Local Compactness and Number Theory, Summaries of Number Theory

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Local Compactness and Number Theory

Sam Mundy

May 1, 2015

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Introduction

The present document is a slightly expanded version of a series of lectures I gave at the University of New Mexico in the spring of 2015. The series has a title: “Local Compactness and Number Theory,” the same as this document. The main idea is to develop number theory from a topological point of view, much like in Weil’s Basic Number Theory [11]. However, the main goal of these lectures is more broad, in that the point here is to moti- vate material and to give an overview of relevant topics, rather than to give all proofs in detail. If I were to do the latter, I may as well just lecture from Weil. The first lecture is on the main apparatus which will be used in most the rest of the lectures, namely the locally compact abelian groups. Such groups come equipped with a natural measure called the Haar measure. This will allow a nice integration theory on such groups, and eventually, a Fourier analysis. But the Fourier theory will come near the end of the lectures. Afterwards, we define the local fields, which will be locally compact by definition. The analysis of these fields will be very topological. We will classify them without proof. Next we will define the global fields and look at them algebraically. (The more enlight- ened readers will recognize our analysis also as geometric at its core). We describe a process which constructs any local field out of some global field. Now the global fields will not be a gluing together of local fields in any obvious way; instead, the gluing together of local fields results in another locally compact group (actually a ring), the adeles. The geometric structure of the adeles (or more accurately, their units, also locally compact) gives rise to rich information about the global fields. Next, we describe class field theory and the theory of curves over finite fields in order to give a taste of what is beyond the basic theory we develop. In fact we go as far as to give a description of the Weil Conjectures for varieties over finite fields. While this subject deviates far from the main focus of the material, it raises an important point: We will not have even touched the theory of zeta functions. This is remedied immediately as we give examples of classical zeta functions in number theory. Here we will touch upon class field theory again. But then we will want to generalize our zeta functions. To do this, we need the aforementioned Fourier analysis. This will lead to a detailed description of Tate’s thesis, where the zeta functions we know, and many more, are con- structed from local analogues, like the adeles are constructed from local fields. But the general theory will be very lucrative: Tate’s thesis will allow us to obtain important ana- lytic information about the zeta functions which can be translated into valuable arithmetic data. When I presented these lectures, I skipped Lectures 6 and 7 in this document because the audience had, for the most part, seen this material in other courses. Since I had fifteen lectures planned, and there are thirteen in this document, this left an extra four lectures at the end of the course. I filled them each with brief overviews of the following topics, in order: Modular forms, Eichler-Shimura theory in weight 2, automorphic representations and the adelization of modular forms, and Langlands functoriality.

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1. and is only used when p is prime. The finite field with q elements is denoted Fq.

Contents

• 1 Locally Compact Abelian Groups
• 2 Local Fields
• 3 Global Fields
• 4 Adeles and Ideles
• 5 Class Field Theory
• 6 Algebraic Varieties
• 7 Function Fields and Curves
• 8 The Weil Conjectures
• 9 Zeta Functions and L-Functions
• 10 Abstract Fourier Analysis
• 11 Tate’s Thesis: Local Zeta Functions
• 12 Tate’s Thesis: Analysis on Adeles and Ideles
• 13 Tate’s Thesis: Global Zeta Functions

LECTURE 1. LOCALLY COMPACT ABELIAN GROUPS 2

consequence of this definition is that the empty set always has measure zero. Now if X is a topological space, we can consider the smallest σ-algebra B which contains the open sets (and hence also the closed sets) of X. The elements of this σ-algebra are called the Borel sets of X. We call a measure μ a Borel measure if it is defined on all Borel sets. Finally, we call a regular Borel measure μ on a topological space X a Radon measure if it Borel and we have, for all E in the σ-algebra on which it is defined, that μ(E) is the supremum over all compacts K in E of the real numbers μ(K) and, from the other perspective, that μ(E) is the infimum over all open sets U containing E, of μ(U ). These conditions are referred to as inner and outer regularity, respectively. Because of inner regularity, Radon measures are determined by their values on compact sets. Now given any topological group G and any subset S ⊂ G and any element g ∈ G, we denote by gS the set {gh | h ∈ S}. We are now ready to state the fundamental result.

Theorem 1.1. Let G be a locally compact group. Then there is a nonzero Radon measure μ on G which is translation invariant, i.e., for all g ∈ G and all measurable E, we have

μ(gE) = μ(E).

Such a measure is called a Haar Measure on G. Furthermore, this measure is as unique as possible, in the following sense: If ν is another Haar measure on G, then there is a c > 0 such μ = cν.

One comment is in order here: We have technically only defined the Haar measure to be left translation invariant. But one can show that this implies right translation invariance. Since we will only care about the Haar measure on locally compact abelian groups, this is immaterial to us. So what is the role of local compactness in all of this? Recall the Riesz Representation Theorem, which says that given a locally compact space X and any continuous linear functional L on the compactly supported continuous functions Cc(X) on X, there is a unique Radon measure μ for which

Lf =

f (x) dμ(x)

for any f ∈ Cc(X). Thus, on a locally compact space X, measures are the same as contin- uous linear functionals on Cc(X). See Rudin [8] for the proof. Here the local compactness is essential, and the way one constructs the Haar measure on a locally compact group G is by constructing a functional H on Cc(G) such that H ◦ Lg = H for all g ∈ G, where Lg : Cc(G) → Cc(G) is the linear operator defined by the formula (Lgf )(h) = f (gh). Before we give examples, we first establish some general facts about the measures of open and compact sets in a locally compact group.

Proposition 1.2. Let G be a locally compact group with Haar measure μ. Then every open set has positive (or infinite) measure and every compact set has finite measure.

Proof. Let E be a set in G with finite positive measure under μ, which exists by the nontriviality of the Haar measure. Then by inner regularity, for any  > 0, there is a compact set K in E with μ(E) − μ(K) < . In particular, we can choose K to have positive

LECTURE 1. LOCALLY COMPACT ABELIAN GROUPS 3

measure. Now let U be an open set and x ∈ U. Then {ax−^1 U | a ∈ K} is an open cover of K, and hence there is a finite subcover, say {a 1 x−^1 U,... , anx−^1 U }, a 1 ,... , an ∈ K. Then 0 < μ(K) ≤ μ(a 1 x−^1 U ) + · · · + μ(anx−^1 U ) = nμ(U ). Thus U has positive measure. On the other hand, using outer regularity, we can get an open set U ′^ containing E with finite positive measure. Let K′^ be any compact set in G. Then like above, we can get a finite cover of K′^ by translates of U ′. Since these translates all have finite measure, so does their union, and hence so does K′. This completes the proof.

Example 1.3. First we consider G = R, under addition, of course. R, with its usual (euclidean) topology is locally compact, so we have a Haar measure. It is easy to see that it must be (a multiple of) the Lebesgue measure, since this is obviously translation invariant.

Example 1.4. We can consider more generally G = Rn. Again this has the Lebesgue measure as a Haar measure. In fact, there is the following theorem.

Theorem 1.5. Let G 1 ,... , Gn be locally compact groups with respective Haar measures μ 1 ,... , μn. Then G 1 × · · · × Gn is a locally compact group with the product topology and Haar measure μ 1 × · · · × μn. This measure is characterized by

μ 1 × · · · × μn(E 1 × · · · × En) = μ 1 (E 1 ) · · · μn(En)

for measurable Ei ⊂ Gi, i = 1,... , n.

Example 1.6. Let p ∈ Z be a prime. Consider the projective system {Z/pnZ}n∈N, with the maps being the natural projections. Then we can take the projective limit,

Zp = lim ←− n

Z/pnZ.

Explicitly,

Zp = {(a 1 , a 2 , a 3 ,... ) ∈

∏^ ∞

n=

Z/pnZ | ai+1 ≡ ai (mod pi)}.

The ring Zp, called the p-adic integers, is an integral domain of characteristic zero; this is an easy exercise, especially in light of the following considerations. The ring structure itself is made more explicit as follows. First write the element (a 1 , a 2 , a 3 ,... ) ∈ Zp instead like

(α 0 + pZ, α 1 p + α 0 + p^2 Z, α 2 p^2 + α 1 p + α 0 + p^3 Z,... ),

where each αi ∈ { 0 , 1 ,... , p − 1 }. We can do this because of the projectivity of the system. Then we write formally ∑∞

i=

αipi

for this element. Then this is like a base p expansion which is infinite in length, and the addition and multiplication are given by carrying in base p. The canonical example of this is the amusing computation

1 +

∑^ ∞

i=

(p − 1)pi^ = 0,

LECTURE 1. LOCALLY COMPACT ABELIAN GROUPS 5

d(a, b) = |a − b| where |

i=m αit i| = q−m (^) if αm 6 = 0. This topology on Fq[[t]] then extends

to its fraction field Fq((t)) via the same base of neighborhoods about 0, or by the same formula for the metric. Similarly to the previous example, Fq[[t]] is compact and hence Fq((t)) is locally compact, but not compact. We leave these details to the reader. Note also that Fq[[t]] is a discrete valuation ring with its only prime ideal the one generated by t. Finally, the Haar measure is determined by its value on Fq[[t]], and the ideals tnFq[[t]] have measure 1/qn^ times that of Fq[[t]].

Lecture 2

Local Fields

This lecture and the next will be devoted to the study of local and global fields. Local fields will turn out to come from global fields by a process which may very well be viewed as a localization. However, we go the other way in these lectures for the sake of the progression of the material. To define a local field, we need

Definition 2.1. Let k be a field. A function | · | : k → R≥ 0 is called an absolute value on k if the following properties hold: (1) [Positive definiteness]. For any a ∈ k, we have that |a| = 0 if and only if a = 0; (2) [Multiplicativity]. For any a, b ∈ k, we have |ab| = |a||b|; (3) [Triangle inequality]. For any a, b ∈ k, we have |a + b| ≤ |a| + |b|.

An absolute value |·| on a field k gives rise to a metric d on k defined by d(a, b) = |b−a|. Thus k also gets a topology, and it is easy to check that addition and multiplication, as well as their respective inversions (excluding 0 in the multiplicative case), are continuous. A field with a topology for which these four operations are continuous is called a topological field. We will always assume that any topological field k we consider does not have the discrete topology. If | · | is an absolute value on k, this is the same as saying that | · | is not trivial, i.e., it is not the case that |a| = 1 for all a ∈ k, a 6 = 0.

Definition 2.2. A field k is called a local field if either of the following two equivalent conditions holds: (1) k is a (nondiscrete) topological field which is locally compact; (2) k has a (nontrivial) absolute value such that the resulting metric topology is complete and locally compact.

These conditions are not obviously equivalent. In fact, we will not prove their equiva- lence here, but we will show now how to construct the absolute value assuming condition (1). Let μ be a Haar measure on a field k satisfying condition (1) of the definition. Let a ∈ k×. Then we define the measure μa to be the measure given by μa(E) = μ(aE) for measurable sets E. The measure μa is again a Haar measure, and so by the uniqueness of the Haar measure, there is a positive constant c such that μa = cμ. Define mod(a) = c.

LECTURE 2. LOCAL FIELDS 8

cosets have equal measure q−^1 M. Now p is the complement of an open set (the union of its cosets different from itself) in a compact (namely Ok), so it is itself compact. Since the module is continuous (which is shown directly in Exercise 2.1), it attains a maximum value on p, say r, which must be smaller than 1 because p = B 1. Therefore p and Cr are actually equal as sets. Let π ∈ k be such that ‖π‖ = r. If a ∈ Ok, then ‖aπ‖ ≤ r. Conversely if ‖aπ‖ ≤ r, then ‖a‖ ≤ r/‖π‖ = 1, and so a ∈ Ok. Thus p = πOk. This proves that p is principal with generator any element with maximal module in p. Furthermore,

q−^1 M = μ(p) = μ(πOk) = ‖π‖μ(Ok) = ‖π‖M,

so q−^1 = ‖π‖. This completes the proof.

An element which generates p is called a prime element, and the field Ok/p is called the residue field of k. The proposition shows that Ok is a discrete valuation ring, which just means it is a local principal ideal domain. But clearly its fraction field is k since if a ∈ k with ‖a‖ ≥ 1, then ‖a−^1 ‖ ≤ 1. It follows from standard facts about discrete valuation rings that all ideals in Ok are of the form pm^ for some integer m ≥ 0. Therefore, given any prime element π, we get a decomposition k×^ ∼= O k× × πZ, where πZ^ is the multiplicative group generated by π. For any a ∈ k, we define v(a) to be the largest integer n for which a ∈ πnOk. This does not depend on the choice of π because the sets πnOk do not. Then ‖a‖ = q−v(a), where q = |Ok/p|. The number v(a) is called the valuation of a. It is the same as the one coming from the discrete valuation ring Ok. Now let l/k be an extension of nonarchimedean local fields. Let ‖ · ‖k and ‖ · ‖l be the respective modules, and let vk and vl be the respective valuations. Let πk be a prime element of k. Then vl(πk) is equal to some integer e = e(l/k), and this is called the ramification index of l/k. This can also be described via the equality

pkOl = pe l (l/k).

Since pk ⊂ pl, we get an extension of residue fields Ok/pk ⊂ Ol/pl. Since both fields are finite, this extension is finite. We denote the degree of this extension by f = f (l/k). Note that this discussion implies that we can find which of the fields Qp or Fp((t)) is contained in a given nonarchimedean local field k: The characteristic of the residue field gives p, and the characteristic of k itself determines whether it is Qp or Fp((t)). Furthermore, none of Qp or Fp((t)) is an extension of another. We finish this lecture by stating a useful theorem about extensions of nonarchimedean local fields.

Theorem 2.5. Let l/k be an extension of nonarchimedean local fields. Then

[l : k] = e(l/k)f (l/k).

Lecture 3

Global Fields

We begin with the main definition.

Definition 3.1. A global field is either a finite extension of Q or a finite separable extension of Fq(t). The former of these fields are called number fields and the latter are called function fields.

Number theory is concerned with the arithmetic of these fields and their extensions. Our first task will be to describe the absolute values on these fields. This will ultimately lead to important arithmetic data of these fields. Let | · | 1 and | · | 2 be absolute values on a field k. We say | · | 1 and | · | 2 are equivalent if they induce the same topology on k. Now if K is a global field, VK will denote the set of absolute values on K up to equiva- lence. An element of VK will be called a place. To describe VK , we need a definition.

Definition 3.2. Let K be a global field. A prime ring in K is a subring P of K which is a discrete valuation ring and whose fraction field is K.

Let us give some examples. Let p be an integer prime, and let Z(p) be the localization of Z at that prime. Then Z(p) is a prime ring in Q, as is easily checked. One can do something similar with Fq(t). We have a subring Fq[t] ⊂ Fq(t) which is a principal ideal domain. The irreducible polynomials in Fq[t] generate prime ideals, and all prime ideals are generated this way. So if P is an irreducible polynomial in Fq[t], we can localize at the ideal it generates and obtain a prime ring in Fq(t). There is a prime ring in Fq(t) which does not come from a prime ideal in Fq[t], but rather from a prime ideal in an isomorphic subring. It is obtained by localizing the ring Fq[t−^1 ] at the ideal generated by t−^1. This is the only prime ring in Fq(t) which does not come from localizing Fq[t] at one of its prime ideals. Now let K be a global field and P a prime ring in K. Just as in the previous lecture, there is a valuation associated to P on K, which is defined explicitly as follows. The maximal ideal in P is generated by one element π, and for every α ∈ K, there is a unique unit u in P and a unique n ∈ Z such that α = uπn. The integer n does not depend on the choice of π, and is called the valuation of α at P. The function which extracts valuations at P is denoted vP. It is a homomorphism K×^ → Z. Now the prime ring P gives rise to a nonarchimedean absolute value on K as follows. Let r be a real number greater than 1. Define |α|r,P = r−vP(α)^ for α ∈ K×, and | 0 |r,P = 0.

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LECTURE 3. GLOBAL FIELDS 11

the | · |σ as we did with the absolute values coming from prime rings. We will obtain either R or C depending on whether σ is, respectively, real or complex. Now for number fields, the theory of the finite places has another important incarnation. To describe it, we need some results and definitions which belong to commutative algebra.

Definition 3.5. Let R be a subring of a ring S. An element b ∈ S is called integral over R if there is are elements a 0 , a 1 ,... , an− 1 ∈ R such that

bn^ + an− 1 bn−^1 + · · · + a 1 b + a 0 = 0.

Proposition 3.6. Let A be a noetherian integral domain, K its field of fractions, and L a finite extension of K. The set B of all elements of L which are integral over A forms a ring, called the integral closure of A in L. In the case L = K, if A = B then A is said to be integrally closed. In any case, no matter what L is, B is integrally closed.

Definition 3.7. An integral domain A is called a Dedekind domain if it is noetherian, integrally closed, and all nonzero prime ideals in A are maximal.

Proposition 3.8. If A is a Dedekind domain with fraction field K and L is a finite exten- sion of K, then the integral closure of A in K is Dedekind.

See Lang [4], or the exercises, for proofs of these facts. As an example, all unique factorization domains are integrally closed. Here is a quick argument: Let A be a unique factorization domain and K its fraction field, and let a/b be a fraction which is fully reduced. If

(a/b)n^ + cn− 1 (a/b)n−^1 + · · · + c 1 (a/b) + c 0 = 0

with c 0 ,... , cn− 1 ∈ A, then upon multiplying by bn, we find that an^ is a multiple of b, contradiction. It follows that Z is integrally closed. Of course, Z is noetherian and all nonzero prime ideals in Z are maximal, so Z is a Dedekind domain.

Definition 3.9. Let K be a number field. The integral closure of Z in K is called the ring of integers in K and is denoted OK.

Let K be a number field and p a nonzero prime (that is, prime ideal) in K. Then the localization (OK )(p) of OK at p is a prime ring in K. This process is reversible: Given a prime ring P with maximal ideal q, the ideal q∩OK is a nonzero prime, and the localization of OK at it is P. This is Exercise 3.18. Thus we get a bijection between prime rings in K and nonzero primes in K, and hence also between these and finite places of K. Now the nonzero primes of K give rise to a very important structure associated to K.

Definition 3.10. Let A be a Dedekind domain with fraction field K. A nonzero finitely generated A-submodule of K is called a fractional ideal. If a, b are fractional ideals, we define their product ab by as the A-submodule of K generated by the products {αβ | α ∈ a, β ∈ b}. It is also finitely generated, and hence is a fractional ideal. We define the inverse a−^1 of a fractional ideal a by a−^1 = {β ∈ K | αa ⊂ A}. This is also finitely generated because, briefly, for any α ∈ a, αa−^1 is in A, and A is noetherian. Hence a−^1 is a fractional ideal. The set of all fractional ideals of A will be denoted J(A). Note that the fractional ideals contained in A are just the (nonzero) ideals.

LECTURE 3. GLOBAL FIELDS 12

The fundamental theorem about fractional ideals is this.

Theorem 3.11. Let A be a Dedekind domain. (1) If a, b are fractional ideals of A, then a ⊂ b if and only if there is an ideal c ⊂ A such that cb = a. In either case, we say that b divides a. (2) The set J(A) is a group with product and inverse as above, with identity A. More- over, (3) J(A) is free abelian on the nonzero prime ideals of A.

This theorem implies that every ideal in a Dedekind domain A factors uniquely into prime ideals. This is important arithmetically because not all rings of integers of number fields are unique factorization domains. In fact, usually they are not. Now, using the theorem above, we can define directly a valuation associated to a prime in a number field. Let K be a number field and let α ∈ K be nonzero. Then we define vp(α) to be the power of p which occurs in the factorization of the fractional ideal αOK. This is the same as the one associated to the prime ring (OK )(p). What about function fields? The ring Fq[t] in Fq(t) is indeed Dedekind, and in fact, all of the theory works for it except for the fact that there is a nonarchimedean place of Fq(t) which does not come from a prime ideal of Fq(t), namely the one associated to the prime ring Fqt−^1 ). But actually, it is worse: An arbitrary function field contains many copies of Fq(t), so the integral closure of some Fq[t] contained in a function field need not be unique. So it is not always clear what one should say is the analogue here of the ring of integers in a number field. However, in any function field K, it is the case that any choice of integral closure of a copy of Fq[t] in K gives a Dedekind domain in K whose localizations provide all but finitely many prime rings of K. It is also the case that no copy of Fq[t] in K will have an integral closure which provides all the prime rings of K. To remedy these complications, we take as an analogue of the group of fractional ideals the free abelian group on the prime rings of K. It is denoted Div(K) and its elements are called divisors. We will study a little the theory of divisors later.

LECTURE 4. ADELES AND IDELES 14

of course. We also topologize AK by declaring a base of neighborhoods about 0 to be   

v∈VK

Uv

Uv ⊂ Kv is open for all v, 0 ∈ Uv for all v, Uv = Ov for almost all v /∈ V∞

The adeles of a global field are locally compact in view of Tychonoff’s theorem. It is easily checked that the adeles also form a topological group. Some remarks: In his thesis, Tate called adeles “valuation vectors,” which gives a nice description for what they are. Philosophically, the adeles should also be viewed as the global analogue of a local field, since they contain all local information of a global field in each component. Now we look at the units in the adeles.

Definition 4.4. Let K be a global field. The ideles of K are the group IK = A× K. So

IK =

(av)v∈VK ∈

v∈VK

K v×

av ∈ O× v for almost all v /∈ V∞

We give it the topology with a base of neighborhoods about 1 given by   

v∈VK

Uv

Uv ⊂ K v× is open for all v, 1 ∈ Uv for all v, Uv = O× v for almost all v /∈ V∞

The ideles of a global field are a locally compact abelian group again by Tychonoff. However, the topology we gave the ideles is not the subspace topology as a subset of the adeles! Let K be a global field. Since K embeds into all of its completions, we have an em- bedding K ⊂ AK which is given by the diagonal α 7 → (α, α,... ). This embedding makes sense in view of the fact that only finitely many valuations v have |α|v 6 = 1 for any α ∈ K×, where | · |v is an absolute value in the class v. We will take this for granted, though see Exercise 4.2. From this it also follows that we get an embedding K×^ ⊂ IK , which we will use later.

Theorem 4.5. With the embedding above, a global field K is discrete in its adeles AK.

We give the idea of the proof, which requires a couple results from the next lecture which do not depend on this theorem. For K a number field, we consider first the open set U ⊂ AK defined by U =

v∈Vf

Ov ×

v∈V∞

Kv.

The elements of K which are in U must then have nonnegative valuation at all finite places. It follows that K ∩ U = OK because, for instance, by the ideal theory discussed in the previous lecture, the fractional ideal generated by an α ∈ K ∩ U contains only positive powers of primes.∏ Now Exercise 4.6 says that OK embeds discretely into the component

v∈V∞ Kv^ of^ U^ , and hence there is an open subset^ U∞^ of^

v∈V∞ Kv^ which contains only

1. Thus the open set U ′^ =

v∈Vf Ov^ ×^ U∞^ is such that^ K^ ∩^ U^ ′ (^) = { 0 }. It follows that K is

discrete in AK. For function fields, we use the following theorem, valid for any global field.

LECTURE 4. ADELES AND IDELES 15

Theorem 4.6 (Product Formula). Let K be a global field and α ∈ K×. Then ∏

v∈VK

‖α‖v = 1,

where ‖ · ‖v is the usual absolute value ‖ · ‖P on K if v comes from a prime ring P in K, ‖ · ‖v = | · |σ if v comes from an real embedding σ, and ‖ · ‖v = | · |^2 τ if v comes from a complex embedding τ.

One proves this theorem by first proving it for K = Fq(t) or K = Q, and then reducing to this case using the theory of extensions of places which we will develop in the next lecture. Also, it should be noted that the product in the theorem makes sense because, as we have said, only finitely many valuations v have ‖α‖v 6 = 1 for any α ∈ K×. Returning to the discreteness of a function field K, let U =

v∈VK Ov. Then^ K

× ∩ U

consists of all elements in K with nonnegative valuation at all places. By the Product Formula, if any valuation of an α ∈ K×^ is positive, another must be negative. Therefore all valuations of an element in K×^ ∩ U are equal to 0. But then one shows that this implies that α is constant, i.e., α is contained in the largest extension of Fq contained in K (this is shown in Exercise 4.3). But such an extension is finite, so U contains only finitely many elements of K. Since the adeles are Hausdorff, K ∩ U is discrete, and so K is discrete. To state our next theorem, we need a definition. We keep the notation ‖ · ‖v of the previous theorem.

Definition 4.7. Let K be a global field and let a = (av)v∈VK ∈ IK. Then we define the norm of a, denoted ‖a‖, via ‖a‖ =

v∈VK

‖av‖v.

We define the group I^1 K to consist of those ideles which have norm 1.

By the Product Formula, K×^ ⊂ I^1 K. We have

Theorem 4.8. Let K be a global field. Then K×^ is discrete in I^1 K.

The main theorem, whose proof is left to the exercises, is the following.

Theorem 4.9. Let K be a global field. (1) AK /K is compact under the quotient topology. (2) I^1 K /K×^ is compact under the quotient topology. Let us conclude by proving the finiteness of the class number. Let K be a number field. For v ∈ Vf , denote also by v the associated valuation on K× v , and denote by pv the prime ideal associated to v. Define a map I^1 K → J(OK ) by (av)v∈VK 7 →

v∈Vf p

v(av ) v. This map is surjective because it is as a map IK → J(OK ) and, if b = (bv) is any idele and c =

v∈Vf ‖bv‖v, then we can always change the infinite part of^ b^ so that^

v∈V∞ ‖bv‖^ =^ c

making it norm 1. Now the kernel U of this map is the set of norm 1 ideles with finite component in

v∈Vf O × v , and so the kernel is open. Hence the quotient I 1 K /U^ is discrete and isomorphic to J(OK ). But then C(OK ) ∼= I^1 K /K×^ · U , and the latter group is discrete and compact, hence finite. So we are done.