Math 129: Algebraic Number Theory

Lecture 22: Properties of the Adeles

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

Thursday, April 29, 2004

The final project and the short take-home final will both be due on Monday, May 24, at NOON. You should turn your final projects in to me via email, since I will be posting all final projects on a web page. Do a good job on your final project, since a few years from now it will probably still be on that web page :-).

You should not worry too much about the final exam, which is worth only 20% of your grade. It will be take home, and you will be able to use all resources except communication with other people. Thus you can search in the library and online for ideas and answer. It will not be something you can really study for, and will probably take several hours.

What is a “final project”? How long should it be? This depends on you. Your goal for the final project should be to write something that you think other people in the course or people browsing the Math 129 web page would find an interesting and reasonably accessible read. Explore a topic, try to understand something interesting about it, and write it up so that somebody with an advanced undergraduate background in mathematics can understand it. A bad project would be one with nothing “interesting” in it that is riddled with errors. In contrast, a good project would be something that people with some interest in algebraic number theory would actually want to read, and is technically correct. Don’t aim so much to impress me with how much you were able to learn, but aim to produce a document that will add some variety to the Math 129 web page.

———

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 the lemma we proved last time. . 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}  (14.3)

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

Proof. We first establish an isomorphism of the two sides of (14.3) as topological spaces. Let ω_{1}, …, ω_{n} be a basis for L/K and let v run through the normalized valuations on K. The left hand side of (14.3), with the tensor product topology, is the restricted product of the tensor products

K_{v} ⊗_{K} L ≅ K_{v} · ω_{1} ⊕ ⋯ ⊕ K_{v} · ω_{n}

with respect to the integers

O_{v} · ω_{1} ⊕ ⋯ ⊕ O_{v} · ω_{n}.  (14.4)

(An element of the left hand side is a finite linear combination ∑ x_{i} ⊗ a_{i} of adeles x_{i} ∈ 𝔸_{K} and coefficients a_{i} ∈ L, and there is a natural isomorphism from the ring of such formal sums to the restricted product of the K_{v} ⊗_{K} L.)

We proved before that

K_{v} ⊗_{K} L ≅ L_{w_{1}} ⊕ ⋯ ⊕ L_{w_{g}},

where w_{1}, …, w_{g} are the normalizations of the extensions of v to L. Furthermore, as we proved using discriminants, the above identification identifies (14.4) with

O_{L_{w_{1}}} ⊕ ⋯ ⊕ O_{L_{wg}} ,

for almost all v. The left hand side of (14.3) is the restricted product of the L_{w_{1}}⊕⋯⊕L_{w_{g}} with respect to the O_{L_{w_{1}}} ⊕ ⋯ ⊕ O_{L_{wg}}. This is canonically isomorphic to the restricted product of all completions L_{w} with respect to O_{w}, which is the right hand side of (14.3). This establishes an isomorphism between the two sides of (14.3) as topological spaces. The map is also a ring homomorphism, so the two sides are algebraically isomorphic, as claimed.

Corollary 14.4. Let 𝔸_{K}^{+} denote the topological group obtained from the additive structure on 𝔸_{K}. Suppose L is a finite seperable extension of K. Then

𝔸_{L}^{+} = 𝔸_{K}^{+} ⊕ ⋯ ⊕ 𝔸_{K}^{+}, ([L : K] summands).

In this isomorphism the additive group L^{+} ⊂ 𝔸_{L}^{+} of the principal adeles is mapped into K^{+} ⊕ ⋯ ⊕ K^{+}.

Proof. For any nonzero ω ∈ L, the subgroup ω · 𝔸_{K}^{+} of 𝔸_{L}^{+} is isomorphic as a topological group to 𝔸_{K}^{+} (the isomorphism is multiplication by 1/ω). By Lemma 14.3, we have isomorphisms

𝔸_{L}^{+} = 𝔸_{K}^{+} ⊗_{K} L ≅ ω_{1} · 𝔸_{K}^{+} ⊕ ⋯ ⊕ ω_{n} · 𝔸_{K}^{+} ≅ 𝔸_{K}^{+} ⊕ ⋯ ⊕ 𝔸_{K}^{+}.

If a ∈ L, write a = ∑ b_{i}ω_{i}, with b_{i} ∈ K. Then a maps via the above map to

x = (ω_{1}(b_{1}), …, ω_{n}(b_{n})),

where (b_{i}) denotes the adele defined by b_{i}. Under the final map, x maps to the tuple

(b_{1}, …, b_{n}) ∈ K ⊕ ⋯ ⊕ K ⊂ 𝔸_{K}^{+} ⊕ ⋯ ⊕ 𝔸_{K}^{+}.

This proves the second claim of the corollary.

Theorem 14.5. The global field K is discrete in 𝔸_{K} and the quotient 𝔸_{K}^{+}/K^{+} of additive groups is compact in the quotient topology.

At this point Cassels remarks

“It is impossible to conceive of any other uniquely defined topology on K. This metamathematical reason is more persuasive than the argument that follows!”

Proof. Corollary 14.4, with K for L and Q or F(t) for K, shows that it is enough to verify the theorem for Q or F(t), and we shall do it here for Q.

To show that Q^{+} is discrete in 𝔸_{Q}^{+} it is enough, because of the group structure, to find an open set U that contains 0 ∈ 𝔸_{Q}^{+}, but which contains no other elements of Q^{+}. (If α ∈ Q^{+}, then U + α is an open subset of 𝔸_{Q}^{+} whose intersection with Q^{+} is {α}.) We take for U the set of x = {x_{v}} ∈ 𝔸_{Q}^{+} with

|x_{∞}|_{∞} < 1  and  |x_{p}|_{p} ≤ 1 (all p),

where | · |_{p} and | · |_{∞} are respectively the p-adic and the usual archimedean absolute values on Q. If b ∈ Q ∩ U, then in the first place b ∈ Z because |b|_{p} ≤ for all p, and then b = 0 because |b|_{∞} < 1. This proves that K^{+} is discrete in 𝔸_{Q}^{+}.

Next we prove that the quotient 𝔸_{Q}^{+}/Q^{+} is compact. Let W ⊂ 𝔸_{Q}^{+} consist of the x = {x_{v}} ∈ 𝔸_{Q}^{+} with

|x_{∞}|_{∞} ≤ \frac{1}{2}  and  |x_{p}|_{p} ≤ 1  for all primes p.

We show that every adele y = {y_{v}} is of the form

y = a + x, a ∈ Q, x ∈ W.

Fix an adele y = {y_{v}} ∈ 𝔸_{Q}^{+}. For each prime p we can find a rational number

r_{p} = \frac{z_{p}}{p^{n_{p}}} (z_{p} ∈ Z, n_{p} ∈ Z_{≥0})

such that

|y_{p} − r_{p}|_{p} ≤ 1,

and

r_{p} ≡ 0 (mod p) for almost all p,

since y is an adele. More precisely, for the finitely many p such that

y_{p} = ∑_{n≥−|s|} a_{n}p^{n} ∉ Z_{p},

choose r_{p} to be a rational number that is the value of an appropriate truncation of the p-adic expansion of y_{p}, and when y_{p} ∈ Z_{p} just choose r_{p} = 0. Hence r = ∑_{p} r_{p} ∈ Q is well defined and

|y_{p} − r|_{p} ≤ 1 for all p.

(The r_{q} for q ≠ p do not mess up the inequality |y_{p} − r|_{p} ≤ 1 since the valuation is non-archimedean and the r_{q} do not have any p in their denominator, by construction.) Now choose s ∈ Z such that

|b_{∞} − r − s| ≤ \frac{1}{2}.

Then a = r + s and x = y − a do what is required, since y − a = y − r − s has the desired property (since s ∈ Z and the p-adic valuations are non-archimedean).

Hence the continuous map W → 𝔸_{Q}^{+}/Q^{+} induced by the quotient map 𝔸_{Q}^{+} → 𝔸_{Q}^{+}/Q^{+} is surjective. But W is compact (being the topological product of the compact spaces |x_{∞}|_{∞} ≤ 1/2 and the Z_{p} for all p), hence 𝔸_{Q}^{+}/Q^{+} is also compact.

As already remarked, 𝔸_{K}^{+} is a locally compact group, so it has an invariant Haar measure. In fact one choice of this Haar measure is the product of the Haar measures on the K_{v}, in the sense described in the previous section.

Corollary 14.6. There is a subset W of 𝔸_{K} defined by inequalities of the type |x_{v}|_{v} ≤ δ_{v}, where δ_{v} = 1 for almost all v, such that every y ∈ 𝔸_{K} can be put in the form

y = a + x, a ∈ K, x ∈ W,

i.e., 𝔸_{K} = K + W.

Proof. We constructed such a set for K = Q when proving Theorem 14.5. For general K the W coming from the proof determines compenent-wise a subset of 𝔸_{K}^{+} ≅ 𝔸_{Q}^{+}⊕⋯⊕𝔸_{Q}^{+} that is a subset of a W with the properties claimed by the corollary.

Corollary 14.7. The quotient 𝔸_{K}^{+}/K^{+} has finite measure in the quotient measure induced by the Haar measure on 𝔸_{K}^{+}.

Remark 14.8. This statement is independent of the particular choice of the multiplicative constant in the Haar measure on 𝔸_{K}^{+}. We do not here go into the question of finding the measure 𝔸_{K}^{+}/K^{+} in terms of our explicitly given Haar measure. (See Tate’s thesis, Chapter XV of Cassels-Frohlich.)

Proof. This can be reduced similarly to the case of Q or F(t) which is immediate, e.g., the W defined above has measure 1 for our Haar measure.

Alternatively, finite measure follows from compactness. To see this, cover 𝔸_{K}/K^{+} with the translates of U, where U is a nonempty open set with finite measure. The existence of a finite subcover implies finite measure.

Remark 14.9. We give an alternative proof of the product formula ∏ |a|_{v} = 1 for nonzero a ∈ K. We have seen that if x_{v} ∈ K_{v}, then multiplication by x_{v} magnifies the Haar measure in K_{v}^{+} by a factor of |x_{v}|_{v}. Hence if x = {x_{v}} ∈ 𝔸_{K}, then multiplication by x magnifies the Haar measure in 𝔸_{K}^{+} by ∏ |x_{v}|_{v}. But now multiplication by a ∈ K takes K^{+} ⊂ 𝔸_{K}^{+} into K^{+}, so gives a well-defined bijection of 𝔸_{K}^{+}/K^{+} onto 𝔸_{K}^{+}/K^{+} which magnifies the measure by the factor ∏ |a|_{v}. Hence ∏ |a|_{v} = 1 Corollary 14.7. (The point is that if μ is the measure of 𝔸_{K}^{+}/K^{+}, then μ = ∏ |a|_{v} · μ, so because μ is finite we must have ∏ |a|_{v} = 1.)

Lemma 14.10. There is a constant C > 0 depending only on the global field K with the following property:

Whenever x = {x_{v}}_{v} ∈ 𝔸_{K} is such that

∏_{v} |x_{v}|_{v} > C,

then there is a nonzero principal adele a ∈ K ⊂ 𝔸_{K} such that

|a|_{v} ≤ |x_{v}|_{v} (all v).

Proof.

Corollary 14.11. Let v_{0} be a normalized valuation and let δ_{v} > 0 be given for all v ≠ v_{0} with δ_{v} = 1 for almost all v. Then there is a nonzero a ∈ K with

|a|_{v} ≤ δ_{v} (all v ≠ v_{0}).

Proof.

Next week we will prove the above two lemmas, then use them to deduce the strong approximation theorem, which is an extreme generalization of the Chinese Remainder Theorem; it asserts that K^{+} is dense (!) in the analogue of the adeles but with one places removed. Then we’ll introduce the ideles 𝔸_{K}^{∗}. Finally on Thursday, we’ll relate ideles to ideals, and use everything so far to give a new interpretation of class groups and their finiteness.