This paper classifies dimension one polynomial formal groups over valuation rings
of local fields, and applies isogenies of such formal groups to construct certain p-power
rank i?-Hopf algebras. The Kummer theory of formal groups is applied to classify
the principal homogeneous spaces over these Hopf algebras.
1991 Mathematics subject classification numbers: 11S31, 16W30,14L05, 14L15
We are interested in (one dimensional) polynomial formal groups denned over a
ring R (commutative, with identity) and the polynomial maps between them. Recall
that if ^{x^y) and Q(x,y) are formal groups defined over R, then a homomorphism
(j: T —* Q is a polynomial in R[x] with zero constant term, so that ^{^{x^y)) =
Q {j{x))j){y)). In particular, if T and Q are degree 2 polynomial formal groups, then
a linear homomorphism j): T Q defined over R is multiplication by an element
of R which we will also call (f. Clearly the homomorphism j is an isomorphism
over R if and only if the element 0 is a unit in R.
Even when R is an algebraically closed field of characteristic zero, the classifi-
cation of polynomial formal groups is interesting. For an initial observation, the
dimension one additive group Ga(x,y) = x+y and the dimension one multiplicative
group Gm(ar, y) = xy + x + y are not isomorphic by a polynomial homomorphism.
For if f)(x) is a polynomial so that
then, as polynomials, j)(x-\-y+xy) = (j)(x)+j)(y). Hence /(0) = 0, and differentiating
with respect to x gives
(j\x + y + xy)(l + y) = /'(x).
Setting y = - 1 yields /'(-l) 0 = /'(x). Thus /(x) = 0. For /(x) = log(l + x) and
Qc R this argument obviously fails!
The object of this paper is to study the dimension one polynomial formal groups,
both to introduce ideas which will be studied further in higher dimensions and to
demonstrate that, even in dimension one, the results are of interest.
We begin by classifying polynomial formal groups over a ring R.
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