Notes on class field theory

The property of a field $K$ of characteristic 0 having trivial Brauer group is useful in the theory of finite group representations: for such a field, any $K$-valued character of a finite group arises from a representation defined over $K\text{.}$ (This follows from Schur’s lemma: the character in question appears within some irreducible $K$-linear representation, whose endomorphism ring is a division algebra; the triviality of the Brauer group forces this to split without any base extension.)

By contrast, for $G=\{\pm 1,\pm i,\pm j,\pm k\}$ the unit quaternion group, the standard 2-dimensional representation of $G$ has a $\mathbb{Q}$-valued character but cannot be realized as a representation over $\mathbb{Q}\text{.}$ In other words, this representation has nontrivial Schur index.

One can also associate Brauer groups to arbitrary rings and even to schemes in algebraic geometry, by extending the definition of Azumaya algebras suitably. One again has a relationship with Galois cohomology (or rather étale cohomology), but there is a bit more subtlety involved. See [15] for a classical reference or [42], section 6.6 for a modern treatment.

By Proposition 7.6.14, the field ${\mathbb{Q}}^{\mathrm{ab}}$ has trivial Brauer group. Since in addition every complex character of a finite group has values in ${\mathbb{Q}}^{\mathrm{ab}}\text{,}$ it follows that every irreducible complex representation of a finite group can be realized over ${\mathbb{Q}}^{\mathrm{ab}}\text{;}$ for a more direct proof of this, see [47], Chapter 12, Theorem 24.

Let $L/K$ be a cyclic extension of number fields. At this point, $r_{L/K}$ kills both principal idèles (by Proposition 6.4.7) and norms (since it does so locally), so it induces a map $C_K/{\mathrm{Norm}}_{L/K}{C}_L\to \mathrm{Gal}(L/K)\text{.}$ By the surjectivity of the Artin map, as deduced from the First Inequality (Proposition 7.3.7), this map is surjective; by comparing orders using the Second Inequality (Theorem 7.2.9), we see that the map is also an isomorphism. This establishes local-global compatibility (Proposition 6.4.7) for cyclic extensions, from which it directly follows also for abelian extensions (using the structure theorem for finite abelian groups).

For general $L/K\text{,}$ $r_{L/K}:I_K\to \mathrm{Gal}(L/K{)}^{\mathrm{ab}}$ agrees with $r_{M/K}:I_K\to \mathrm{Gal}(M/K)$ when $M/K$ is the maximal abelian subextension of $L/K\text{.}$ In particular, $r_{L/K}$ once again vanishes on $K^{\ast }\text{.}$ Hooray again!

One can extend this conclusion to the case where $L/K$ is abelian as follows. In this case, local reciprocity (Theorem 4.1.2) and Remark 7.6.17 together imply that we have a well-defined map. Using the cyclic case, we may see that this map is surjective; by Corollary 7.2.7 (a side effect of our proof of the Second Inequality), the map is forced to be an isomorphism.

For a general extension $L/K$ with maximal abelian subextension $M/K\text{,}$ we have from Remark 7.6.17 a well-defined map $r_{L/K}:C_K\to \mathrm{Gal}(L/K{)}^{\mathrm{ab}}$ which coincides with $r_{M/K}:C_K\to \mathrm{Gal}(M/K)\text{.}$ However, now we must be careful because the completion of $L$ at some place might have maximal abelian subextension strictly larger than the corresponding completion of $M\text{;}$ consequently, we cannot directly apply local norm limitation to deduce global norm limitation. In particular, we have an induced map $r_{L/K}:C_K/{\mathrm{Norm}}_{L/K}{C}_L\to \mathrm{Gal}(L/K{)}^{\mathrm{ab}}$ but we currently only know that it is surjective. To resolve this, see Proposition 7.6.19 and Remark 7.6.20.

Returning to Remark 7.6.18, we see that the map $r_{L/K}:C_K/{\mathrm{Norm}}_{L/K}{C}_L\to \mathrm{Gal}(L/K{)}^{\mathrm{ab}}$ that we constructed from global reciprocity is a surjective map between two groups which are now known to be of the same order! We conclude that this map is an isomorphism, thus confirming Theorem 6.4.3 and Theorem 6.4.5. (Note that we avoided having to check that the isomorphism in the previous paragraph is compatible with the isomorphism coming from local-global compatibility.)

At this point, the reader who is avoiding abstract class field theory can now return to Section 7.4 for the proof of the existence theorem: the argument given therein (Theorem 7.4.8) depends only on the formulation of global reciprocity, not any particular method of proof.

Let $X$ be an algebraic variety over a number field $K\text{.}$ For each place $v$ of $K\text{,}$ one can ask whether the set of $K_v$-rational points $X(K_v)$ is empty. If this holds for some $v\text{,}$ then evidently $X(K)$ is also empty. However, the converse is generally not true; that is, there is no local-to-global principle for the existence of rational points on a general algebraic variety.

It was first observed by Manin that this failure can sometimes be overcome using the Brauer group $\mathrm{Br}(X)$ (Remark 7.6.7). The key relevant point for now is that the construction of $\mathrm{Br}(X)$ is contravariantly functorial, so any $K$-rational point $x\in X(K)$ gives rise to a homomorphism $\mathrm{Br}(X)\to \mathrm{Br}(K)\text{.}$ We can view this construction as a “pairing” $X(K)\times \mathrm{Br}(X)\to \mathrm{Br}(K)\text{,}$ although it is only linear in the second argument as there is no natural group structure on $X(K)$ in general.

For each place $v$ of $K\text{,}$ we can form the base extension $X_v:=X{\times }_K{K}_v\text{,}$ so that $X_v(K_v)=X(K_v)\text{.}$ In particular, for each $x\in X(K)\text{,}$ we get a corresponding point $x_v\in K_v\text{.}$ Meanwhile, for each class $y\in \mathrm{Br}(X)\text{,}$ and define $y_v\in \mathrm{Br}(X_v)$ as the image of $y$ under the functoriality map $\mathrm{Br}(X_v)\to \mathrm{Br}(X)\text{.}$ We again get a local pairing $X(K_v)\times \mathrm{Br}(X_v)\to \mathrm{Br}(K_v)\text{.}$ Note that the map $X(K_v)\to \mathrm{Br}(K_v)$ induced by pairing with a given $x$ is locally constant for the $v$-adic topology on $X(K_v)$ ([42], Proposition 8.2.9).

Now let ${\mathrm{inv}}_v:\mathrm{Br}(K_v)\to \mathbb{Q}/\mathbb{Z}$ be the local invariant map at the place $v\text{.}$ The key point is now to apply Theorem 7.6.12: given a class $y\in \mathrm{Br}(X)\text{,}$ a necessary condition for a tuple $(x_v{)}_v\in \prod _{v}X(K_v)$ to come from a global point $x\in X(K)$ is that the sum of the local invariants ${\mathrm{inv}}_v(x_v,y_n)$ must vanish. That is, each $y$ gives rise to a commutative diagram as in Figure 7.6.22 in which the vertical maps are evaluation maps and the bottom row is the fundamental exact sequence.

Applying this condition for all $y\in \mathrm{Br}(X)$ picks out a subset of $\prod _{v}X(K_v)\text{,}$ the Brauer set, which must contain all of $X(K)\text{;}$ when this happens, $X$ is said to have a Brauer-Manin obstruction forcing $X(K)=\mathrm{\varnothing }\text{.}$ More commonly, one says that $X$ admits a Brauer-Manin obstruction to the local-global principle if each $X(K_v)$ to be nonempty but the Brauer set is empty. See [42], Chapter 8 for some examples where this occurs.