10 Miniproject: Fujisaki’s Lemma
10.1 The goal
There is an idelic compactness statement which encapsulates both finiteness of the class group of a number field and Dirichlet’s units theorem about the rank of the unit group. In fact there is even a noncommutative version of this statement. In John Voight’s book [ 16 ] this is Main Theorem 27.6.14(a) and Voight calls it Fujisaki’s lemma. I know nothing of the history but I’m happy to adopt this name. In the quaternion algebra miniproject we will use this compactness result to prove finite-dimensionality of a space of quaternionic modular forms.
10.2 Initial definitions
Let \(K\) be a field. A central simple \(K\)-algebra is a \(K\)-algebra \(B\) (not necessarily commutative) with centre \(K\) such that \(B\) has exactly two two-sided ideals, namely \({0}\) and \(B\) (or \(\bot \) and \(\top \), as Lean would call them). We will be concerned only with central simple \(K\)-algebras which are finite-dimensional as \(K\)-vector spaces, and when \(K\) is clear we will just refer to them as central simple algebras. We remark that a 4-dimensional central simple algebra is called a quaternion algebra; we will have more to say about these later on.
Matrix algebras \(M_n(K)\) are examples of finite-dimensional central simple \(K\)-algebras. If \(K=\mathbb {C}\) (or more generally if \(K\) is algebraically closed) then matrix algebras are the only finite-dimensional examples up to isomorphism. There are other examples over the reals: for example Hamilton’s quaternions \(\mathbb {H}:=\mathbb {R}\oplus \mathbb {R}i\oplus \mathbb {R}j\oplus \mathbb {R}k\) with the usual rules \(i^2=j^2=k^2=-1\), \(ij=-ji=k\) etc, are an example of a central simple \(\mathbb {R}\)-algebra (and a quaternion algebra), and matrix algebras over \(\mathbb {H}\) are other central simple \(\mathbb {R}\)-algebras. For a general field \(K\) one can make an analogue of Hamilton’s quaternions \(K\oplus Ki\oplus Kj\oplus Kk\) with the same multiplication rules (\(i^2=-1\) and so on) to describe the multiplication, and if the characteristic of \(K\) isn’t 2 then this is a quaternion algebra (which may or may not be isomorphic to \(M_2(K)\) in this generality).
Some central simple algebras \(B\) are division algebras, meaning that they are division rings, or equivalently that every nonzero \(b\in B\) has a two-sided inverse. For example Hamilton’s quaternions are a division algebra over \(\mathbb {R}\), because \((x+yi+zj+tk)(x-yi-zj-tk)=x^2+y^2+z^2+t^2\), so the inverse of a nonzero \(x+yi+zj+tk\) is \((x-yi-zj-tk)/(x^2+y^2+z^2+t^2)\). However \(2\times 2\) matrices over a field \(K\), whilst being a central simple algebra over \(K\), are never a division algebra (even if \(K=\mathbb {C}\)) because a nonzero matrix with determinant zero such as \(\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}\) has no inverse.
10.3 Enter the adeles
The adeles of a number field are discussed in far more detail in the adele miniproject 8. We just recall here that if \(K\) is a number field then there are two huge commutative topological \(K\)-algebras called the finite adeles \(\mathbb {A}_K^\infty \) and the adeles \(\mathbb {A}_K\) of \(K\), and that they’re both locally compact as topological spaces. We also know from theorem 8.47 that \(\mathbb {A}_K\cong K\otimes _{\mathbb {Q}}\mathbb {A}_{\mathbb {Q}}K\) (both topologically and algebraically), meaning that if \(R\) is a \(K\)-algebra then \(R_{\mathbb {A}} := R\otimes _K\mathbb {A}_K\) is naturally isomorphic to \(R\otimes _{\mathbb {Q}}\mathbb {A}_{\mathbb {Q}}\). One can furthermore check that if \(R\) is a finite \(K\)-algebra then the \(\mathbb {A}_K\)-module topologies and \(\mathbb {A}_{\mathbb {Q}}\)-module topologies on \(R_{\mathbb {A}}\) coincide. Indeed, the topology on \(\mathbb {A}_K\) is the \(\mathbb {A}_{\mathbb {Q}}\)-module topology, as \(\mathbb {A}_K=\mathbb {A}_{\mathbb {Q}}\otimes _{\mathbb {Q}}K\) as topological \(\mathbb {A}_{\mathbb {Q}}\)-algebras, where the right hand side has the \(\mathbb {A}_{\mathbb {Q}}\)-module topology by definition. So the claim follows from the fact that if \(A\) is a topological ring, \(B\) is a topological \(A\)-algebra finite as an \(A\)-module and with the \(A\)-module topology, and if \(M\) is a topological \(B\)-module (and hence a topological \(A\)-module), then the \(A\)-module and \(B\)-module topologies on \(M\) coincide (this is moduleTopology.trans in the repo, not yet PRed to mathlib).
Let \(K\) be a number field and let \(D/K\) be a finite-dimensional central simple \(K\)-algebra (later on \(D\) will be a division algebra (hence the name) but we do not need this yet). Then \(D_{\mathbb {A}}:=D\otimes _K\mathbb {A}_K\) is an \(\mathbb {A}_K\)-algebra which is free of finite rank, and if we give \(D_{\mathbb {A}}\) the \(\mathbb {A}_K\)-module topology then it is a topological ring (by results in mathlib). Furthermore \(D_{\mathbb {A}}\) is free of finite rank over the locally compact topological ring \(\mathbb {A}_K\) and is thus also locally compact. So by the theory of Haar characters (see Chapter 9) there is a canonical character \(\delta _{D_{\mathbb {A}}}:D_{\mathbb {A}}^\times \to \mathbb {R}_{{\gt}0}\) measuring how left multiplication by an element of \(D_{\mathbb {A}}^\times \) changes the additive Haar measure on \(D_{\mathbb {A}}\). Let \(D_{\mathbb {A}}^{(1)}\) denote the kernel of \(\delta _{D_{\mathbb {A}}}\), and give it the subspace topology coming from \(D_{\mathbb {A}}^\times \). Corollary 9.39 from the Haar character miniproject shows that \(D^\times \) (regarded as a subgroup of \(D_{\mathbb {A}}^\times \) via the map \(d\mapsto d\otimes 1\)) is contained within \(D_{\mathbb {A}}^{(1)}\), thus the below theorem typechecks.
If \(D\) is a division algebra then the quotient \(D^\times \backslash D_{\mathbb {A}}^{(1)}\) with its quotient topology coming from \(D_{\mathbb {A}}^{(1)}\), is compact.
The rest of this miniproject is devoted to a proof of this theorem.
10.4 The proof
We prove the theorem via a series of lemmas.
There’s a compact subset \(E\) of \(D_{\mathbb {A}}\) with the property that for all \(x\in D_{\mathbb {A}}^{(1)}\), the obvious map \(xE\to D\backslash D_{\mathbb {A}}\) is not injective.
We know that if we pick a \(\mathbb {Q}\)-basis for \(D\) of size \(d\) then this identifies \(D\) with \(\mathbb {Q}^d\), \(D_{\mathbb {A}}\) with \(\mathbb {A}_{\mathbb {Q}}^d\), and \(D\backslash D_{\mathbb {A}}\) with \((\mathbb {Q}\backslash \mathbb {A}_{\mathbb {Q}})^d\). Now \(\mathbb {Q}\) is discrete in \(\mathbb {A}_{\mathbb {Q}}\) by theorem 8.51, and the quotient \(\mathbb {Q}\backslash \mathbb {A}_{\mathbb {Q}}\) is compact by theorem 8.52. Hence \(D\) is discrete in \(D_{\mathbb {A}}\) and the quotient \(D\backslash D_{\mathbb {A}}\) is compact.
Fix a Haar measure \(\mu \) on \(D_{\mathbb {A}}\) and push it forward to \(D\backslash D_{\mathbb {A}}\); by compactness this quotient has finite and positive measure, say \(m\in \mathbb {R}_{{\gt}0}\). Choose any compact \(E\subseteq D_{\mathbb {A}}\) with measure \({\gt} m\) (for example, choose a \(\mathbb {Z}\)-lattice \(L\cong \mathbb {Z}^d\) in \(D\cong \mathbb {Q}^d\), define \(E_f:=\prod _p L_p\in D\otimes _{\mathbb {Q}}\mathbb {A}_{\mathbb {Q}}^\infty \), and define \(E_{\infty }\subseteq D\otimes _{\mathbb {Q}}\mathbb {R}\cong \mathbb {R}^n\) to be a huge closed ball, large enough to ensure the measure of \(E:=E_f\times E_{\infty }\) is bigger than \(m\)). Then \(\mu (xE)=\mu (E){\gt}m\) so the map can’t be injective.
We let \(E\) denote any compact set satisfying the hypothesis of the previous lemma.
Define \(X:=E-E:=\{ e-f:e,f\in E\} \subseteq D_{\mathbb {A}}\).
Define \(Y:=X.X:=\{ xy:x,y\in X\} \subseteq D_{\mathbb {A}}\).
\(X\) is a compact subset of \(D_{\mathbb {A}}\).
It’s the continuous image of the compact set \(E\times E\).
\(Y\) is a compact subset of \(D_{\mathbb {A}}\).
It’s the continuous image of the compact set \(X\times X\).
If \(\beta \in D_{\mathbb {A}}^{(1)}\) then \(\beta X\cap D^\times \not=\emptyset \).
Indeed by lemma 10.2, the map \(\beta E\to D\backslash D_{\mathbb {A}}\) isn’t injective, so there are distinct \(\beta e_1,\beta e_2\in \beta E\) with \(e_i\in E\) and \(\beta e_1-\beta e_2=b\in D\). Now \(b\not=0\) and \(D\) is a division algebra, so \(b\in D^\times \). And \(e_1-e_2\in X\) so \(b=\beta (e_1-e_2)\in \beta X\), so we’re done.
Similarly, if \(\beta \in D_{\mathbb {A}}^{(1)}\) then \(X\beta ^{-1}\cap D^\times \not=\emptyset \).
Indeed, \(\beta ^{-1}\in D_{\mathbb {A}}^{(1)}\), and so left multiplication by \(\beta ^{-1}\) doesn’t change Haar measure on \(D_{\mathbb {A}}\), so neither does right multiplication (by theorem ??). So the same argument works: \(E\beta ^{-1}\to D\backslash D_{\mathbb {A}}\) is not injective so choose \(e_1\beta ^{-1}\not=e_2\beta ^{-1}\) with difference \(b\in D\) and then \((e_1-e_2)\beta ^{-1}\in D-{0}=D^\times \).
Let \(T:=Y\cap D^\times \).
\(T\) is finite.
It suffices to prove that \(Y\cap D\) is finite. But \(D\subseteq D_{\mathbb {A}}\) is a discrete additive subgroup, and hence closed. And \(Y\subseteq D_{\mathbb {A}}\) is compact. So \(D\cap Y\) is compact and discrete, so finite.
Define \(C:= (T^{-1}.X) \times X\subset D_{\mathbb {A}}\times D_{\mathbb {A}}\).
\(C\) is compact.
\(X\) is compact and \(T\) is finite.
For every \(\beta \in D_{\mathbb {A}}^{(1)}\), there exists \(b\in D^\times \) and \(\nu \in D_{\mathbb {A}}^{(1)}\) such that \(\beta =b\nu \) and \((\nu ,\nu ^{-1})\in C.\)
By lemma 10.8, \(\beta X\cap D^\times \not=\emptyset \), and lemma 10.9, \(X\beta ^{-1}\cap D^\times \not=\emptyset \), so we can write \(\beta x_1=b_1\) and \(x_2\beta ^{-1}=b_2\) with \(b_i\in D^\times \) and \(x_i\in X\). Note that \(\beta \in D_{\mathbb {A}}^{(1)}\) and \(b_i\in D^{\times }\subseteq D_{\mathbb {A}}^{(1)}\) by corollary 9.39, so \(x_i\in D_{\mathbb {A}}^{(1)}\) as well. In particular \(x_i\in D_{\mathbb {A}}^\times \) so \(x_1^{-1}\) makes sense.
Multiplying the equations defining the \(x_i\) and \(b_i\) we deduce that \(x_2x_1=b_2b_1\in Y\cap D^\times =T\) (recall that \(Y=X.X\) and \(T=Y\cap D^\times \) is finite); call this element \(t\). Then \(x_1^{-1}=t^{-1}x_2\in T^{-1}.X\), and \(x_1\in X\), so if we set \(\nu =x_1^{-1}\in D_{\mathbb {A}}^{(1)}\) and \(b=b_1\in D^\times \) then we have \(\beta =b\nu \) and \((\nu ,\nu ^{-1})\in C := (T^{-1}.X)\times X\). We are done!
We can now prove Fujisaki’s theorem 10.1.
Indeed, if \(M\) is the preimage of \(C\) under the inclusion \(D_{\mathbb {A}}^{(1)} \to D_{\mathbb {A}}\times D_{\mathbb {A}}\) sending \(\nu \) to \((\nu ,\nu ^{-1})\), then \(M\) is a closed subspace of a compact space so it’s compact (note that \(\delta _{D_{\mathbb {A}}}\) is continuous, by theorem 9.13, so \(D_{\mathbb {A}}^{(1)}\) is a closed subset of \(D_{\mathbb {A}}^\times \) which is itself a closed subset of \(D_{\mathbb {A}}\times D_{\mathbb {A}}\)). Lemma 10.14 shows that \(M\) surjects onto \(D^\times \backslash D_{\mathbb {A}}^{(1)}\) which is thus also compact.
We note here some useful consequences.
\(D^\times \backslash (D\otimes _K\mathbb {A}_K^\infty )^\times \) is compact.
There’s a natural map \(\alpha \) from \(D^\times \backslash D_{\mathbb {A}}^{(1)}\) to \(D^\times \backslash (D\otimes _K \mathbb {A}_K^\infty )^\times \). We claim that it’s surjective. Granted this claim, we are home, because if we put the quotient topology on \(D^\times \backslash (D\otimes _K \mathbb {A}_K^\infty )^\times \) coming from \((D\otimes _K \mathbb {A}_K^\infty )^\times \) then it’s readily verified that \(\alpha \) is continuous, and the continuous image of a compact space is compact.
As for surjectivity: say \(x\in (D\otimes _K \mathbb {A}_K^\infty )^\times \). We need to extend \(x\) to an element \((x,y)\in (D\otimes _K \mathbb {A}_K^\infty )^\times \times (D\otimes _K K_\infty )^\times \) which is in the kernel of \(\delta _{D_{\mathbb {A}}}\). Because \(\delta _{D_{\mathbb {A}}}(x,1)\) is some positive real number, it will suffice to show that if \(r\) is any positive real number then we can find \(y\in (D\otimes _K \mathbb {A}_K^\infty )^\times =(D\otimes _{\mathbb {Q}}\mathbb {R})^\times \) with \(\delta _{D_{\mathbb {A}}}(1,y)=r\), or equivalently (setting \(D_{\mathbb {R}}=D\otimes _{\mathbb {Q}}\mathbb {R}\)) that \(\delta _{D_{\mathbb {R}}}(y)=r\). But \(D\not=0\) as it is a division algebra,and hence \(\mathbb {Q}\subseteq D\), meaning \(\mathbb {R}\subseteq D_{\mathbb {R}}\), and if \(x\in \mathbb {R}^\times \subseteq D_{\mathbb {R}}^\times \) then \(\delta (x)=|x|^d\) with \(d=\dim _{\mathbb {Q}}(D)\), as multiplication by \(x\) is just scaling by a factor of \(x\) on \(D_{\mathbb {R}}\cong \mathbb {R}^d\). In particular we can set \(x=y^{1/d}\).
In this generality the quotient might not be Hausdorff.
If \(U\) is an open subgroup of \((D\otimes _K \mathbb {A}_K^\infty )^\times \) then the double coset space \(D^\times \backslash (D\otimes _K \mathbb {A}_K^\infty )^\times /U\) is finite.
The double cosets give a disjoint open cover of \((D\otimes _K \mathbb {A}_K^\infty )\) which descends to a disjoint open cover of the quotient space \(D^\times \backslash (D\otimes _K \mathbb {A}_K^\infty )^\times \). However this space is compact by theorem 10.15.