The Lubin-Tate Formal Group Law

A (one-dimensional, commutative) formal group law f(X,Y) over some ring A is a formal power series in two variables with coefficients in A satisfying the following axioms that among other things makes it behave like an abelian group law:

  • f(X,Y)=X+Y+\text{higher order terms}
  • f(X,Y)=f(Y,X)
  • f(f(X,Y),Z)=f(X,f(Y,Z))

A homomorphism of formal group laws g:f_{1}(X,Y)\to f_{2}(X,Y) is another formal power series in two variable such f_{1}(g(X,Y))=g(f_{2}(X,Y)). An endomorphism of a formal group law is a homomorphism of a formal group law to itself.

As basic examples of formal group laws, we have the additive formal group law \mathbb{G}_{a}(X,Y)=X+Y, and the multiplicative group law \mathbb{G}_{m}(X,Y)=X+Y+XY. In this post we will focus on another formal group law called the Lubin-Tate formal group law.

Let F be a nonarchimedean local field and let \mathcal{O}_{F} be its ring of integers. Let A be an \mathcal{O}_{F}-algebra with i:\mathcal{O}_{F}\to A its structure map. A formal \mathcal{O}_{F}-module law over A over A is a formal group law f(X,Y) such that for every element a of \mathcal{O}_{F} we have an associated endomorphism [a] of f(X,Y), and such that the linear term of this endomorphism as a power series is i(a)X.

Let \pi be a uniformizer (generator of the unique maximal ideal) of \mathcal{O}_{F}. Let q=p^{f} be the cardinality of the residue field of \mathcal{O}_{F}. There is a unique (up to isomorphism) formal \mathcal{O}_{F}-module law over \mathcal{O}_{F} such that as a power series its linear term is \pi X and such that it is congruent to X^{q} mod \pi. It is called the Lubin-Tate formal group law and we denote it by \mathcal{G}(X,Y).

The Lubin-Tate formal group law was originally studied by Jonathan Lubin and John Tate for the purpose of studying local class field theory (see Some Basics of Class Field Theory). The results of local class field theory state that the Galois group of the maximal abelian extension of F is isomorphic to the profinite completion \widehat{F}^{\times}. This profinite completion in turn decomposes into the product \mathcal{O}_{F}^{\times}\times \pi^{\widehat{\mathbb{Z}}}.

The factor isomorphic to \mathcal{O}_{F}^{\times} fixes the maximal unramified extension F^{\text{nr}} of F, the factor isomorphic to \pi^{\widehat{\mathbb{Z}}} fixes an infinite, totally ramified extension F_{\pi} of F, and we have that F=F^{\text{nr}}F_{\pi}. The theory of the Lubin-Tate formal group law was developed to study F_{\pi}, taking inspiration from the case where F=\mathbb{Q}_{p}. In this case \pi=p and the infinite totally ramified extension F_{p} is obtained by adjoining to \mathbb{Q}_{p} all p-th power roots of unity, which is also the p-th power torsion of the multiplicative group \mathbb{G}_{m}. We want to generalize \mathbb{G}_{m}, and this is what the Lubin-Tate formal group law accomplishes.

Let \mathcal{G}[\pi^{n}] be the set of all elements in the maximal ideal of some separable extension \mathcal{O}_{F} such that its image under the endomorphism [\pi^{n}] is zero. This takes the place of the p-th power roots of unity, and adjoining to F all the \mathcal{G}[\pi^{n}] for all n gives us the field F_{\pi}.

Furthermore, Lubin and Tate used the theory they developed to make local class field theory explicit in this case. We define the \pi-adic Tate module T_{\pi}(\mathcal{G}) as the inverse limit of \mathcal{G}[\pi^{n}] over all n. This is a free \mathcal{O}_{F}-module of rank 1 and its automorphisms are in fact isomorphic to \mathcal{O}_{F}^{\times}. Lubin and Tate proved that this is isomorphic to the Galois group of F_{\pi} over F and explicitly described the reciprocity map of local class field theory in this case as the map from F^{\times } to \text{Gal}(F_{\pi}/F) sending \pi to the identity and an element of \mathcal{O}_{F}^{\times} to the image of its inverse under the above isomorphism.

To study nonabelian extensions, one must consider deformations of the Lubin-Tate formal group. This will lead us to the study of the space of these deformations, called the Lubin-Tate space. This is intended to be the subject of a future blog post.

References:

Lubin-Tate Formal Group Law on Wikipedia

Formal Group Law on Wikipedia

The Geometry of Lubin-Tate Spaces by Jared Weinstein

A Rough Introduction to Lubin-Tate Spaces by Zhiyu Zhang

Formal Groups and Applications by Michiel Hazewinkel

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