Math 129: Algebraic Number Theory

Lecture 21: Global Fields and Adeles

William Stein (based closely on Cassels’s Global Fields article in Cassels-Fröhlich)

Tuesday, April 27, 2004

12 Global Fields

Definition 12.1 (Global Field). A global field is a number field or a finite separable extension of F(t), where F is a finite field, and t is transcendental over F.

Below we will focus attention on number fields leaving the function field case to the reader.

The following lemma essentially says that the denominator of an element of a global field is only “nontrivial” at a finite number of valuations.

Lemma 12.2. Let a ∈ K be a nonzero element of a global field K. Then there are only finitely many inequivalent valuations | · | of K for which

|a| > 1.

Proof. If K = Q or F(t) then the lemma follows by Ostrowski’s classification of all the valuations on K. For example, when a = \frac{n}{d} ∈ Q, with n, d ∈ Z, then the valuations where we could have |a| > 1 are the archimedean one, or the p-adic valuations | · |_{p} for which p | d.

Suppose now that K is a finite extension of Q, so a satisfies a monic polynomial

a^{n} + c_{n−1}a^{n−1} + ⋯ + c_{0} = 0,

for some n and c_{0}, …, c_{n−1} ∈ Q. If | · | is a non-archimedean valuation on K, we have

|a|^{n} = |−(c_{n−1}a^{n−1} + ⋯ + c_{0})|
≤ max(1, |a|^{n−1}) · max(|c_{0}| , …, |c_{n−1}|).

Dividing each side by |a|^{n−1}, we have that

|a| ≤ max(|c_{0}| , …, |c_{n−1}|),

so in all cases we have

|a| ≤ max(1, |c_{0}| , …, |c_{n−1}|)^{1/(n−1)}.  (12.1)

We know the lemma for Q, so there are only finitely many valuations | · | on Q such that the right hand side of (12.1) is bigger than 1. Since each valuation of Q has finitely many extensions to K, and there are only finitely many archimedean valuations, it follows that there are only finitely many valuations on K such that |a| > 1.

Any valuation on a global field is either archimedean, or discrete non-archimedean with finite residue class field, since this is true of Q and F(t) and is a property preserved by extending a valuation to a finite extension of the base field. Hence it makes sense to talk of normalized valuations. Recall that the normalized p-adic valuation on Q is |x|_{p} = p^{− ord_{p}(x)}, and if v is a valuation on a number field K equivalent to an extension of | · |_{p}, then the normalization of v is the composite of the sequence of maps

K ↪ K_{v} ⟶^{Norm} Q_{p} ⟶^{| · |_{p}} R,

where K_{v} is the completion of K at v.

Example 12.3. Let K = Q(\sqrt{2}), and let p = 2. Because \sqrt{2} ∉ Q_{2}, there is exactly one extension of | · |_{2} to K, and it sends a = 1/\sqrt{2} to

|Norm_{Q_{2}(\sqrt{2})/Q_{2}}(1/\sqrt{2})|_{2}^{1/2} = \sqrt{2}.

Thus the normalized valuation of a is 2.

There are two extensions of | · |_{7} to Q(\sqrt{2}), since Q(\sqrt{2}) ⊗_{Q} Q_{7} ≅ Q_{7} ⊕ Q_{7}, as x^{2} − 2 = (x − 3)(x − 4) (mod 7). The image of \sqrt{2} under each embedding into Q_{7} is a unit in Z_{7}, so the normalized valuation of a = 1/\sqrt{2} is, in both cases, equal to 1. More generally, for any valuation of K of characteristic an odd prime p, the normalized valuation of a is 1.

Since K = Q(\sqrt{2}) ↪ R in two ways, there are exactly two normalized archimedean valuations on K, and both of their values on a equal 1/\sqrt{2}. Notice that the product of the absolute values of a with respect to all normalized valuations is

2 · \frac{1}{\sqrt{2}} · \frac{1}{\sqrt{2}} · 1 · 1 · 1 ⋯ = 1.

This “product formula” holds in much more generality, as we will now see.

Theorem 12.4 (Product Formula). Let a ∈ K be a nonzero element of a global field K. Let | · |_{v} run through the normalized valuations of K. Then |a|_{v} = 1 for almost all v, and

∏_{all v} |a|_{v} = 1 (the product formula).

Proof. By Lemma 12.2, we have |a|_{v} ≤ 1 for almost all v. Likewise, 1/ |a|_{v} = |1/a|_{v} ≤ 1 for almost all v, so |a|_{v} = 1 for almost all v.

Let w run through all normalized valuations of Q (or of F(t)), and write v | w if the restriction of v to Q is equivalent to w. Then (by the previous section),

∏_{v} |a|_{v} = ∏_{w} (∏_{v|w} |a|_{v}) = ∏_{w} |Norm_{K/Q}(a)|_{w} ,

so it suffices to prove the theorem for K = Q.

By multiplicativity of valuations, if the theorem is true for b and c then it is true for the product bc and quotient b/c (when c ≠ 0). The theorem is clearly true for −1, which has valuation 1 at all valuations. Thus to prove the theorem for Q it suffices to prove it when a = p is a prime number. Then we have |p|_{∞} = p, |p|_{p} = 1/p, and for primes q ≠ p that |p|_{q} = 1. Thus

∏_{v} |p|_{v} = p · \frac{1}{p} · 1 · 1 · 1 ⋯ = 1,

as claimed.

If v is a valuation on a field K, recall that we let K_{v} denote the completion of K with respect to v. Also when v is non-archimedean, let

O_{v} = O_{K,v} = {x ∈ K_{v} : |x| ≤ 1}

be the ring of integers of the completion.

Definition 12.5 (Almost All). We say a condition holds for almost all elements of a set if it holds for all but finitely many elements.

We will use the following lemma later to prove that formation of the adeles of a global field is compatible with base change.

Lemma 12.6. Let ω_{1}, …, ω_{n} be a basis for L/K, where L is a finite separable extension of the global field K of degree n. Then for almost all normalized non-archimedean valuations v on K we have

ω_{1}O_{v} ⊕ ⋯ ⊕ ω_{n}O_{v} = O_{w_{1}} ⊕ ⋯ ⊕ O_{w_{g}} ⊂ K_{v} ⊗_{K} L,  (12.2)

where w_{1}, …, w_{g} are the extensions of v to L. Here we have identified a ∈ L with its canonical image in K_{v} ⊗_{K} L, and the direct sum on the left is the sum taken inside the tensor product (so directness means that the intersections are trivial).

Proof. The proof proceeds in two steps. First we deduce easily from Lemma 12.2 that for almost all v the left hand side of (12.2) is contained in the right hand side. Then we use a trick involving discriminants to show the opposite inclusion for all but finitely many primes.

Since O_{v} ⊂ O_{w_{i}} for all i, the left hand side of (12.2) is contained in the right hand side if |ω_{i}|_{w_{j}} ≤ 1 for 1 ≤ i ≤ n and 1 ≤ j ≤ g. Thus by Lemma 12.2, for all but finitely many v the left hand side of (12.2) is contained in the right hand side. We have just eliminated the finitely many primes corresponding to “denominators” of some ω_{i}, and now only consider v such that ω_{1}, …, ω_{n} ∈ O_{w} for all w | v.

For any elements a_{1}, …, a_{n} ∈ K_{v} ⊗_{K} L, consider the discriminant

D(a_{1}, …, a_{n}) = Det(Tr(a_{i}a_{j})) ∈ K_{v},

where the trace is induced from the L/K trace. Since each ω_{i} is in each O_{w}, for w | v, the traces lie in O_{v}, so

d = D(ω_{1}, …, ω_{n}) ∈ O_{v}.

Also note that d ∈ K since each ω_{i} is in L. Now suppose that

α = ∑_{i=1}^{n} a_{i}ω_{i} ∈ O_{w_{1}} ⊕ ⋯ ⊕ O_{w_{g}},

with a_{i} ∈ K_{v}. Then by properties of determinants for any m with 1 ≤ m ≤ n, we have

D(ω_{1}, …, ω_{m−1}, α, ω_{m+1}, …, ω_{n}) = a_{m}^{2}D(ω_{1}, …, ω_{n}).  (12.3)

The left hand side of (12.3) is in O_{v}, so the right hand side is well, i.e.,

a_{m}^{2} · d ∈ O_{v}, (for m = 1, …, n),

where d ∈ K. Since ω_{1}, …, ω_{n} are a basis for L over K and the trace pairing is nondegenerate, we have d ≠ 0, so by Theorem 12.4 we have |d|_{v} = 1 for all but finitely many v. Then for all but finitely many v we have that a_{m}^{2} ∈ O_{v}. For these v, that a_{m}^{2} ∈ O_{v} implies a_{m} ∈ O_{v} since a_{m} ∈ K_{v}, i.e., α is in the left hand side of (12.2).

Example 12.7. Let K = Q and L = Q(\sqrt{2}). Let ω_{1} = 1/3 and ω_{2} = 2\sqrt{2}. In the first stage of the above proof we would eliminate | · |_{3} because ω_{2} is not integral at 3. The discriminant is

d = D (\frac{1}{3}, 2\sqrt{2}) = Det (\begin{matrix}\frac{2}{9} & 0 \\ 0 & 16 \end{matrix}) = \frac{32}{9}.

As explained in the second part of the proof, as long as v ≠ 2, 3, we have equality of the left and right hand sides in (12.2).

13 Restricted Topological Products

In this section we describe a topological tool, which we need in order to define adeles (see Definition 14.1).

Definition 13.1 (Restricted Topological Products). Let X_{λ}, for λ ∈ Λ, be a family of topological spaces, and for almost all λ let Y_{λ} ⊂ X_{λ} be an open subset of X_{λ}. Consider the space X whose elements are sequences x = {x_{λ}}_{λ∈Λ}, where x_{λ} ∈ X_{λ} for every λ, and x_{λ} ∈ Y_{λ} for almost all λ. We give X a topology by taking as a basis of open sets the sets ∏ U_{λ}, where U_{λ} ⊂ X_{λ} is open for all λ, and U_{λ} = Y_{λ} for almost all λ. We call X with this topology the restricted topological product of the X_{λ} with respect to the Y_{λ}.

Corollary 13.2. Let S be a finite subset of Λ, and let X_{S} be the set of x ∈ X with x_{λ} ∈ Y_{λ} for all λ ∉ S, i.e.,

X_{S} = ∏_{λ∈S} X_{λ} × ∏_{λ∉S} Y_{λ} ⊂ X.

Then X_{S} is an open subset of X, and the topology induced on X_{S} as a subset of X is the same as the product topology.

The restricted topological product depends on the totality of the Y_{λ}, but not on the individual Y_{λ}:

Lemma 13.3. Let Y′_{λ} ⊂ X_{λ} be open subsets, and suppose that Y_{λ} = Y′_{λ} for almost all λ. Then the restricted topological product of the X_{λ} with respect to the Y′_{λ} is canonically isomorphic to the restricted topological product with respect to the Y_{λ}.

Lemma 13.4. Suppose that the X_{λ} are locally compact and that the Y_{λ} are compact. Then the restricted topological product X of the X_{λ} is locally compact.

Proof. For any finite subset S of Λ, the open subset X_{S} ⊂ X is locally compact, because by Lemma 13.2 it is a product of finitely many locally compact sets with an infinite product of compact sets. (Here we are using Tychonoff’s theorem from topology, which asserts that an arbitrary product of compact topological spaces is compact (see Munkres’s Topology, a first course, chapter 5).) Since X = ∪_{S}X_{S}, and the X_{S} are open in X, the result follows.

Definition 13.5 (Product Measure). For all λ ∈ Λ, suppose μ_{λ} is a measure on X_{λ} with μ_{λ}(Y_{λ}) = 1 when Y_{λ} is defined. We define the product measure μ on X to be that for which a basis of measurable sets is

∏_{λ} M_{λ}

where each M_{λ} ⊂ X_{λ} has finite μ_{λ}-measure and M_{λ} = Y_{λ} for almost all λ, and where

μ (∏_{λ} M_{λ}) = ∏_{λ} μ_{λ}(M_{λ}).

14 The Adele Ring

Let K be a global field. For each normalization | · |_{v} of K, let K_{v} denote the completion of K. If | · |_{v} is non-archimedean, let O_{v} denote the ring of integers of K_{v}.

Definition 14.1 (Adele Ring). The adele ring 𝔸_{K} of K is the topological ring whose underlying topological space is the restricted topological product of the K_{v} with respect to the O_{v}, and where addition and multiplication are defined componentwise:

(xy)_{v} = x_{v}y_{v}  (x + y)_{v} = x_{v} + y_{v}  for x, y ∈ 𝔸_{K}.  (14.1)

It is readily verified that (i) this definition makes sense, i.e., if x, y ∈ 𝔸_{K}, then xy and x + y, whose components are given by (14.1), are also in 𝔸_{K}, and (ii) that addition and multiplication are continuous in the 𝔸_{K}-topology, so 𝔸_{K} is a topological ring, as asserted. Also, 𝔸_{K} is locally compact because the K_{v} are locally compact, and the O_{v} are compact.

There is a natural continuous ring inclusion

K ↪ 𝔸_{K}  (14.2)

that sends x ∈ K to the adele every one of whose components is x. This is an adele because x ∈ O_{v} for almost all v, by Lemma 12.2. The map is injective because each map K → K_{v} is an inclusion.

Definition 14.2 (Principal Adeles). The image of (14.2) is the ring of principal adeles.

It will cause no trouble to identify K with the principal adeles, so we shall speak of K as a subring of 𝔸_{K}.

Formation of the adeles is compatibility with base change, in the following sense.

Lemma 14.3. Suppose L is a finite (separable) extension of the global field K. Then

𝔸_{K} ⊗_{K} L ≅ 𝔸_{L}

both algebraically and topologically. Under this isomorphism, L ≅ K ⊗_{K} L ⊂ 𝔸_{K} ⊗_{K} L maps isomorphically onto L ⊂ 𝔸_{L}.

*  *  *

Next time we will start by proving the above lemma. Here are some further highlights of what will come in the final three lectures.

Theorem 14.4. The global field K is discrete in 𝔸_{K} and the quotient 𝔸_{K}^{+}/K^{+} of additive groups is compact in the quotient topology.

If we remove even one valuation, then the situation changes dramatically:

Theorem 14.5 (Strong Approximation). Let v_{0} be any valuation of the global field K. Define 𝔸′_{K} to be the restricted topological product of the K_{v} with respect to the O_{v}, where v runs through all normalized valuations v ≠ v_{0}. Then K is dense in 𝔸′_{K}.

Definition 14.6 (Idele Group). The idele group 𝕀_{K} of K is the group 𝔸_{K}^{∗} of invertible elements of the adele ring 𝔸_{K}.

The subgroup 𝕀_{K}^{1} is the subgroup of ideles x = {x_{v}} such that ∏_{v} |x_{v}|_{v} = 1. Note

that K^{∗} ⊂ 𝕀_{K}^{1}.

Theorem 14.7. The quotient 𝕀_{K}^{1}/K^{∗} with the quotient topology is compact.

Theorem 14.8. The ideal class group of K (with the discrete topology) is canonically isomorphic to 𝕀_{K}^{1}/K^{∗}.

Since a discrete compact group is finite, this proves that the ideal class group of a global field is finite.

I think we will likely stop here, and not do sections 18 and 19. This will be a nice conclusion, because you’ll finish the class having learned the basic theorems and objects of algebraic number theory, both from the classical and adelic points of view, and will have seen a nontrivial result proved from both directions. You have also learned something about the structure of local fields (completions of global fields).