8 Miniproject: Adeles
8.1 Status
This is an active miniproject.
8.2 The goal
There are several goals to this miniproject.
Define the adeles \(\mathbb {A}_K\) of a number field \(K\) and give them the structure of a \(K\)-algebra (status: now in mathlib thanks to Salvatore Mercuri);
Prove that \(\mathbb {A}_K\) is a locally compact topological ring (status: also proved by Mercuri but not yet in mathlib);
Base change: show that if \(L/K\) is a finite extension of number fields then the natural map \(L\otimes _K\mathbb {A}_K\to \mathbb {A}_L\) is an isomorphism; (status: not formalized yet, but there is a plan – see the project dashboard);
Prove that \(K \subseteq \mathbb {A}_K\) is a discrete subgoup and the quotient is compact (status: not formalized yet, but there is a plan – see the project dashboard);
Get this stuff into mathlib (status: (1) done, (2)–(4) not done).
We briefly go through the basic definitions. A cheap definition of the finite adeles \(\mathbb {A}_K^\infty \) of \(K\) is \(K\otimes _{\mathbb {Z}}\widehat{\mathbb {Z}}\), where \(\widehat{\mathbb {Z}}\) is the profinite completion of the integers. A cheap definition of the infinite adeles \(K_\infty \) of \(K\) is \(K\otimes _{\mathbb {Q}}\mathbb {R}\), and a cheap definition of the adeles of \(K\) is \(\mathbb {A}_K^\infty \times K_\infty \).
However in the literature different definitions are often given. The finite adeles of \(K\) are usually defined in the books as the so-called restricted product \(\prod '_{\mathfrak {p}}K_{\mathfrak {p}}\) over the completions \(K_{\mathfrak {p}}\) of \(K\) at all maximal ideals \(\mathfrak {p}\subseteq \mathcal{O}_K\) of the integers of \(K\). Here the restricted product is the subset of \(\prod _{\mathfrak {p}}K_{\mathfrak {p}}\) consisting of elements which are in \(\mathcal{O}_{K,\mathfrak {p}}\) for all but finitely many \(\mathfrak {p}\). This is the definition given in mathlib. Mathlib also has the proof that they’re a topological ring; furthermore the construction of the finite adeles in mathlib works for any Dedekind domain (this was pointed out to me by Maria Ines de Frutos Fernandez; the adeles are an arithmetic object, but the finite adeles are an algebraic object).
Similarly the infinite adeles of a number field \(K\) are usually defined as \(\prod _v K_v\), the product running over the archimedean completions of \(K\), and this is the mathlib definition.
The adeles of a number field \(K\) are the product of the finite and infinite adeles, and mathlib knows that they’re a \(K\)-algebra and a topological ring.
8.3 Local compactness
As mentioned above, Salvatore Mercuri has a complete formalisation of the proof that the adele ring is locally compact. His work is in his own repo which I don’t want to have as a dependency of FLT, because this work should all be in mathlib.
The adeles of a number field are locally compact.
See this line in Mercuri’s repo.
8.4 Base change
The “theorem” we want is that if \(L/K\) is a finite extension of number fields, then \(\mathbb {A}_L=L\otimes _K\mathbb {A}_K\). This isn’t a theorem though, this is actually a definition (the map between the two objects) and a theorem about the definition (that it’s an isomorphism). Before we can prove the theorem, we need to make the definition.
Recall that the adeles \(\mathbb {A}_K\) of a number field is a product \(\mathbb {A}_K^\infty \times K_\infty \) of the finite adeles and the infinite adeles. So our “theorem” follows immediately from the “theorem”s that \(\mathbb {A}_L^\infty =L\otimes _K\mathbb {A}_K^\infty \) and \(L_\infty =L\otimes _KK_\infty \). We may thus treat the finite and infinite results separately.
8.4.1 Base change for finite adeles
As pointed out above, the theory of finite adeles works fine for Dedekind domains. So Let \(A\) be a Dedekind domain. Recall that the height one spectrum of \(A\) is the nonzero prime ideals of \(A\). Note that because we stick to the literature, rather than to common sense, fields are Dedekind domains in mathlib, and the height one spectrum of a field is empty. The reason I don’t like allowing fields to be Dedekind domains is that geometrically the standard definition of Dedekind domain is “smooth affine curve, or a point”. But many theorems in algebraic geometry begin “let \(C\) be a smooth curve”, rather than “let \(C\) be a smooth curve or a point”.
Let \(K\) be the field of fractions of \(A\). If \(v\) is in the height one spectrum of \(A\), then we can put the \(v\)-adic topology on \(A\) and on \(K\), and consider the completions \(A_v\) and \(K_v\). The finite adele ring \(\mathbb {A}_{A,K}^\infty \) is defined to be the restricted product of the \(K_v\) with respect to the \(A_v\), as \(v\) runs over the height one spectrum of \(A\).
Now let \(L/K\) be a finite separable extension, and let \(B\) be the integral closure of \(A\) in \(L\). If \(w\) is a nonzero prime ideal of \(B\) lying under \(v\), a prime ideal of \(A\), then we can put the \(w\)-adic topology on \(L\) and the \(v\)-adic topology on \(K\).
If \(i:K\to L\) denotes the inclusion then \(e\times w(i(k))=v(k)\), where \(e\) is the ramification index of \(w/v\).
Standard.
There’s a natural continuous \(K\)-algebra homomorphism map \(K_v\to L_w\). It is defined by completing the inclusion \(K\to L\) at the finite places \(v\) and \(w\), which can be done by the previous lemma.
Now say \(v\) is in the height one spectrum of \(A\). The map above induces a continuous \(K\)-algebra homomorphism \(K_v\to \prod _{w|v}L_w\), where the product runs over the height one primes of \(B\) which pull back to \(v\).
The induced continuous \(L\)-algebra homomorphism \(L\otimes _KK_v\to \prod _{w|v}L_w\) is an isomorphism.
See Theorem 5.12 on p21 of these notes.
The isomorphism \(L\otimes _KK_v\to \prod _{w|v}L_w\) induces a topological isomorphism \(B\otimes _AA_v\to \prod _{w|v}B_w\) for all but finitely many \(v\) in the height one spectrum of \(A\).
Certainly the image of the integral elements are integral. The argument in the other direction is more delicate. One approach (following Cassels–Froehlich, Cassels’ article “Global fields”, section 12 lemma, 61) is the following. Choose a \(K\)-basis \(\omega _1,\omega _2,\ldots ,\omega _n\) for \(L/K\) with all \(\omega _i\in A\). Then \(\omega _1A\oplus \cdots \oplus \omega _nA\) and \(B\) are two \(A\)-lattices in \(L\), so they agree at almost all primes. In particular \(\omega _1A_v\oplus \omega _2A_v\oplus \cdots \oplus \omega _nA_v=B\otimes _AA_v\) for all but finitely many primes \(v\) of \(A\). If we define the discriminant map \(D\) on \(L/K\) by \(D(\gamma _1,\gamma _2,\ldots ,\gamma _n)=det_{i,j}(trace_{L/K}(\gamma _i\gamma _j))\) then it’s well-known that \(d:=D(\omega _1,\omega _2,\ldots ,\omega _n)\) is nonzero (here we use separability), and hence a \(v\)-adic unit for all but finitely many \(v\). Furthermore if \(\gamma _i\in \prod _{w|v}B_w\) for all \(i\) then \(D(\gamma _1,\gamma _2,\ldots ,\gamma _n)\in A_v\) as all of the traces are in \(A_v\). Now say we have an element of \(\prod _{w|v}B_w\), and write it as \(\sum _i b_i\omega _i\) with \(b_i\in K_v\). Then for each \(i\) we have \(D(\omega _1,\omega _2,\ldots ,\omega _{i-1},b,\omega _{i+1},\ldots ,\omega _n)\in A_v\) but it is also \(b_i^2d\). Because \(d\) is a \(v\)-adic unit for almost all \(v\), we see that the \(b_i\) must also be in \(A_v\) for almost all \(v\).
We can take the product of the maps \(K_v\to \prod _{w|v}L_w\) over \(v\).
There’s a natural \(K\)-algebra homomorphism \(\prod _v K_v\to \prod _w L_w\), where the products run over the height one spectra of \(A\) and \(B\) respectively.
Note that we make no claim about continuity (for reasons which will become clear below).
This map induces a natural continuous \(K\)-algebra homomorphism \(\mathbb {A}_{A,K}^\infty \to \mathbb {A}_{B,L}^\infty \).
Note that the restricted product does not have the subspace topology.
If we give \(L\otimes _K\mathbb {A}_{A,K}^\infty \) the “module topology”, coming from the fact that \(L\otimes _K\mathbb {A}_{A,K}^\infty \) is an \(\mathbb {A}_{A,K}^\infty \)-module, then the induced \(L\)-algebra morphism \(L\otimes _K\mathbb {A}_{A,K}^\infty \to \mathbb {A}_{B,L}^\infty \) is a topological isomorphism.
Follows from theorem 8.5.
8.4.2 Base change for infinite adeles
Recall that if \(K\) is a number field then the infinite adeles of \(K\) are defined to be the product \(\prod _{v\mid \infty } K_v\) of all the completions of \(K\) at the infinite places.
The result we need here is that if \(L/K\) is a finite extension of number fields, then the map \(K\to L\) extends to a continuous \(K\)-algebra map \(K_\infty \to L_\infty \), and thus to a continuous \(L\)-algebra isomorphism \(L\otimes _KK_\infty \to L_\infty \).
8.4.3 Base change for adeles
From the previous results we deduce immediately that if \(L/K\) is a finite extension of number fields then there’s a natural (topological and algebraic) isomorphism \(L\otimes _K\mathbb {A}_K\to \mathbb {A}_L\).
8.5 Discreteness and compactness
We need that if \(K\) is a number field then \(K\subseteq \mathbb {A}_K\) is discrete, and the quotient (with the quotient topology) is compact. Here is a proposed proof.
There’s an open subset of \(\mathbb {A}_{\mathbb {Q}}\) whose intersection with \(\mathbb {Q}\) is \(\{ 0\} \).
Use \(\prod _p{\mathbb {Z}_p}\otimes (-1,1)\). Any rational \(q\) in this set is a \(p\)-adic integer for all primes \(p\) and hence (writing it in lowest terms as \(q=n/d\)) satisfies \(p\nmid d\), meaning that \(d=\pm 1\) and thus \(q\in \mathbb {Z}\). The fact that \(q\in (-1,1)\) implies \(q=0\).
The rationals \(\mathbb {Q}\) are a discrete subgroup of \(\mathbb {A}_{\mathbb {Q}}\).
If \(q\in \mathbb {Q}\) and \(U\) is the open subset in the previous lemma, then it’s easily checked that \(\mathbb {Q}\cap U=\{ 0\} \) implies \(\mathbb {Q}\cap (U+q)=\{ q\} \), and \(U+q\) is open.
The additive subgroup \(K\) of \(\mathbb {A}_K\) is discrete.
By a previous result, we have \(\mathbb {A}_K=K\otimes _{\mathbb {Q}}\mathbb {A}_{\mathbb {Q}}\). Choose a basis of \(K/\mathbb {Q}\); then \(K\) can be identified with \(\mathbb {Q}^n\subseteq (\mathbb {A}_{\mathbb {Q}})^n\) and the result follows from the previous theorem.
For compactness we follow the same approach.
The quotient \(\mathbb {A}_{\mathbb {Q}}/\mathbb {Q}\) is compact.
The space \(\prod _p\mathbb {Z}_p\times [0,1]\subseteq \mathbb {A}_{\mathbb {Q}}\) is a product of compact spaces and is hence compact. I claim that it surjects onto \(\mathbb {A}_{\mathbb {Q}}/\mathbb {Q}\). Indeed, if \(a\in \mathbb {A}_{\mathbb {Q}}\) then for the finitely many prime numbers \(p\in S\) such that \(a_p\not\in \mathbb {Z}_p\) we have \(a_p\in \frac{r_p}{p^{n_p}}+\mathbb {Z}_p\) with \(r_p/p^{n_p}\in \mathbb {Q}\), and if \(q=\sum _{p\in S}\frac{r_p}{p^{n_p}}\in \mathbb {Q}\) then \(a-q\in \prod _p\mathbb {Z}_p\times \mathbb {R}\). Now just subtract \(\lfloor a_{\infty }-q\rfloor \) to move into \(\prod _p\mathbb {Z}_p\times [0,1)\) and we are done.
The quotient \(\mathbb {A}_K/K\) is compact.
We proceed as in the discreteness proof above, by reducing to \(\mathbb {Q}\). As before, choosing a \(\mathbb {Q}\)-basis of \(K\) gives us \(\mathbb {A}_K/K\cong (\mathbb {A}_{\mathbb {Q}}/\mathbb {Q})^n\) so the result follows from the previous theorem.