Formal Groups and Applications

Formal groups are a cool idea. Suppose you want to define a group law, so that z is the “product” of x and y. One might imagine doing this via a power series in the variables x and y, so that z = F(x, y). In order to have an identity element, we require F(x, 0) = x and F(0, y) = y. For associativity, we need F(a, F(b, c)) = F(F(a, b), c). It turns out that we can assume that

F(x, y) = x + y + higher order terms

and as a consequence we see that we don’t need to add a condition for inverses to exist: we can always find a formal power series i(x) such that F(x, i(x)) = F(i(x), x) = 0.

The power series F(x, y) is sometimes referred to as itself a formal group; it’s more precise to say it’s a one-dimensional formal group law. The two basic examples are the additive group, given by

F(x, y) = x + y

and the multiplicative formal group,

F(x, y) = x + y + xy

(for the latter, notice that we need 0 to be the identity element, so we write elements of the multiplicative group in the form 1 + x).

The neat thing about all this is that it is entirely formal: F(x, y) is just a power series, and we can play with it directly. And this simple idea turns out to have lots of applications, as the size of this book shows.

If we replace the variable x by an n-tuple (x₁, x₂, …, x_n), y by (y₁, y₂, …, y_n) and the power series F(x, y) by an n-tuple of power series in 2n variables, we get the definition of a n-dimensional formal group law. The first two chapters of Hazewinkel’s book give many examples of such things.

The idea of a formal group comes from the theory of Lie groups. Since a Lie group is an n-dimensional manifold, we can choose coordinates (x₁, x₂, …,x_n) centered on the identity element, so that the identity has coordinates (0, 0, …, 0). Since multiplication on a Lie group can be expressed as a power series, we get a formal group law. In characteristic zero, this formal group law is completely captured by the Lie algebra (this is one way to “read” the Baker-Campbell-Hausdorff formula), but this is not the case in characteristic p. It was to study the situation in characteristic p that Dieudonné used formal group laws, which serve as a kind of intermediate object between the group and its Lie algebra.

One typically works with formal groups over a commutative ring which is complete with respect to some topology. In that case, a formal group law yields an actual group whose elements are the topologically nilpotent elements of the base ring. Or we can work in a larger ring which is an algebra over the base ring and take topologically nilpotent elements of that algebra. The functor this defines is the actual “formal group.”

Hazewinkel’s account of the theory and applications of formal group laws first appeared in 1978 and quickly became the standard reference on the subject. It has long been out of print, however, and fairly hard to find. Anyone interested in the subject will be delighted that AMS/Chelsea has brought it back into print. The author has made a few corrections, which he says are mostly minor. They are incorporated in the text of the new printing whenever possible, but also appear as addenda in a few places. In fact, then, this is a new and improved version of a classic reference.

Hazewinkel mentions, in the introduction to the new edition, that his first idea was to update the book to reflect what has happened since 1978. After digging a little through the literature, he discovered that so much has happened that a whole new book is warranted, and he promises us that book in a year or two. Meanwhile, we have his classic account of the theory, once more available to those who want to explore the subject.

Fernando Q. Gouvêa first learned about formal groups as a graduate student. That was after 1978, but not much after. He is now Carter Professor of Mathematics at Colby College in Waterville, ME.