Lubin–Tate formal group law

Let Z_p 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 + x^p is an endomorphism of F, in other words

${\displaystyle e(F(x,y))=F(e(x),e(y)).\ }$

More generally, the choice for e may be any power series such that

e(x) = px + higher-degree terms and

e(x) = x^p mod p.

All such group laws, for different choices of e satisfying these conditions, are strictly isomorphic.[1] 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 Z_p 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 Z_p on the Lubin–Tate formal group law.

There is a similar construction with Z_p replaced by any complete discrete valuation ring with finite residue class field, where p is replaced by a choice of uniformizer.[2]

We outline here a formal group equivalent of the Frobenius element, which is of great importance in class field theory,[3] generating the maximal unramified extension as the image of the reciprocity map.

For this example we need the notion of an endomorphism of formal groups, which is a formal group homomorphism f where the domain is the codomain. A formal group homomorphism from a formal group F to a formal group G is a power series over the same ring as the formal groups which has zero constant term and is such that:

${\displaystyle f(F(X,Y))=G(f(X),f(Y))}$

Consider a formal group F(X,Y) with coefficients in the ring of integers in a local field (for example Z_p). Taking X and Y to be in the unique maximal ideal gives us a convergent power series and in this case we define F(X,Y) = X +_F Y and we have a genuine group law. For example if F(X,Y)=X+Y, then this is the usual addition. This is isomorphic to the case of F(X,Y)=X+Y+XY, where we have multiplication on the set of elements which can be written as 1 added to an element of the prime ideal. In the latter case f(S) = (1 + S)^p-1 is an endomorphism of F and the isomorphism identifies f with the Frobenius element.

Lubin–Tate theory is important in explicit local class field theory. The unramified part of any abelian extension is easily constructed, Lubin–Tate finds its value in producing the ramified part. This works by defining a family of modules (indexed by the natural numbers) over the ring of integers consisting of what can be considered as roots of the power series repeatedly composed with itself. The compositum of all fields formed by adjoining such modules to the original field gives the ramified part.

A Lubin–Tate extension of a local field K is an abelian extension of K obtained by considering the p-division points of a Lubin–Tate group. If g is an Eisenstein polynomial, f(t) = t g(t) and F the Lubin–Tate formal group, let θ_n denote a root of gf^n-1(t)=g(f(f(⋯(f(t))⋯))). Then K(θ_n) is an abelian extension of K with Galois group isomorphic to U/1+pⁿ where U is the unit group of the ring of integers of K and p is the maximal ideal.[2]

Lubin and Tate studied the deformation theory of such formal groups. A later application of the theory has been in the field of stable homotopy theory, with the construction of a particular extraordinary cohomology theory associated to the construction for a given prime p. As part of general machinery for formal groups, a cohomology theory with spectrum is set up for the Lubin–Tate formal group, which also goes by the names of Morava E-theory or completed Johnson–Wilson theory.[4]

• Lurie, J. (2010), Lubin–Tate theory (PDF)