Cohomology of the idèles II: the "Second Inequality"

Section 7.2 Cohomology of the idèles II: the “Second Inequality”

Reference.

[37] VII.5; [38] VI.4. See also this blog post by Akhil Mathew

amathew.wordpress.com/2010/06/05/the-algebraic-proof-of-the-second-inequality-i/

In Section 7.1, we proved that for \(L/K\) a cyclic extension of number fields, the Herbrand quotient \(h(C_L)\) of the idèle class group of \(L\) is equal to \([L:K]\) (Theorem 7.1.1) and deduced that \(\#H^0_T(\Gal(L/K), C_L) \geq [L:K]\) (the “First Inequality”; Theorem 7.1.2). This time we’ll prove the reverse inequality, and even a somewhat stronger statement (see Theorem 7.2.9 below).

For this step, we have no local analogue to draw upon: the corresponding assertion in local class field theory is covered by Theorem 90 (Lemma 1.2.3). Unfortunately, there seems to be no direct approach to computing either \(H^{-1}_T(\Gal(L/K), C_L)\) or \(H^1(\Gal(L/K), C_L)\) (or for that matter \(H^0_T(\Gal(L/K), L^*)\text{;}\) see Theorem 7.2.10), so some alternate strategy is needed.

We take an analytic approach motivated by the proof of Dirichlet’s theorem on primes in arithmetic progressions; see Lemma 7.2.5. There is also an algebraic approach, but we prefer to postpone discussing it until we are ready to tackle the existence theorem, as these two topics share similar ideas; see Theorem 7.4.14.

Subsection Back to ideals

For the analytic proof, we need to recast the Second Inequality back into classical, ideal-theoretic language. In this argument, there is no need to assume that \(L/K\) is cyclic.

Definition 7.2.1.

Let \(L/K\) be a finite Galois extension and \(\gothm\) a formal product of places of \(K\text{.}\) As in Definition 2.2.3, let \(J^{\gothm}_K\) be the group of fractional ideals of \(K\) coprime to \(\gothm\text{;}\) similarly, let \(J^{\gothm}_L\) be the group of fractional ideals of \(L\) coprime to \(\gothm\text{.}\)

Let \(I^{\gothm}_K\) be the subset of \(\alpha \in I_K\) such that:

for each finite prime \(\gothp\) of \(K\text{,}\) \(\alpha_v \equiv 1 \pmod{\gothp^e}\) where \(e\) is the exponent of \(\gothp\) in \(\gothm\text{;}\)
for each real place \(v\) in \(\gothm\text{,}\) \(\alpha_v > 0\text{.}\)

Define \(I^{\gothm}_L\) similarly.

Let \(P^{\gothm}_K\) be the subgroup of \(J^{\gothm}_K\) consisting of principal ideals admitting a generator \(\alpha \in K^* \cap I^{\gothm}_L\text{;}\) define \(P^{\gothm}_L\) similarly. In this notation,

\begin{equation*} \Cl^{\gothm}(K) = J^{\gothm}_K/P^{\gothm}_K, \qquad \Cl^{\gothm}(L) = J^{\gothm}_L / P^{\gothm}_L\text{.} \end{equation*}

The homomorphism \(I_K \to J_K\) from Definition 6.2.5 restricts to a homomorphism \(I_K^{\gothm} \to J_K^{\gothm}\text{,}\) which in turn induces a surjective homomorphism \(I_K^{\gothm}/(K^* \cap I_K^{\gothm}) \to \Cl^{\gothm}(K)\text{.}\) On the other hand, as indicated in Remark 6.2.6, we have \(K^* I_K^{\gothm} = I_K\) and hence \(I_K^{\gothm}/(K^* \cap I_K^{\gothm}) \cong C_K\text{,}\) yielding a surjection \(C_K \to \Cl^{\gothm}(K)\text{.}\)

Lemma 7.2.2.

With notation as in Definition 7.2.1, the composition

\begin{equation*} C_K / \Norm_{L/K} C_L \cong I_K^{\gothm}/(K^* \Norm_{L/K} I_L^{\gothm}) \to J^{\gothm}_K / P^{\gothm}_{K} \Norm_{L/K} J^{\gothm}_L \end{equation*}

is an isomorphism for some \(\gothm\text{.}\)

Proof.

The map in question is surjective because \(I_K^{\gothm} \to J_K^{\gothm}\) is; we thus need to check injectivity for suitable \(\gothm\text{.}\) Let \(S\) be the set of finite places of \(K\) which ramify in \(L\text{.}\) For each \(v \in S\text{,}\) apply local class field theory (see Theorem 4.1.5) to see that for \(w\) a place of \(L\) above \(v\text{,}\) the image of \(\Norm_{L_w/K_v} L_w^*\) is an open subgroup \(U_v\) of \(K_v^*\) of finite index. We may then choose \(\gothm\) to include every real place and each place in \(S\text{,}\) and to ensure that for each \(v \in S\text{,}\) \((I_K^{\gothm})_v \subseteq U_v\text{.}\)

We now prove the claim for such a choice of \(\gothm\text{.}\) Given an element of \(I_K^{\gothm}\) whose image in \(J^{\gothm}_K\) belongs to \(P^{\gothm}_{K} \Norm_{L/K} J^{\gothm}_L\text{,}\) we can factor it as an element of \(K^* \cap I_K^{\gothm}\) times an element of \(\Norm_{L/K} I_L^{\gothm}\) times an element \(\alpha \in I_K^{\gothm}\) such that for each finite place \(v\text{,}\) \(\alpha_v \in \gotho_{K_v}^*\text{.}\) We see that \(\alpha \in \Norm_{L/K} I_L^{\gothm}\) by looking separately at real places (which are okay because these places appear in \(\gothm\)), complex places (which are okay for trivial reasons), finite places in \(S\) (which are okay by our choice of \(\gothm\)), and finite places not in \(S\) (which are okay because these places are unramified in \(L\)).

With Lemma 7.2.2 in hand, we can reduce the Second Inequality to proving that

\begin{equation*} [J^{\gothm}_K : P^{\gothm}_{K} \Norm_{L/K} J^{\gothm}_L] \leq [L:K]\text{.} \end{equation*}

Subsection A special case of Chebotaryov density

We will need a special case of the Chebotaryov density theorem, which fortunately we can prove without already having all of class field theory. We use the notion of Dirichlet density for sets of prime ideals in a number field; see Definition 2.4.8 and the remainder of the discussion in Section 2.4.

Proposition 7.2.3.

Let \(L\) be a finite extension of \(K\) and let \(M/K\) be its Galois closure. Then the set \(S\) of prime ideals of \(K\) that split completely in \(L\) has Dirichlet density \(\frac{1}{[M:K]}\) (in the sense of Definition 2.4.8).

Proof.

A prime of \(K\) splits completely in \(L\) if and only if it splits completely in \(M\text{,}\) so we may assume \(L=M\) is Galois. Recall that the set \(T\) of unramified primes \(\gothq\) of \(L\) of absolute degree 1 has Dirichlet density 1 (see Exercise 1 and Exercise 2); each such prime lies over an unramified prime \(\gothp\) of \(K\) of absolute degree 1 which splits completely in \(L\text{.}\)

The set \(T\) having Dirichlet density 1 means that

\begin{equation*} \sum_{\gothq \in T} \frac{1}{\Norm(\gothq)^s} \sim \frac{1}{s-1} \qquad s \to 1^-\text{.} \end{equation*}

If we group the primes in \(T\) by which prime of \(S\) they lie over, then we get

\begin{equation*} [L:K] \sum_{\gothp \in S} \frac{1}{\Norm(\gothp)^s} \sim \frac{1}{s-1} \qquad s \to 1^-\text{.} \end{equation*}

That is, the Dirichlet density of \(S\) is \(\frac{1}{[L:K]}\text{.}\)

Example 7.2.4.

For \(L/\QQ\) a quadratic extension, Proposition 7.2.3 states that the set of prime ideals of \(\QQ\) that split completely in \(L\) has Dirichlet density \(\frac{1}{2}\text{.}\) As this splitting is governed by a congruence condition thanks to quadratic reciprocity, this assertion also follows from Dirichlet’s theorem on primes in arithmetic progressions.

Subsection The Second Inequality

We are now ready to prove the Second Inequality.

Lemma 7.2.5.

With notation as in Definition 7.2.1, we have

\begin{equation*} [J^{\gothm}_K : P^{\gothm}_K \Norm_{L/K} J^{\gothm}_{L}] \leq [L:K]\text{.} \end{equation*}

Proof.

Define the group

\begin{equation*} H := P^{\gothm}_K \Norm_{L/K} J^{\gothm}_L \subseteq J^{\gothm}_K\text{.} \end{equation*}

The group \(H\) includes every prime of \(K\) coprime to \(\gothm\) that splits completely in \(L\text{,}\) since such a prime is the norm of any prime of \(L\) lying over it; thus on one hand, by Proposition 7.2.3 the set of primes in \(H\) has Dirichlet density at least \(\frac{1}{[L:K]}\text{.}\) On the other hand, by Dirichlet’s theorem for number fields (Remark 2.4.9), the same set has density \(\frac{1}{[J^{\gothm}_K:H]}\text{.}\) We conclude that \([J^{\gothm}_K:H] \leq [L:K]\text{,}\) as desired.

Remark 7.2.6.

Recall that Remark 2.4.9 depends on the nonvanishing of \(L(1, \chi)\) when \(\chi\) is a nontrivial character of a generalized ideal class group (Theorem 2.4.7). Without this, we can still deduce Lemma 7.2.5: the discrete Fourier analysis shows that the Dirichlet density of the set of primes in \(H\) equals

\begin{equation*} \frac{1 - \sum_\chi m(\chi)}{[J^{\gothm}_K:H]} \end{equation*}

where \(\chi\) runs over all of the nontrivial characters of \(\Cl^{\gothm}(K)\) which are trivial on \(H\) and \(m(\chi)\) denotes the order of vanishing of \(L(s,\chi)\) at \(s=1\text{.}\) From the proof of Lemma 7.2.5, we must have \(1 - \sum_\chi m(\chi) \gt 0\text{,}\) whence \(m(\chi) = 0\) for all \(\chi\) which are nontrivial on \(H\text{.}\)

Turning this around, we can now deduce Theorem 2.4.7 provided that we can choose \(L/K\) (depending on \(\gothm\)) so that \(\Norm_{L/K} J^{\gothm}_L \subseteq P^{\gothm}_K\text{,}\) as then \(\chi\) runs over all nontrivial characters of \(\Cl^{\gothm}(K)\text{.}\) This will eventually follow from the adelic existence theorem (Theorem 6.4.4).

Corollary 7.2.7.

Let \(L/K\) be a Galois extension of number fields. Then

\begin{equation*} \# H^0_T(\Gal(L/K), C_L) \leq [L:K]\text{.} \end{equation*}

Proof.

This follows from Lemma 7.2.5 by translating back into the language of idèles using Lemma 7.2.2.

Remark 7.2.8.

We do not consider Corollary 7.2.7 to be a component of the Second Inequality because it is not needed in order to verify the class field axiom (and we will not reproduce it in the algebraic approach). In fact, once we complete the proofs of the reciprocity law (Theorem 6.4.3) and the norm limitation theorem (Theorem 6.4.5), Corollary 7.2.7 will also follow from those two statements together.

On the other hand, if one wants to avoid abstract class field theory, then it is helpful to have Corollary 7.2.7 in hand. See Remark 7.6.18.

Theorem 7.2.9. Second Inequality.

Let \(L/K\) be a Galois extension of number fields. Then:

the group \(H^1(\Gal(L/K), C_L)\) is trivial;
the group \(H^2(\Gal(L/K), C_L)\) is finite of order at most \([L:K]\text{.}\)

Proof.

For \(L/K\) cyclic, combining Corollary 7.2.7 with the periodicity of Tate groups (Theorem 3.4.1) shows that \(\#H^2(\Gal(L/K), C_L) \leq [L:K]\text{.}\) Combining with the First Inequality (Theorem 7.1.2) yields that \(H^1(\Gal(L/K), C_L)\) is trivial and \(\#H^2(\Gal(L/K), C_L) = [L:K]\text{.}\)

For \(L/K\) solvable, we may proceed by induction on \([L:K]\text{.}\) If \([L:K]\) is not cyclic, choose an intermediate Galois subextension \(K'/K\text{.}\) By the induction hypothesis, \(H^1(\Gal(L/K'), C_{L})\) vanishes, so we may apply the inflation-restriction exact sequence (Corollary 4.2.15) to see that for \(i=1,2\text{,}\)

\begin{equation*} 0 \to H^i(\Gal(K'/K), C_{K'}) \stackrel{\Inf}{\to} H^i(\Gal(L/K), C_L) \stackrel{\Res}{\to} H^i(\Gal(L/K'), C_L) \end{equation*}

is exact. This allows us to complete the induction.

For \(L/K\) general, put \(G := \Gal(L/K)\text{,}\) let \(p\) be a prime, and let \(G_p\) be a Sylow \(p\)-subgroup of \(G\text{.}\) Then for any \(i>0\text{,}\) \(H^i(G,C_L)\) is killed by the order of \(G\) and

\begin{equation*} \Res\colon H^i(G, C_L) \to H^i(G_p, C_L) \end{equation*}

is injective on \(p\)-primary components (both by the relationship between restriction and corestriction, from Example 3.2.24). Since \(G_p\) is solvable, we may deduce both assertions from the solvable case.

Subsection Aside: the Hasse norm theorem

We record a byproduct of the Second Inequality (not needed in what follows).

Theorem 7.2.10. Hasse norm theorem.

Let \(L/K\) be a cyclic extension of number fields. Then an element \(x \in K^*\) belongs to \(\Norm_{L/K} L^*\) if and only if for each (finite or infinite) place \(v\) of \(K\text{,}\) for some (and hence every) place \(w\) of \(L\) lying over \(K\text{,}\) \(x \in \Norm_{L_w/K_v} L_w^*\text{.}\)

Proof.

We may interpret the statement as saying that

\begin{equation*} H^0_T(G, L^*) \to H^0_T(G, I_L) \end{equation*}

is injective. (For all but finitely places \(v\) of \(K\text{,}\) we have \(x \in \gotho_{K_v}^*\) and so any \(y \in L_w^*\) with \(\Norm_{L_w/K_v}(y) = x\) itself belongs to \(\gotho_{L_w}^*\text{.}\)) We may deduce this injectivity from the fact that \(H^{-1}_T(G, C_L) = 1\) thanks to the Second Inequality (Theorem 7.2.9) plus periodicity (Theorem 3.4.1).

Remark 7.2.11.

The conclusion of Theorem 7.2.10 fails completely if \(L/K\) is abelian but not cyclic. See Exercise 1.

Remark 7.2.12.

Another related fact is the Grunwald-Wang theorem. It was originally announced (and published) in an incorrect form by Grunwald [16], who asserted that for \(K\) a number field and \(n\) a positive integer, an element \(x \in K^*\) is an \(n\)-th power if and only if it is an \(n\)-th power in \(K_v\) for all but finitely many places \(v\) of \(K\text{.}\)

It was then shown by Wang [56] that this statement fails in the following way: for any number field \(K\text{,}\) the element \(16\) is an \(8\)th power in \(K_v\) for any place \(v\) not lying above \(2\) (see Exercise 2) but need not be an \(8\)th power in \(K\text{.}\) Worse yet, for some choices of \(K\text{,}\) \(16\) is an \(8\)th power in \(K_v\) for every place \(v\) even though \(16\) is not an \(8\)th power in \(K\) (see Exercise 3).

Finally, Wang [57] established a corrected version of the theorem, which shows that the original statement is “nearly” true. For example, it holds as written whenever \(n\) is odd.

Subsection The Albert-Brauer-Hasse-Noether theorem

We record another byproduct of the Second Inequality. We will sharpen this result later; see Theorem 7.6.12.

Theorem 7.2.13. Albert-Brauer-Hasse-Noether theorem.

For any number field \(K\text{,}\) the map

\begin{equation*} H^2(\Gal(\overline{K}/K), \overline{K}^*) \to \bigoplus_v H^2(\Gal(\overline{K_v}/K_v), \overline{K_v}^*) \end{equation*}

is injective, where the sum runs over places \(v\) of \(K\text{.}\)

Proof.

By Theorem 7.2.9, for each Galois extension \(L/K\) of number fields, we have the exact sequence

\begin{equation*} 1 = H^1(\Gal(L/K), C_L) \to H^2(\Gal(L/K), L^*) \to \bigoplus_v H^2(\Gal(L_w/K_v), L_w^*) \end{equation*}

for any Galois extension \(L/K\text{,}\) where \(w\) denotes some place of \(L\) above \(K\text{.}\) Taking the union over \(K\) and using the injectivity of the inflation maps (Lemma 1.2.3 plus Proposition 4.2.13) yields the stated conclusion.

Exercises Exercises

1.

Show that the Hasse norm theorem (Theorem 7.2.10) fails for \(K = \QQ\text{,}\) \(L = \QQ(\sqrt{13}, \sqrt{17})\text{.}\) (This example is due to Serre and Tate.)

Hint.

Prove that every square in \(L\) is a local norm, but \(5^2\) is not a global norm.

2.

Show that in any field \(K\) of characteristic not equal to 2, \(16\) is an \(8\)th power in \(K\) if and only if one of \(-1, 2, -2\) is a square in \(K\text{.}\) Then deduce that for \(K\) a number field, \(16\) is an \(8\)th power in \(K_v\) for any place \(v\) not lying above \(2\text{,}\) even though it is not always an \(8\)th power in \(K\text{.}\)

3.

Put \(K = \QQ(\sqrt{7})\text{.}\) Show that \(16\) is an \(8\)th power in every completion of \(K\text{,}\) but not in \(K\) itself.

Hint.

Use Exercise 2 to deal with the plcaes not above 2. For the places above 2, check that \(\QQ_2(\sqrt{7}) = \QQ_2(\sqrt{-1})\text{.}\)

4.

Let \(K\) be a number field and choose \(a,b,c \in K^*\text{.}\) Prove that the equation \(ax^2 + by^2 + c^2 = 0\) has a solution with \(x,y,z \in K\) not all zero if and only if for each place \(v\) of \(K\text{,}\) there exists a solution with \(x,y,z \in K_v\) not all zero. (This is a special case of the Hasse-Minkowski theorem.)

Hint.

The equation has a solution in \(K\) if and only if \(-c\) is a norm from \(K(\sqrt{-b/a})\) to \(K\text{.}\)

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