20 1. ALGEBRAIC THEORY OF COMPLEX MULTIPLICATION
18.104.22.168. Theorem. Let L be a global field. There is an exact sequence
where the direct sum is taken over all places of L and the first map is defined via
extension of scalars.
Proof. This is [120, §9.7, §11.2].
For a global field L and central division algebra Δ over L, invv(Δ) denotes
invLv (ΔLv ). Theorem 22.214.171.124 says that a central division algebra Δ over a global
field L is uniquely determined up to isomorphism by its invariants invv(Δ), and
that these may be arbitrarily assigned subject to the conditions invv(Δ) = 0 for all
but finitely many v and
invv(Δ) = 0. Moreover, the order of [Δ] in Br(L) is the
least common “denominator” of the local invariants invv(Δ) ∈ Q/Z.
If K is any field then for a class c ∈ Br(K) its period is its order and its index is
[Δ : K] with Δ the unique central division algebra over K representing the class
c. It is a classical fact that the period divides that index and that these integers
have the same prime factors (see [107, X.5], especially Lemma 1 and Exercise 3),
but in general equality does not hold. For example, there are function fields of
complex 3-folds for which some order-2 elements in the Brauer group cannot be
represented by a quaternion algebra; examples are given in [61, §4], and there are
examples with less interesting fields as first discovered by Brauer. We have noted
above that over local fields there is equality of period and index (the archimedean
case being trivial). The following deep result is an analogue over global fields.
126.96.36.199. Theorem. For a central division algebra Δ over a global field L, the order
of [Δ] in Br(L) is [Δ : L].
As a special (and very important) case, elements of order 2 in Br(L) are pre-
cisely the Brauer classes of quaternion division algebras for a global field L; as noted
above, this fails for more general fields. Since Theorem 188.8.131.52 does not seem to
be explicitly stated in any of the standard modern references on class field theory
(though there is an allusion to it at the end of [4, Ch. X, §2]), and the structure
theory of endomorphism algebras of abelian varieties rests on it, here is a proof.
Proof. Let Δ have degree n2 over L and let d be the order of [Δ] in Br(L), so d|n.
Note that d is the least common multiple of the local orders dv of [ΔLv ] ∈ Br(Lv)
for each place v of L, with dv = 1 for complex v, dv|2 for real v, and dv = 1 for all
but finitely many v. Using these formal properties of the dv’s, we may call upon
the full power of global class field theory via Theorem 6 in [4, Ch. X] to infer the
existence of a cyclic extension L /L of degree d such that [Lv : Lv] is a multiple of
dv for every place v of L (here, v is any place on L over v, and the constraint on the
local degree is only non-trivial when dv 1). In the special case d = 2 (the only
case we will require) one only needs weak approximation and Krasner’s Lemma
rather than class field theory: take L to split a separable quadratic polynomial
over L that closely approximates ones that define quadratic separable extensions of
Lv for each v such that dv = 2.
By (184.108.40.206), restriction maps on local Brauer groups induce multiplication by
the local degree on the local invariants, so ΔL is locally split at all places of L .
Thus, by the injectivity of the map from the global Brauer group into the direct