Lubin–Tate formal group law
In mathematics, the Lubin–Tate formal group law is a formal group law introduced by Lubin and Tate (1965) to isolate the local field part of the classical theory of complex multiplication of elliptic functions. In particular it can be used to construct the totally ramified abelian extensions of a local field. It does this by considering the (formal) endomorphisms of the formal group, emulating the way in which elliptic curves with extra endomorphisms are used to give abelian extensions of global fields
Let Zp be the ring of p-adic integers. 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, the choice for e may be any power series such that
All such group laws, for different choices of e satisfying these conditions, are strictly isomorphic. We choose these conditions so as to ensure that they reduce modulo the maximal ideal to Frobenius and the derivative at the origin is the prime element.
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. This gives an action of the ring Zp on the Lubin–Tate formal group law.
There is a similar construction with Zp replaced by any complete discrete valuation ring with finite residue class field, where p is replaced by a choice of uniformizer.
We outline here a formal group equivalent of the Frobenius element, which is of great importance in class field theory, generating the maximal unramified extension as the image of the reciprocity map.
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