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 \(\gothm_K\text{.}\) For any two \(f,g \in \calF_\pi\text{,}\) 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 \(\gothm_K\text{.}\) For any \(f \in \calF_\pi\text{,}\) 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. Note that this notation was used previously in Remark 4.1.9, and we will eventually show that these usages coincide (Theorem 4.6.16).

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.

Subsection Local reciprocity from formal groups

We now fulfill a promise made for \(K = \QQ_p\) in Example 4.1.4, and use the Lubin-Tate construction to completely pin down the local reciprocity map. This can be done without even assuming the existence of a local reciprocity map, but we will not bother with that here.

Lemma 4.6.15.

Let \(K\) be a local field. Let \(\pi_1, \pi_2\) be two uniformizers of \(K\) and set \(a := \pi_2/\pi_1 \in \gotho_K^\times\text{.}\) For \(i=1,2\text{,}\) pick \(f_i \in \calF_{\pi_i}\) and let \(F_i\) be the formal group associated to \(f_i\) via Corollary 4.6.8. Let \(L/K\) be the maximal unramified extension, let \(R\) be the completion of \(\gotho_L\text{,}\) let \(\sigma\colon R \to R\) be the Frobenius automorphism, and choose \(\epsilon \in R^\times\) such that \(\epsilon^\sigma = a \epsilon\) (see Exercise 4). Then as formal groups over \(R\text{,}\) there is a unique isomorphism \(\theta\) from \(F_1\) to \(F_2\) such that \(\theta \equiv \epsilon T \pmod{T^2}\) and \(\theta^\sigma = \theta \circ [a]_{f_1}\) (where \(\theta^\sigma\) means apply \(\sigma\) to each coefficient of \(\theta\)).

Proof.

See Exercise 3.

Theorem 4.6.16. 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 inverse of 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 \(\Norm_{K(\alpha)/K}(\alpha) = \pi\text{.}\) Consequently, our proposed formula for \(\phi_K\) has the property that \(\phi_K(\pi)\) fixes \(K_\pi\text{.}\) It will thus suffice to show that \(\phi_K(\varpi)\) fixes \(K_{\varpi}\) for any other uniformizer \(\varpi\) of \(K\text{,}\) as then we can conclude as in Remark 4.1.9. (In other words, we need to show that our formula does not depend on the choice of \(\pi\text{.}\))

Set \(a := \varpi/\pi \in \gotho_K^\times\text{.}\) For compatibility with the notation in Lemma 4.6.15, set \(\pi_1 := \pi\text{,}\) \(\pi_2 := \varpi\text{.}\) Then \(K_\varpi\) is generated by \(\theta(\alpha)\) for \(\alpha\) running over \(K_\pi\text{,}\) and

\begin{align*} \phi_{K,2}(\pi_1)(\theta(\alpha)) &= [a^{-1}]_{f_1}(\theta(\alpha))^\sigma \\ &= [a^{-1}]_{f_1}(\theta^\sigma(\alpha))\\ &= (\theta^\sigma)([a^{-1}]_{f_1}(\alpha))\\ &= (\theta \circ [a]_{f_1})([a^{-1}]_{f_1}(\alpha)) = \theta(\alpha)\text{.} \end{align*}

(Compare [37], I.3.)

Remark 4.6.17.

We recall Remark 2.2.10: 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]). When \(K\) is a function field then there does exist such an analogue: for \(K = \FF_p(t)\) this was discovered by Carlitz, and this was (rediscovered and) generalized to arbitrary \(K\) 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.

3.

Prove Lemma 4.6.15.

Hint.

Use an inductive argument in the manner of Proposition 4.6.7, but this time making repeated use of Exercise 4 to match up increasingly many coefficients of \(\theta^\sigma\) and \(\theta \circ [a]_{f_1}\text{.}\) For more details, see [37], Proposition I.3.10.

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