being pointed or connected).
In the smooth case, choosing coordinates is equivalent to taking a distinguished basis of the formal group ring.
The Lubin–Tate formal group law is the unique (1-dimensional) formal group law F such that e(x) = px + xp is an endomorphism of F, in other words More generally we can allow e to be any power series such that e(x) = px + higher-degree terms and e(x) = xp mod p. All the group laws for different choices of e satisfying these conditions are strictly isomorphic.
[8] For each element a in Zp there is a unique endomorphism f of the Lubin–Tate formal group law such that f(x) = ax + higher-degree terms.
There is a similar construction with Zp replaced by any complete discrete valuation ring with finite residue class field.
[9] This construction was introduced by Lubin & Tate (1965), in a successful effort to isolate the local field part of the classical theory of complex multiplication of elliptic functions.