Abelian extensions via the Lubin-Tate construction

Section 4.6 Abelian extensions via the Lubin-Tate construction

Reference.

[37], I.2 and I.3; [4], IV.

Subsection Formal group laws

The idea here is to imitate Kronecker-Weber by identifying some kind of group object whose torsion points generate abelian extensions of a local field. We define the group object first.

Definition 4.6.1.

A one-parameter commutative formal group law over a commutative ring \(A\) is a formal power series \(F \in A \llbracket X,Y \rrbracket\) such that:

\(F \equiv X + Y \pmod{(X,Y)^2}\text{;}\)
\(F(X, F(Y,Z)) = F(F(X,Y),Z)\) (associativity);
there exists a unique series \(i_F \in X A \llbracket X \rrbracket\) such that \(F(X, i_F(X)) = 0\) (existence of inverses);
\(F(X,Y) = F(Y,X)\) (commutativity).

Note that throughout this definition, we are using the fact that we can substitute a power series with constant term 0 (in any number of variables) into another power series to get a meaningful result.

Suppose now that \(A = \gotho_K\) where \(K\) is a local field. Then we can also substitute elements of the maximal ideal \(\gothm_K\) into \(F\text{,}\) and this will define an exotic group structure on this set.

Example 4.6.2.

Take \(F = X + Y\) (the formal additive group). When \(A = \gotho_K\text{,}\) this recovers the usual group structure on \(\gothm_K\text{.}\)

Example 4.6.3.

Take \(F = X+Y+XY\) (the formal multiplicative group). When \(A = \gotho_K\text{,}\) this corresponds to the group structure on \(1+\gothm_K\) via the bijection \(x \mapsto x+1\) from \(\gothm_K\) to \(1+\gothm_K\text{.}\)

Subsection Lubin-Tate formal groups

We next introduce a condition on a formal group law analogous to complex multiplication for an elliptic curve.

Definition 4.6.4.

Let \(F_1, F_2\) be a formal group law over a ring \(A\text{.}\) A homomorphism from \(F_1\) to \(F_2\) is a power series \(h \in T A \llbracket T \rrbracket\) such that

\begin{equation*} h(F_1(X,Y)) = F_2(h(X), h(Y))\text{.} \end{equation*}

When \(F_1 = F_2 = F\) we call this an endomorphism of \(F\text{.}\)

There is a natural notion of addition of endomorphisms: if \(f_1\) and \(f_2\) are endomorphisms of \(F\text{,}\) then so is

\begin{equation*} f_1 +_F f_2 = F(f_1, f_2)\text{.} \end{equation*}

There is also a notion of multiplication of endomorphisms, which coincides with composition of formal power series. In particular this is in general not commutative.

Example 4.6.5.

For any formal group law \(F\text{,}\) there exist unique endomorphisms \([n]\) for \(n \in \ZZ\) such that:

\([0] = 0\) and \([1] = T\text{;}\)
\([n_1] + [n_2] = [n_1 + n_2]\text{;}\)
\([n_1] \circ [n_2] = [n_1 n_2]\text{.}\)

Definition 4.6.6.

Let \(K\) be a local field with residue field of order \(q\) and fix a generator \(\pi\) of \(\gothm_K\text{.}\) Let \(\calF_\pi\) be the subset of \(f \in T \gotho_K \llbracket T \rrbracket\) satisfying the following conditions:

\(f \equiv \pi T \pmod{T^2}\text{;}\)
\(f \equiv T^q \pmod{\pi}\text{.}\)

Proposition 4.6.7.

Let \(K\) be a local field with residue field of order \(q\) and fix a generator \(\pi\) of \(\gothm_K\text{.}\) Let \(f,g \in T \gotho_K \llbracket T \rrbracket\) be two series such that \(f,g \equiv \pi T \pmod{T^2}\) and \(f,g \equiv T^q \pmod{\pi}\text{.}\) Then for any homogeneous linear polynomial \(\phi_1\) in the variables \(X_1,\dots,X_n\) over \(\gotho_K\text{,}\) there exists a unique power series \(\phi \in \gotho_K \llbracket X_1,\dots,X_n \rrbracket\) such that:

\(\phi \equiv \phi_1 \pmod{(X_1,\dots,X_n)^2}\text{;}\)
\(f(\phi(X_1,\dots,X_n)) = \phi(g(X_1),\dots,g(X_n))\text{.}\)

Proof.

Writing \(\phi = \sum_{i=1}^\infty \phi_i\) where \(\phi_i\) is homogeneous of degree \(i\text{,}\) we construct the \(\phi_i\) inductively. Suppose that \(\phi_1,\dots,\phi_i\) are already specified (initially with \(i=1\)), and set \(\phi_{\leq i} = \phi_1 + \cdots + \phi_i\text{.}\) By looking at Taylor expansions, we obtain

\begin{align*} f((\phi_{\leq_i} + \phi_{i+1})(X_1, \dots, X_N)) &\equiv f(\phi_{\leq i}(X_1,\dots,X_n)) + \pi \phi_{i+1} \pmod{(X_1,\dots,X_n)^{i+2}}\\ (\phi_{\leq_i} + \phi_{i+1})(g(X_1),\dots,g(X_n)) &\equiv \phi_{\leq i}(g(X_1),\dots,g(X_n)) + \pi^{i+1} \phi_{i+1} \pmod{(X_1,\dots,X_n)^{i+2}}\text{.} \end{align*}

This means that we must ensure that

\begin{equation*} \phi_{i+1} \equiv \frac{f(\phi_{\leq i}(X_1,\dots,X_n)) - \phi_{\leq i}(g(X_1),\dots,g(X_n))}{(\pi^i-1)\pi} \pmod{(X_1,\dots,X_n)^{i+2}}. \end{equation*}

But in fact the numerator of the fraction is divisible by \(\pi\text{:}\) modulo \(\pi\) both \(f\) and \(g\) are congruent to \(T^q\) and this map is a ring homomorphism. We thus obtain a unique choice of \(\phi_{i+1}\) that allows the construction to continue.

Corollary 4.6.8.

Let \(K\) be a local field with residue field of order \(q\) and fix a generator \(\pi\) of \(\gothm_K\text{.}\) Let \(f \in T \gotho_K \llbracket T \rrbracket\) be a series such that \(f \equiv \pi T \pmod{T^2}\) and \(f \equiv T^q \pmod{\pi}\text{.}\) Then there exists a unique formal group \(F\) over \(\gotho_K\) admitting \(f\) as an endomorphism. Moreover, for every \(a \in \gotho_K\text{,}\) \(F\) admits a unique endomorphism \([a] \equiv \alpha T \pmod{T^2}\) which commutes with \(f\) (in particular, \([\pi] = f\)), and these satisfy

\begin{equation*} [a+b] = [a] +_F [b], \qquad [ab] = [a] \circ [b]\text{.} \end{equation*}

Proof.

We get \(F\) by applying Proposition 4.6.7 with \(g=f\) and \(\phi_1(X,Y) = X+Y\text{.}\) Since \(\phi_1(X,Y)\) also equals \(Y+X\text{,}\) the uniqueness aspect of Proposition 4.6.7 implies commutativity. We deduce existence of inverses by similarly taking \(\phi_1(X) = -X\text{.}\) We deduce associativity by similarly taking \(\phi_1(X,Y,Z) = X+Y+Z\) and invoking the uniqueness aspect. We produce \([a]\) by similarly taking \(\phi_1(X) = aX\text{;}\) we check that \([a+b] = [a] +_F [b]\) and \([ab] = [a] \circ [b]\) by invoking the uniqueness aspect. (Compare [37], Proposition I.2.12 and I.2.14.)

Remark 4.6.9.

For each \(f \in \calF_\pi\text{,}\) we can find a unit \(u \in \gotho_K \llbracket T \rrbracket^\times\) with \(u \equiv 1 \pmod{\pi}\) such that \(uf\) (which is also in \(\calF_\pi)\) is a monic polynomial of degree \(q\text{.}\) This follows for example from the Weierstrass preparation theorem.

Similarly, if we were to relax the second condition in Definition 4.6.6 to require only that \(f \equiv T^q \pmod{\pi, T^{q+1}}\text{,}\) then we could again always find a unit \(u \in \gotho_K \llbracket T \rrbracket^\times\) with \(u \equiv 1 \pmod{\pi,T}\) such that \(uf\) is a monic polynomial of degree \(q\text{.}\) (This relaxation would correspond to the definition of the height of a formal group.)

Subsection Abelian extensions from a Lubin-Tate formal group

Definition 4.6.10.

Choose \(K, f, F\) as in Corollary 4.6.8. Let \(\Lambda_f\) be the \(\gotho_K\)-module whose underlying set consists of all \(\alpha \in \overline{K}\) of absolute value less than 1, with addition given by \(\alpha + \beta := F(\alpha,\beta)\) and the action of \(a \in \gotho_K\) on \(\beta\) given by \([a](\beta)\text{.}\) Let \(K_\pi\) be the subfield of \(\overline{K}\) generated by the \([\pi^k]\)-torsion elements of \(\Lambda_f\) as \(k\) runs over all positive integers.

Remark 4.6.11.

When applying Definition 4.6.10, we may use the fact that by Proposition 4.6.7, for any other series \(g\) satisfying \(g \equiv \pi T \pmod{T^2}\) and \(g \equiv T^q \pmod{\pi}\text{,}\) we get a canonical isomorphism \(\Lambda_f \cong \Lambda_g\) of \(\gotho_K\)-modules; in particular, this isomorphism is compatible with the action of \(\Gal(\overline{K}/K)\text{.}\) This lets us simplify some arguments by reducing to the case where \(f\) is a polynomial, or even to the case \(f = \pi T + T^q\text{.}\)

Lemma 4.6.12.

With notation as in Definition 4.6.10, the \([\pi^k]\)-torsion submodule of \(\Lambda_f\) is isomorphic to \(\gotho_K/\pi^k\text{.}\)

Proof.

For \(k=1\) we see that the \([\pi]\)-torsion submodule consists of \(q\) elements (by Remark 4.6.11 we need only check this for \(f = \pi T + T^q\) for which it is obvious), and hence is a one-dimensional \(\gotho_K/(\pi)\)-vector space. Since \([\pi]\colon \Lambda_f \to \Lambda_f\) is surjective, the general case now follows using Nakayama’s lemma (or see [37], lemma I.3.3).

Corollary 4.6.13.

The extension \(K_\pi/K\) is abelian and totally ramified; more precisely, there is a canonical isomorphism \(\Gal(K_\pi/K) \to \gotho_K^\times\text{.}\)

Proof.

The inclusion \(\Gal(K_\pi/K) \to \gotho_K^\times\) is immediate from Lemma 4.6.12. To check surjectivity and the totally ramified property, we check that the torsion points of exact order \(\pi^k\) are the roots of an Eisenstein polynomial over \(\gotho_K\text{.}\) Again by Remark 4.6.11 we need only check this for \(f = \pi T + T^q\text{,}\) in which case we can write

\begin{equation*} [\pi]^k = \sum_{i=0}^k \pi^{k-i} T^{q^i} + \cdots \end{equation*}

and the omitted monomials do not contribute to the Newton polygon (each one is strictly divisible by one of the summands). By factoring according to the slopes of the Newton polygons (using Hensel’s lemma), we see that the roots of \([\pi^k]\) which are not roots of \([\pi^{k-1}]\) are the roots of an Eisenstein polynomial over \(\gotho_K\text{.}\)

Example 4.6.14.

For \(K = \QQ_p\text{,}\) we can take \(\pi=p\) and \(f = (1+T)^p - 1\text{.}\) In this case \(K_\pi = \bigcup_k \QQ(\zeta_{p^k})\) is the \(p\)-part of the cyclotomic tower.

For the following statement, we will use local reciprocity but not the existence theorem. It is also possible to give an “elementary” proof more in the style of the proof of local Kronecker-Weber (Theorem 1.1.5), but we will not do this here.

Theorem 4.6.15. Lubin–Tate.

With notation as in Definition 4.6.10, \(K^{\ab} = K_\pi K^{\unr}\text{.}\) Moreover, the local reciprocity map may be written in the form

\begin{equation*} \phi_K\colon \gotho_K^\times \times \pi^\ZZ \to \Gal(K_\pi/K) \times \Gal(K^{\unr}/K) \end{equation*}

with the first factor mapping via the isomorphism \(\gotho_K^\times \to \Gal(K_\pi/K)\) from Corollary 4.6.13 and the second factor mapping \(\pi\) to the Frobenius in \(\Gal(K^{\unr}/K)\text{.}\)

Proof.

By Corollary 4.6.13, \(K_\pi\) is a totally ramified abelian extension of \(K\) which is linearly disjoint from \(K^{\unr}\) with Galois group \(\gotho_K^\times\text{.}\) Hence \(K_\pi K^{\unr}\) is an abelian extension of \(K\) with Galois group \(\ZZ \times \gotho_K^\times\text{;}\) using local reciprocity (Theorem 4.1.2) we see that this extension must be all of \(K^{\ab}\text{.}\)

Now note that if \(\alpha \in \Lambda_f\) is a \(\pi^k\)-torsion point, then \(\Norm_{K(\alpha)/K}(\alpha) = \pi\text{.}\) Meanwhile, we see from the congruence \(f \equiv T^q \pmod{\pi}\) that the action of the local reciprocity map on \(\gotho_K^\times\) agrees with the map from Corollary 4.6.13.

Remark 4.6.16.

In general, there is no analogue of the Lubin-Tate construction over a number field, except when \(K = \QQ\) for which the multiplicative group can be used (recovering the Kronecker-Weber theorem) or when \(K\) is imaginary quadratic for which an elliptic curve with complex multiplication by \(K\) can be used (see [10]). However, when \(K\) is a function field then there does exist such an analogue, which was originally discovered by Carlitz and then rediscovered (and extended) by Drinfeld (see [19]).

Exercises Exercises

1.

Prove that in Definition 4.6.1, the existence of inverses is a consequence of the other conditions.

2.

Prove that in Definition 4.6.1, if the ring \(A\) has characteristic 0, then commutativity is a consequence of the other conditions.

Hint.

First note that we can check commutativity after replacing \(A\) by \(A\text{,}\) so we may assume that \(A\) is a \(\QQ\)-algebra. In that case, show that \(F\) is isomorphic to the formal additive group.

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