THE IDELIC APPROACH TO NUMBER THEORY

TOM WESTON

1. Introduction

I have tried in this paper to emphasize the topological details in these constructions, and hopefully have not ignored any important points. We will assume some familiarity with number fields, at the level of [3, Chapter 1], [4, Chapters 1-3] or [8, Chapter 1].

We fix the following notation throughout this paper: we let k be a number field (that is, a finite extension of ℚ) of degree n over ℚ. We let 𝔬_{k} be the ring of integers of k. If 𝔭 is a prime ideal of 𝔬_{k} with 𝔭 ∩ ℤ = (p), we write e_{𝔭} for the ramification degree and f_{𝔭} for the inertial degree of 𝔭 over p. That is, e_{𝔭} is the largest power of 𝔭 dividing p𝔬_{k}, and f_{𝔭} is the degree of the residue field extension 𝔬_{k}/𝔭 over ℤ/(p). If the prime 𝔭 is clear from context, then we will just write e = e_{𝔭} and f = f_{𝔭}.

Part 1. Classical Algebraic Number Theory

2. Local Fields : Algebraic Description

Recall that the local ring 𝔬_{𝔭} ⊆ k is a discrete valuation ring. Let π be a uniformizing element of 𝔬_{𝔭}; that is, π generates the unique non-zero prime ideal 𝔭𝔬_{𝔭} of 𝔬_{𝔭}. Then any α ∈ k^{∗} can be written as α = uπ^{m} for a unique integer m and some unit u of 𝔬_{𝔭}^{∗}. We say that this m is the 𝔭-adic valuation v_{𝔭}(α) of α. Setting v_{𝔭}(0) = ∞, we have defined a discrete valuation

v = v_{𝔭} : k → ℤ ∪ {∞},

which is easily checked to satisfy the usual properties of a discrete valuation:

1. v_{𝔭}(αβ) = v_{𝔭}(α) + v_{𝔭}(β) for all α, β ∈ k;

2. v_{𝔭}(α + β) ≥ min{v_{𝔭}(α), v_{𝔭}(β)} for all α, β ∈ k;

3. v_{𝔭}(α) = ∞ if and only if α = 0. We use the standard conventions for arithmetic with ∞, including ∞ · 0 = ∞. 𝔬_{𝔭} is precisely the set of elements of k with nonnegative valuation; the units 𝔬_{𝔭}^{∗} are precisely the elements of k^{∗} with valuation 0. We also have

𝔬_{𝔭}/𝔭𝔬_{𝔭} = (𝔬_{k}/𝔭)_{𝔭𝔬_{k}/𝔭} = (𝔬_{k}/𝔭)_{(0)} = 𝔬_{k}/𝔭

since 𝔬_{k}/𝔭 is already a field.

Note that p = uπ^{e} for some unit u ∈ 𝔬_{𝔭}^{∗}, so that

v_{𝔭}(p) = e,

where e is the ramification degree of 𝔭.

We will now construct the completion of 𝔬_{𝔭}. Let A_{m} = 𝔬_{𝔭}/(π^{m}). Then we have natural maps

φ_{m} : A_{m} → A_{m−1}

given by simply considering a residue class modulo π^{m} modulo π^{m−1}. The A_{m} form an inverse system, and we define the 𝔭-adic integers 𝔬_{v} to be the inverse limit of the A_{m}. That is, 𝔬_{v} is the subset of elements (a_{1}, a_{2}, …) of the infinite product

∏_{m=1}^{∞} A_{m}

such that

φ_{m}(a_{m}) = a_{m−1}

for all m. It is easy to see that if (a_{m}) and (b_{m}) lie in 𝔬_{v}, then so do (a_{m} + b_{m}) and (a_{m}b_{m}), so 𝔬_{v} inherits a ring structure from ∏_{m=1}^{∞} A_{m}.

Note that we have natural inclusions

𝔬_{k} ↪ 𝔬_{𝔭} ↪ 𝔬_{v}

given by sending α ∈ 𝔬_{𝔭} to (ᾱ, ᾱ, …) ∈ 𝔬_{v}. These are clearly ring homomorphisms, so 𝔬_{k} and 𝔬_{𝔭}, are naturally subrings of 𝔬_{v}.

We would like to have a better way to think of elements of 𝔬_{v}. To do this, first fix representatives 0 = c_{0}, …, c_{p^{f} −1} ∈ 𝔬_{𝔭} of the residue classes in the finite field 𝔬_{𝔭}/(π) ≅ 𝔬_{k}/𝔭. (In the case where 𝔬_{𝔭}/(π) = 𝔽_{p} we have the standard representatives 0, 1, …, p − 1, but in general there are no such obvious choices.) Now, take any element a = (a_{m}) of 𝔬_{v}. There is then a unique c_{i_{0}} such that

a_{1} ≡ c_{i_{0}} (mod π).

Next consider a − c_{i_{0}} = (a_{m} − c_{i_{0}}) ∈ 𝔬_{v}, The first coordinate of a − c_{i_{0}} will be 0, and thus divisible by π. It follows from the compatibility of the (a_{m} − c_{i_{0}}) under the φ_{m} that each a_{m} − c_{i_{0}} is then divisible by π. Thus, we can write

a − c_{i_{0}} = (a_{m} − c_{i_{0}})
= (πa′_{m})
= π(a′_{m})
= πa′

for some a′ = (a′_{m}) in 𝔬_{v}. Repeating this procedure, we can write

a′ − c_{i_{1}} = πa′′

for some unique c_{i_{1}} and some a′′ ∈ 𝔬_{v}.

Continuing this and combining all of the terms, we have

a = c_{i_{0}} + (a − c_{i_{0}})
= c_{i_{0}} + πa′
= c_{i_{0}} + π (c_{i_{1}} + (a′ − c_{i_{1}}))
= c_{i_{0}} + c_{i_{1}}π + π(π(c_{i_{2}} + (a′′ − c_{i_{2}})))
= c_{i_{0}} + c_{i_{1}}π + c_{i_{2}}π^{2} + ⋯

for some uniquely determined c_{i_{0}}, c_{i_{1}}, c_{i_{2}}, …, each chosen from our fixed set of representatives. Further, it is clear that any such expression

∑_{m=0}^{∞} c_{i_{m}}π^{m}

corresponds to some a = (a_{m}) ∈ 𝔬_{v} by setting

a_{m} ≡ c_{i_{0}} + c_{i_{1}}π + ⋯ + c_{i_{m−1}}π^{m−1} (mod π^{m}).

Thus, we can think of the elements of 𝔬_{v} as some sort of infinite power series in the uniformizing element π, with coefficients chosen from representatives of the finite field 𝔬_{𝔭}/(π) = 𝔬_{k}/𝔭. We will think of the c_{i_{m}} as the digits of x ∈ 𝔬_{v}.

Writing elements of 𝔬_{v} in this form, it is now clear how to extend our discrete valuation v_{𝔭} to 𝔬_{v}: given any element

a = c_{i_{0}} + c_{i_{1}}π + c_{i_{2}}π^{2} + ⋯

of 𝔬_{v}, we define v_{𝔭}(a) = m, where π^{m} is the first power of π with a non-zero coefficient. It is easy to see that v_{𝔭} agrees with our original discrete valuation on the image of 𝔬_{𝔭} in 𝔬_{v}, and that it still satisfies the axioms of a discrete valuation. Therefore, 𝔬_{v} is a discrete valuation ring. The maximal ideal 𝔭_{v} of 𝔬_{v} is generated by any element of valuation 1; for example, our original uniformizer π. Thus, 𝔭_{v} = 𝔭𝔬_{v}. Further, it is clear from our expansion of elements of 𝔬_{v} in terms of π that

𝔬_{v}/𝔭_{v} ≅ 𝔬_{𝔭}/𝔭𝔬_{𝔭} ≅ 𝔬_{k}/𝔭,

so that all of the residue fields coincide.

The units 𝔬_{v}^{∗} of 𝔬_{v} are precisely those elements of 𝔬_{v} of valuation 0, and the non-units 𝔭 are precisely those elements of 𝔬_{v} divisible by π. Since any element of 𝔬_{v} can be written as uπ^{m} for some u ∈ 𝔬_{v}^{∗} and some nonnegative integer n, the field of fractions of 𝔬_{v} is obtained simply by inverting π. We write k_{v} for this field; then we have

k_{v} = 𝔬_{v}[π^{−1}].

Therefore, elements x of k_{v} have the form

x = ∑_{m=m_{0}}^{∞} c_{i_{m}}π^{m}

for some integer m_{0}. If we require that c_{i_{m_{0}}} ≠ 0, then m_{0} is just v_{𝔭}(x).

Since k is the field of fractions of 𝔬_{𝔭}, and 𝔬_{𝔭} embeds in 𝔬_{v}, k must embed in k_{v}.

3. Local Fields : Topological Description

We remain in the situation of the preceding section; that is, we have a number field k and a prime 𝔭 of the ring of integers 𝔬_{k} lying over a prime p of ℤ, and we let v = v_{𝔭} be the corresponding discrete valuation on k. In this section we will give a topological description of the completion k_{v}, in terms of the valuation v_{𝔭}.

First, observe that the valuation v induces a norm ∥·∥_{v} on k by

∥α∥_{v}= p^{−fv(α)}.

(We could have set ∥α∥_{v}= c^{v(α)} for any real constant c between 0 and 1; it does not matter for the purposes of this section. However, the choice c = p^{−f} is necessary for the product formula of Section 4. For the moment, note that the residue class ring 𝔬_{𝔭}/(α) has size 1/ ∥α∥_{v}, so this choice of c has at least something recommending it.) It follows immediately from the axioms of a discrete valuation that ∥·∥_{v} satisfies the axioms of a norm :

1. ∥αβ ∥_{v}=∥α∥_{v}∥β ∥_{v} for all α, β ∈ k;

2. ∥α + β ∥_{v}≤∥α∥_{v} + ∥β ∥_{v} for all α, β ∈ k. In fact, ∥·∥_{v} satisfies the stronger axiom

3. ∥α + β ∥_{v}≤ min{∥α∥_{v}, ∥β ∥_{v}} for all α, β ∈ k. For this reason we call ∥·∥_{v} a non-archimedean norm.

We can now in the usual way use ∥·∥_{v} to give us a metric on k, defined by

d_{v}(α, β) =∥α − β ∥_{v} .

This then gives us a topology on k. One can now define the completion of k with respect to v to be the usual completion of a metric space, in terms of equivalence classes of Cauchy sequences. This in fact gives rise to exactly the field k_{v} we constructed in Section 2. We will not go through the construction here; for a detailed exposition of it, see [2, Chapter 1, Section 4]. We will instead show that our algebraically constructed field of Section 2 has the necessary topological properties.

Note that we can use the discrete valuation on our field k_{v} of Section 2 to define a metric on it, which will agree with the metric defined above on the subfield k. We will show that under this topology k_{v} is a complete, locally compact topological ring, with compact subsets 𝔬_{v} and 𝔬_{v}^{∗}.

Let us describe the basic open sets on k_{v}; these are simply the open balls

B(x, r) = {y ∈ k_{v}; ∥x − y ∥_{v}< r}.

First, note that since our metric takes on only a discrete set of values, for any r there is some small ε so that

B(x, r) = \overline{B(x, r − ε)},

if ε is chosen small enough so that the metric does not take on any values between r − ε and r. Thus every basic open set is also closed. (Of course, it does not follow that every open set is also closed.)

Next, let us consider in more detail the open balls around 0 ∈ k_{v}. It is enough to consider the balls

B(0, p^{−fs})

for integral s, since these are the only values the metric takes. An element y of k_{v} satisfies ∥ y − 0 ∥_{v}< p^{−fs} if and only if v_{𝔭}(y) > s; that is, if and only if y has the form

∑_{m=s+1}^{∞} c_{i_{m}}π^{m}.

Thus, the ball B(0, p^{−fs}) consists precisely of those y which have coefficients 0 up to (and including) the coefficient of π^{s}. In the same way, for any x ∈ k_{v}, the ball B(x, p^{−fs}) consists of those y ∈ k_{v} agreeing with x up to (and including) the coefficient of π^{s}.

In particular, we see that

𝔬_{v} = B(0, p^{f})

and

𝔭_{v} = B(0, 1)

are open and closed subsets of k_{v}. Then

𝔬_{v}^{∗} = 𝔬_{v} − 𝔭_{v}

is open and closed as well.

We can actually describe the induced topology on 𝔬_{v} in a different way. Recall that we defined 𝔬_{v} to be a subset of an infinite product

∏_{m=1}^{∞} A_{m}.

Each set A_{m} is finite, and we give it the discrete topology. We then give ∏_{m=1}^{∞} A_{m} the product topology, and 𝔬_{v} the topology it inherits as a subspace.

Let us now try to describe the basic open sets in this topology. They have the form

∏_{m=1}^{∞} B_{m} ∩ 𝔬_{v}

where B_{m} is any subset of A_{m}, and B_{m} = A_{m} for all but finitely many m. Equivalently, they are the sets of the form

( ∏_{m=1}^{m_{0}} B_{m} × ∏_{m>m_{0}} A_{m}) ∩ 𝔬_{v} (1)

for positive integers m_{0}, where the B_{m} are any subsets of A_{m}. Now, consider any set U of the form (1), and pick any point x in it. It is then clear that

x ∈ B(x, p^{−f(m_{0}−1)}) ⊆ U,

so that U is open in the metric topology on 𝔬_{v}.

Conversely, consider any open ball B(x, p^{−fs}) in the metric topology. Such a set is actually of the form (1), taking m_{0} = s + 1 and the first s + 1 B_{m} to be a single point. We therefore have established:

Proposition 3.1. The two topologies we have defined on 𝔬_{v} are the same.

We are now ready to give the fundamental topological description of k_{v}.

Theorem 3.2. k_{v} is a locally compact, complete topological field, with compact open and closed subsets 𝔬_{v} and 𝔬_{v}^{∗}.

Proof. To say that k_{v} is a topological field is to say that addition, negation, multiplication and inversion are all continuous maps. The proofs of these facts are nearly identical to the proofs of the corresponding facts for the real numbers ℝ; in fact, the non-archimedean nature of our norm makes the proofs even easier.

Next we will show that 𝔬_{v} is compact. This is easy, using our alternate description of the topology on 𝔬_{v}: each finite set A_{m} is compact, so by the Tychonoff theorem, ∏_{m=1}^{∞} A_{m} is compact. Thus it is enough to show that 𝔬_{v} is closed as a subspace of ∏_{m=1}^{∞} A_{m}. So take (a_{m}) ∉ 𝔬_{v}. Then there is an m_{0} with φ_{m_{0}}(a_{m_{0}}) ≠ a_{m_{0}−1}. Consider the set

{a_{1}} × {a_{2}} × ⋯ × {a_{m_{0}}} × ∏_{m>m_{0}} A_{m}.

This is open in the product topology, and it is disjoint from 𝔬_{v} since every point in it has the same incompatible beginning. Thus, the complement of 𝔬_{v} is open, so 𝔬_{v} is closed, and thus compact.

Since 𝔬_{v}^{∗} is a closed subset of the compact set 𝔬_{v}, it is also compact. In fact, every basic open subset of k_{v} is compact, since we have a homeomorphism

ψ : B(x, p^{−fs}) → B(0, p^{f}) = 𝔬_{v}

given by ψ(y) = π^{−s−1}(y −x). (ψ is continuous since k_{v} is a topological field, and it is easy to see that it is bijective.) This shows that k_{v} is locally compact. Finally, since k_{v} is locally compact, it is complete.

Proposition 3.3. 𝔬_{v}^{∗}, 𝔬_{v} and k_{v} are the topological closures of 𝔬_{k}^{∗}, 𝔬_{k} and k respectively.

Proof. Since we clearly have 𝔬_{k}^{∗} ⊆ 𝔬_{v}^{∗}, 𝔬_{k} ⊆ 𝔬_{v} and k ⊆ k_{v} and the larger sets are all closed, it is enough to show that any element of the larger set can be arbitrarily closely approximated by an element of the smaller set. But this is clear; for example, if we pick our representatives c_{i} to all lie in 𝔬_{k} (which we can do since 𝔬_{𝔭}/𝔭𝔬_{𝔭} = 𝔬_{k}/𝔭), as well as π, then given x = ∑_{m=0}^{∞} c_{i_{m}}π^{m} ∈ 𝔬_{v}, the sequence

(c_{i_{0}}, c_{i_{0}} + c_{i_{1}}π, c_{i_{0}} + c_{i_{1}}π + c_{i_{2}}π^{2}, …)

of elements of 𝔬_{k} converges to x.

Note that since 𝔬_{k}^{∗} ⊆ 𝔬_{𝔭}^{∗} ⊆ 𝔬_{v}^{∗} and 𝔬_{k} ⊆ 𝔬_{𝔭} ⊆ 𝔬_{v}, we also have \overline{𝔬_{𝔭}^{∗}} = 𝔬_{v}^{∗} and \overline{𝔬_{𝔭}} = 𝔬_{v}.

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