Strong approximation

Abstract

We have already seen in several places in this book how theorems about quaternion algebras over global fields are often first investigated locally, and then a global result is recovered using some form of approximation. Approximation provides a way to transfer analytic properties (encoded in congruences or bounds) into global elements. In this chapter, we develop robust approximation theorems and investigate their arithmetic applications.

1 $\triangleright$ Beginnings

We have already seen in several places in this book how theorems about quaternion algebras over global fields are often first investigated locally, and then a global result is recovered using some form of approximation. Approximation provides a way to transfer analytic properties (encoded in congruences or bounds) into global elements. In this chapter, we develop robust approximation theorems and investigate their arithmetic applications.

We begin by reviewing weak and strong approximation over $\mathbb Q$, taking a breath in preparation for the idelic efforts to come.

28.1.1

The starting point is the Sun Zi theorem (CRT): given a finite, nonempty set ${{{\texttt {\textit{S}}}}}$ of primes, and for each $p \in {{{\texttt {\textit{S}}}}}$ an exponent $n_p \in \mathbb Z _{\ge 1}$ and an element $x_p \in \mathbb Z /p^{n_p}\mathbb Z$, there exists $x \in \mathbb Z$ such that $x \equiv x_p \pmod {p^{n_p}}$ for all $p \in {{{\texttt {\textit{S}}}}}$. These congruences can be equivalently formulated in the p-adic metric by lifting to $x_p \in \mathbb Z _p$ and asking for $|x-x_p \,|<p^{-n_p}$ for $p \in {{{\texttt {\textit{S}}}}}$; or equivalently, the map $\mathbb Z \rightarrow \prod _{p \in {{{\texttt {\textit{S}}}}}} \mathbb Z _p$ has dense image for any finite, nonempty set of primes ${{{\texttt {\textit{S}}}}}$ (giving the target the product topology). We may therefore think of the CRT as an approximation theorem, in the sense that it allows us to find an integer that simultaneously approximates a finite number of p-adic integers arbitrarily well.

We may generalize 28.1.1 (recalling notation from section 27.1, in particular 27.1.8) as follows.

Theorem 28.1.2

(Weak approximation). Let ${{{\texttt {\textit{S}}}}}\subseteq {{\,\mathrm{Pl}\,}}\mathbb Q$ be a finite, nonempty set of places of $\mathbb Q$. Then the image of $\mathbb Q \hookrightarrow \mathbb Q _{{{{\texttt {\textit{S}}}}}} :=\prod _{v \in {{{\texttt {\textit{S}}}}}} \mathbb Q _v$ is dense.

Proof. Let $x_v \in \mathbb Q _v$ for each $v \in {{{\texttt {\textit{S}}}}}$, and let $\epsilon >0$. We want to show

$$\begin{aligned} \text {there exists }x \in \mathbb Q \text { such that }|x-x_v \,|_v<\epsilon \text { for all }v \in {{{\texttt {\textit{S}}}}}\text {.} \end{aligned}$$

(28.1.3)

We proceed by considering increasingly more general cases; our intent is to communicate (concrete, effective) meaning behind the symbols and to set us up to prove strong approximation below; for a short, uniform proof (which reproves the CRT), see Exercise 28.1.

Case 1. Suppose $\infty \not \in {{{\texttt {\textit{S}}}}}$ and $x_p \in \mathbb Z _p$ for all $p \in {{{\texttt {\textit{S}}}}}$. Then (28.1.3) holds by the CRT, as in 28.1.1; in fact, we have infinitely many $x \in \mathbb Z$ for this purpose.

Case 2. Suppose $\infty \not \in {{{\texttt {\textit{S}}}}}$, but $x_p \in \mathbb Q _p$ for $p \in {{{\texttt {\textit{S}}}}}$. We employ continuity of multiplication to reduce to the previous case, as follows. We consider the least common denominator:

$$\begin{aligned} d :=\prod _{p \in {{{\texttt {\textit{S}}}}}} p^{\max (0,-v_p(x_p))} \in \mathbb Z _{>0}. \end{aligned}$$

(28.1.4)

Then $dx_p \in \mathbb Z _p$ for all $p \in {{{\texttt {\textit{S}}}}}$. By the case just established by the CRT, there exists $x' \in \mathbb Z$ such that $|x'-dx_p \,|_p<\epsilon |d \,|_p$ for all $p \in {{{\texttt {\textit{S}}}}}$, so taking $x :=x'/d$ and dividing through we conclude that (28.1.3) holds.

Case 3. To conclude, suppose $\infty \in {{{\texttt {\textit{S}}}}}$. We employ an additive translation: we find a rational number close to $x_\infty$ and add to it a small solution to the previous case, as follows. Since $\mathbb Q \subseteq \mathbb R$ is dense, there exists $y \in \mathbb Q$ such that $|y-x_\infty \,|_\infty <\epsilon /2$. Let $y_p :=x_p-y$. From case 2, we find $y' \in \mathbb Q$ such that $|y'-y_p \,|_p < \epsilon$ for all $p \in {{{\texttt {\textit{S}}}}}$. By case 1, there exist infinitely many $m \in \mathbb Z$ such that $|1-m \,|_p < \min (1,|y'-y_p \,|_p)$ for all $p \in {{{\texttt {\textit{S}}}}}$; for such m, we have $m \equiv 1 ~(\text{ mod } ~{p})$ so $|m \,|_p=1$ and

$$\begin{aligned} |(y'/m)-y_p \,|_p=\frac{|y'-my_p \,|_p}{|m \,|_p} = |y'-y_p+(1-m)y_p \,|_p = |y'-y_p \,|_p < \epsilon \end{aligned}$$

(28.1.5)

by the ultrametric inequality. By choosing m large enough, we may ensure that $|y'/m \,|_\infty < \epsilon /2$. Let $x :=y'/m + y$. Then by (28.1.5)

$$\begin{aligned} |x-x_p \,|_p = |(y'/m) + y - (y_p+y) \,|_p = |(y'/m)-y_p \,|_p < \epsilon \end{aligned}$$

(28.1.6)

and

$$\begin{aligned} |x-x_\infty \,|_\infty \le |x-y \,|_\infty + |y-x_\infty \,|_\infty = |y'/m \,|_\infty + \epsilon /2 < \epsilon \end{aligned}$$

(28.1.7)

proving (28.1.3).$\square$

Theorem 28.1.8

(Strong approximation). Let ${{{\texttt {\textit{S}}}}}\subseteq {{\,\mathrm{Pl}\,}}\mathbb Q$ be a nonempty set of places. Then the image of is dense.

28.1.9

Written out in the standard basis of open sets, strong approximation is equivalent to: given a finite set ${{{\texttt {\textit{T}}}}}\subseteq {{\,\mathrm{Pl}\,}}\mathbb Q$ disjoint from ${{{\texttt {\textit{S}}}}}$, elements $x_v \in \mathbb Q _v$ for $v \in {{{\texttt {\textit{T}}}}}$, and $\epsilon >0$, there exists $x \in \mathbb Q$ such that $|x-x_v \,|_v<\epsilon$ for all $v \in {{{\texttt {\textit{T}}}}}$ and $x \in \mathbb Z _p$ for all $p \not \in {{{\texttt {\textit{S}}}}}\sqcup {{{\texttt {\textit{T}}}}}$ with $p \ne \infty$.

Weak approximation follows from strong approximation by forgetting ${{{\texttt {\textit{S}}}}}$ and weakly approximating $x_v$ for $v \in {{{\texttt {\textit{T}}}}}$. Indeed, the difference between the ‘weak’ and the ‘strong’ is meaningful here. In weak approximation, we satisfy only a finite number of conditions, with no control over the rational number at places $v \not \in {{{\texttt {\textit{S}}}}}$. By contrast, in strong approximation, the role of the set ${{{\texttt {\textit{S}}}}}$ is switched, and have specified conditions at all primes $v \not \in {{{\texttt {\textit{S}}}}}$, either by approximation at finitely many places in ${{{\texttt {\textit{T}}}}}$ as in 28.1.9 or by the assertion of integrality at the rest. (This results in some asymmetry in the conclusion for the real place; we could restore this by defining $\mathbb Z _\infty :=(-1,1)$, but this would not add content as we could just include this interval in ${{{\texttt {\textit{T}}}}}$.)

Proof of Theorem 28.1.8.

We prove the statement in its formulation 28.1.9. Naturally, we return to the proof of weak approximation. Without loss of generality (proving a stronger statement), we may assume $\#{{{\texttt {\textit{S}}}}}=1$.

If ${{{\texttt {\textit{S}}}}}=\{\infty \}$, we apply step 2 of weak approximation over the set ${{{\texttt {\textit{T}}}}}$: the result $x = x'/d$ already has $x \in \mathbb Z _q$ for $q \not \in {{{\texttt {\textit{T}}}}}$, since then $q \not \mid d$.

So suppose ${{{\texttt {\textit{S}}}}}=\{\ell \}$ with $\ell$ prime. We return to case 3. To define y, we note instead that $\mathbb Z [1/p] \subseteq \mathbb R$ is dense, so we may take $y \in \mathbb Z [1/\ell ]$, so in particular $y \in \mathbb Z _q$ for $q \ne \ell$. We just showed that we may take $y' \in \mathbb Z _q$ for $q \not \in {{{\texttt {\textit{T}}}}}$. And for the integers m, we claim we may take $m=\ell ^k$ for $k \in \mathbb Z _{\ge 0}$: indeed, we are applying case 1 (CRT) and, as in 28.1.1, this asks for $m \equiv 1 \pmod {p^{n_p}}$ for $p \in {{{\texttt {\textit{T}}}}}$ (with $n_p \in \mathbb Z _{\ge 1}$, and $\ell \ne p$), so we just need to take k to be a common multiple of the orders of $\ell \in (\mathbb Z /p^{n_p}\mathbb Z )^\times$. With this strengthening, we have $x = y'/m + y \in \mathbb Z _q$ for all $q \not \in {{{\texttt {\textit{S}}}}}\sqcup {{{\texttt {\textit{T}}}}}$. $\square$

28.1.10

As is hopefully evident from the proof of strong approximation when ${{{\texttt {\textit{S}}}}}=\{\infty \}$, aside from continuity of multiplication, the key statement needed was that the map $\mathbb Z \rightarrow \mathbb Z /m\mathbb Z$ is surjective for all $m \in \mathbb Z$, as provided by the CRT. Or more zippily, what is needed is that the image of $\mathbb Z \hookrightarrow \widehat{\mathbb{Z }}$ is dense.

28.1.11

In weak approximation, we can replace the additive group $\mathbb Q$ with with the multiplicative group $\mathbb Q ^\times$: the image of $\mathbb Q ^\times \hookrightarrow \prod _{v \in {{{\texttt {\textit{S}}}}}} \mathbb Q _v^\times$ is dense a fortiori.

However, the embedding is not dense: that is to say, we do not have strong approximation for $\mathbb Q ^\times$. Indeed, taking ${{{\texttt {\textit{S}}}}}=\{\infty \}$ we have ; and since $\widehat{\mathbb{Z }}^\times \cap \mathbb Q ^\times = \mathbb Z ^\times =\{\pm 1\}$, the open set $\widehat{\mathbb{Z }}^\times \smallsetminus \{\pm 1\}$ is disjoint from $\mathbb Q ^\times$. In view of 28.1.10, the problem is also indicated by the fact that $\mathbb Z ^\times =\{\pm 1\}$ does not surject onto $(\mathbb Z /m\mathbb Z )^\times$ for $m \ge 7$.

2 $\triangleright$ Strong approximation for ${{\,\mathrm{SL}\,}}_2(\mathbb Q )$

We now consider approximation in the noncommutative context. For motivation in this second phase of the introduction, we consider the simplest case where $B={{\,\mathrm{M}\,}}_2(\mathbb Q )$ and take ${{{\texttt {\textit{S}}}}}=\{\infty \}$, and $\underline{\mathbb{Q }}_{\not {S}} = \widehat{\mathbb{Q }}$; by analogy, this is like considering a noncommutative generalization of the CRT.

28.2.1

Recall that $B_v={{\,\mathrm{M}\,}}_2(\mathbb Q _v) \simeq \mathbb Q _v^4$ has the coordinate topology (see section 13.5); therefore weak and strong approximation for $B={{\,\mathrm{M}\,}}_2(\mathbb Q )$ follow from these statements for $\mathbb Q$, and weak approximation for ${{\,\mathrm{GL}\,}}_2(\mathbb Q )$ follows as the determinant is continuous.

28.2.2

We should not expect the embedding ${{\,\mathrm{GL}\,}}_2(\mathbb Q ) \hookrightarrow {{\,\mathrm{GL}\,}}_2(\widehat{\mathbb{Q }})$ to be dense any more than it was for $\mathbb Q ^\times ={{\,\mathrm{GL}\,}}_1(\mathbb Q )$, as in 28.1.11. In fact, we rediscover the same issue by taking determinants: the map ${{\,\mathrm{GL}\,}}_2(\mathbb Z ) \rightarrow {{\,\mathrm{GL}\,}}_2(\mathbb Z /m\mathbb Z )$ is not surjective, because $\det ({{\,\mathrm{GL}\,}}_2(\mathbb Z ))=\pm 1$ whereas $\det ({{\,\mathrm{GL}\,}}_2(\mathbb Z /m\mathbb Z ))=(\mathbb Z /m\mathbb Z )^\times$.

Once we restrict to the subgroup of determinant 1, we find a dense subgroup once again.

Theorem 28.2.3

The image of ${{\,\mathrm{SL}\,}}_2(\mathbb Q ) \hookrightarrow {{\,\mathrm{SL}\,}}_2(\widehat{\mathbb{Q }})$ is dense.

Theorem 28.2.3 is known as strong approximation for the group ${{\,\mathrm{SL}\,}}_2(\mathbb Q )$. We give a quick proof of Theorem 28.2.3 in two steps. In preparation, we recall from 27.2.6 that

$$\begin{aligned} \widehat{\mathbb{Q }}^\times = \mathbb Q ^\times \widehat{\mathbb{Z }}^\times \end{aligned}$$

(“denominators can be handled globally”) and prove an analogous decomposition.

Lemma 28.2.4

We have

$$\begin{aligned} \begin{aligned} {{\,\mathrm{GL}\,}}_2(\widehat{\mathbb{Q }})&= {{\,\mathrm{GL}\,}}_2(\mathbb Q ){{\,\mathrm{GL}\,}}_2(\widehat{\mathbb{Z }}) \\ {{\,\mathrm{SL}\,}}_2(\widehat{\mathbb{Q }})&={{\,\mathrm{SL}\,}}_2(\mathbb Q ){{\,\mathrm{SL}\,}}_2(\widehat{\mathbb{Z }}). \end{aligned} \end{aligned}$$

(28.2.5)

Proof. We begin with the first statement. The inclusion ${{\,\mathrm{GL}\,}}_2(\mathbb Q ){{\,\mathrm{GL}\,}}_2(\widehat{\mathbb{Z }}) \subseteq {{\,\mathrm{GL}\,}}_2(\widehat{\mathbb{Q }})$ holds; we prove the other containment. Let $\widehat{\alpha }\in {{\,\mathrm{GL}\,}}_2(\widehat{\mathbb{Q }})$. Consider the collection of lattices $(L_p)_p$ with $L_p=\alpha _p \mathbb Z _p^2 \subseteq \mathbb Q _p^2$. Since $\alpha _p \in {{\,\mathrm{GL}\,}}_2(\mathbb Z _p)$ for all but finitely many p, we have $L_p=\mathbb Z _p^2$ for all but finitely many p. By the local-global dictionary for lattices (Theorem 9.4.9), there exists a unique lattice $L \subseteq \mathbb Q ^2$ whose completions are $L_p$. We now rephrase this adelically (and succinctly): letting $\widehat{L}=\widehat{\alpha }\widehat{\mathbb{Z }}^2 \subseteq \widehat{\mathbb{Q }}^2$, we take $L=\widehat{L} \cap \mathbb Q ^2$. Choose a basis for L and put the columns in a matrix $\alpha$, so $L=\alpha \mathbb Z ^2$. Then $\widehat{L}=\alpha \widehat{\mathbb{Z }}^2 = \widehat{\alpha }\widehat{\mathbb{Z }}^2$, and there exists $\gamma \in {{\,\mathrm{GL}\,}}_2(\widehat{\mathbb{Z }})$ such that $\widehat{\alpha }= \alpha \widehat{\gamma }$. This completes the inclusion.

To get down to ${{\,\mathrm{SL}\,}}_2$, we take determinants. Let $\widehat{\alpha }\in {{\,\mathrm{SL}\,}}_2(\widehat{\mathbb{Q }})$ and write it as $\widehat{\alpha }= \alpha \widehat{\gamma }$ with $\alpha \in {{\,\mathrm{GL}\,}}_2(\mathbb Q )$ and $\widehat{\gamma }\in {{\,\mathrm{GL}\,}}_2(\widehat{\mathbb{Z }})$. Then

$$\begin{aligned} 1=\det (\widehat{\alpha }) = \det (\alpha )\det (\widehat{\gamma }) \in \mathbb Q ^\times \widehat{\mathbb{Z }}^\times = \widehat{\mathbb{Q }}^\times \end{aligned}$$

but $\mathbb Q ^\times \cap \widehat{\mathbb{Z }}^\times = \{\pm 1\}$; multiplying both $\alpha ,\widehat{\gamma }$ by $\begin{pmatrix} -1 &{} 0 \\ 0 &{} 1 \end{pmatrix}$ on the right and left respectively, if necessary, we may take $\det (\alpha )=\det (\widehat{\gamma })=1$, i.e., $\alpha \in {{\,\mathrm{SL}\,}}_2(\mathbb Q )$ and $\widehat{\gamma }\in {{\,\mathrm{SL}\,}}_2(\widehat{\mathbb{Z }})$.$\square$

Now for the slightly magical second step.

Theorem 28.2.6

The map

$$\begin{aligned} {{\,\mathrm{SL}\,}}_2(\mathbb Z ) \rightarrow {{\,\mathrm{SL}\,}}_2(\mathbb Z /m\mathbb Z ) \end{aligned}$$

is surjective for all $m \in \mathbb Z$; equivalently, the image of ${{\,\mathrm{SL}\,}}_2(\mathbb Z ) \hookrightarrow {{\,\mathrm{SL}\,}}_2(\widehat{\mathbb{Z }})$ is dense.

The statement is nontrivial: a matrix modulo m can certainly be lifted to a matrix in $\mathbb Z$ whose determinant will be congruent to 1 modulo m, but the hard part is to ensure that the lifted matrix has determinant equal to 1.

Proof of Theorem 28.2.6.

Let $\alpha \in {{\,\mathrm{M}\,}}_2(\mathbb Z )$ be such that $\alpha$ maps to the desired matrix in ${{\,\mathrm{SL}\,}}_2(\mathbb Z /m\mathbb Z )$, so in particular $\det (\alpha ) \equiv 1 \pmod {m}$. By the theory of elementary divisors (Smith normal form), there exist matrices $\mu ,\nu \in {{\,\mathrm{SL}\,}}_2(\mathbb Z )$ such that $\mu \alpha \nu$ is diagonal; so without loss of generality, we may suppose that $\alpha =\begin{pmatrix} a &{} 0 \\ 0 &{} b \end{pmatrix}$ with $ab \equiv 1 \pmod {m}$. Let

$$\begin{aligned} \alpha ' = \begin{pmatrix} a &{} -(1-ab) \\ 1-ab &{} b(2-ab) \end{pmatrix}. \end{aligned}$$

(28.2.7)

Then $\alpha ' \equiv \alpha \pmod {m}$ and

$$\begin{aligned} \det (\alpha ') = ab(2-ab)+(1-ab)^2 = 1 \end{aligned}$$

(28.2.8)

so $\alpha ' \in {{\,\mathrm{SL}\,}}_2(\mathbb Z )$, as claimed. (Compare Shimura [Shi71, Lemma 1.38].) $\square$

Remark 28.2.9. The proof of Theorem 28.2.6 extends in two ways. First, we can replace $\mathbb Z$ with a PID and the same proof works. Second, arguing by induction, one can show that the map ${{\,\mathrm{SL}\,}}_n(\mathbb Z ) \rightarrow {{\,\mathrm{SL}\,}}_n(\mathbb Z /m\mathbb Z )$ is surjective for all $n \ge 2$ and $m \in \mathbb Z$.

We are now ready to prove strong approximation for ${{\,\mathrm{SL}\,}}_2(\mathbb Q )$.

Proof of Theorem 28.2.3.

Consider the closure of ${{\,\mathrm{SL}\,}}_2(\mathbb Q )$ in ${{\,\mathrm{SL}\,}}_2(\widehat{\mathbb{Q }})$ in the idelic topology; we obtain a closed subgroup. Since ${{\,\mathrm{SL}\,}}_2(\mathbb Q ) \ge {{\,\mathrm{SL}\,}}_2(\mathbb Z )$, by Theorem 28.2.6 the closure contains ${{\,\mathrm{SL}\,}}_2(\widehat{\mathbb{Z }})$, but then by Lemma 28.2.4, it contains all of ${{\,\mathrm{SL}\,}}_2(\mathbb Q ){{\,\mathrm{SL}\,}}_2(\widehat{\mathbb{Z }})={{\,\mathrm{SL}\,}}_2(\widehat{\mathbb{Q }})$! Therefore ${{\,\mathrm{SL}\,}}_2(\mathbb Q ) \le {{\,\mathrm{SL}\,}}_2(\widehat{\mathbb{Q }})$ is dense. $\square$

With the preceding context, we are now ready to state a more general formulation of strong approximation for indefinite quaternion algebras over $\mathbb Q$. The following theorem is a special case of Main Theorem 28.5.3.

Theorem 28.2.10

(Strong approximation). Let B be an indefinite quaternion algebra over $\mathbb Q$. Then $B^1$ is dense in $\widehat{B}^1$.

If B is definite, then $\mathcal {O}^1$ is a finite group () and so cannot be dense in the infinite idelic group $\widehat{B}^1$. This theorem has the following important applications.

Theorem 28.2.11

Let $\mathcal {O}$ be an Eichler order in an indefinite quaternion algebra over $\mathbb Q$. Then the following statements hold.

• (a) Every locally principal right $\mathcal {O}$-ideal is in fact principal, i.e., $\#{{\,\mathrm{Cls}\,}}\mathcal {O}=1$.

• (b) Every order $\mathcal {O}'$ locally isomorphic to $\mathcal {O}$ is in fact isomorphic to $\mathcal {O}$, i.e., $\#{{\,\mathrm{Typ}\,}}\mathcal {O}=1$.

• (c) For any integer m, the reduction map $\mathcal {O}^1 \rightarrow (\mathcal {O}/m\mathcal {O})^1$ is surjective.

Proof. Specialize Main Theorem 28.5.3, Corollary 28.5.6, and Corollary 28.5.14, respectively, using 28.5.16.$\square$

3 Elementary matrices

Before embarking on our more general idelic quest, we pause to give a second proof of strong approximation for ${{\,\mathrm{SL}\,}}_2$ using elementary matrices.

28.3.1

Let R be a domain. An elementary matrix  (or transvection) in ${{\,\mathrm{SL}\,}}_n(R)$ is a matrix which differs from the identity in one off-diagonal entry; such a matrix acts by an elementary row operation (add a multiple of a row to a different row) on the left and by an elementary column operation on the right. For $n=2$, the elementary matrices are those of the form $\begin{pmatrix} 1 &{} b \\ 0 &{} 1 \end{pmatrix}$ or $\begin{pmatrix} 1 &{} 0 \\ c &{} 1 \end{pmatrix}$ with $b,c \in F$.

28.3.2

If F is a field, then ${{\,\mathrm{SL}\,}}_n(F)$ is generated by elementary matrices by the theory of echelon forms (Exercise 28.3).

Lemma 28.3.3

Let R be a Euclidean domain. Then ${{\,\mathrm{SL}\,}}_2(R)$ is generated by elementary matrices.

Proof. The calculation

$$\begin{aligned} \begin{pmatrix} 1 &{} 1 \\ 0 &{} 1 \end{pmatrix} \begin{pmatrix} 1 &{} 0 \\ -1 &{} 1 \end{pmatrix} \begin{pmatrix} 1 &{} 1 \\ 0 &{} 1 \end{pmatrix} = \begin{pmatrix} 0 &{} 1 \\ -1 &{} 0 \end{pmatrix} =:\eta \end{aligned}$$

shows that $\eta$ is in the subgroup of elementary matrices.

We now follow the usual proof of the elementary divisor theorem. Let $\alpha =\begin{pmatrix} a &{} b \\ c &{} d \end{pmatrix} \in {{\,\mathrm{SL}\,}}_2(R)$. First, suppose $b=0$. Then $\det (\alpha )=ad=1$; adding a times the second row to the first, we may suppose $b=1$; then multiplying by $\eta$ on the right we may suppose $a=1$; elementary row and column operations then give $b=c=0$, and then $d=1$. Similarly, if $a=0$, multiplying by $\eta$ gives $b=0$ and we repeat.

So we may suppose $a,b \ne 0$. By the Euclidean algorithm under the norm N, there exists $q,r \in R$ such that $a=bq+r$ and $N(r)<N(b)$. Applying the elementary matrix which adds $-q$ times the second column to the first, we may suppose $a=0$ or $N(a)<N(b)$. If $a=0$, we are done by the previous paragraph; otherwise, we multiply on the right by $\eta$ which swaps columns, and repeat. Because N takes nonnegative integer values, this procedure terminates after finitely many steps.$\square$

Remark 28.3.4. Lemma 28.3.3 holds for general $n \ge 2$, and it follows from the above by induction: see Exercise 28.4.

This theory of elementary matrices has the following striking consequence.

Proposition 28.3.5

Let R be a Dedekind domain. Then for all ideals $\mathfrak m \subseteq R$, the map

$$\begin{aligned} {{\,\mathrm{SL}\,}}_2(R) \rightarrow {{\,\mathrm{SL}\,}}_2(R/\mathfrak m ) \end{aligned}$$

is surjective.

Proof. We may suppose $\mathfrak m$ is nonzero. Then by the CRT, $R/\mathfrak m$ is a finite product of local Artinian principal ideal rings. Therefore $R/\mathfrak m$ is Euclidean and by a generalization of Lemma 28.3.3, the group ${{\,\mathrm{SL}\,}}_2(R/\mathfrak m )$ is generated by elementary matrices: see Exercise 28.6. Every elementary matrix in ${{\,\mathrm{SL}\,}}_2(R/\mathfrak m )$ lifts to an elementary matrix in ${{\,\mathrm{SL}\,}}_2(R)$, and the statement follows.$\square$

Corollary 28.3.6

Let F be a global field, and let $R \subseteq F$ be a global ring with eligible set ${{{\texttt {\textit{S}}}}}$. Then the image of the map

is dense.

Proof. For brevity, we write and $\widehat{R}=\prod _{v\not \in {{{\texttt {\textit{S}}}}}} R_v$. We first show that ${{\,\mathrm{SL}\,}}_2(R) \hookrightarrow {{\,\mathrm{SL}\,}}_2(\widehat{R})$ is dense. If $U \subseteq {{\,\mathrm{SL}\,}}_2(\widehat{R})$ is open, then U contains a standard open neighborhood of the form

$$\begin{aligned} \{\widehat{\beta }\in {{\,\mathrm{SL}\,}}_2(\widehat{R}) : \widehat{\beta }\equiv \alpha _\mathfrak m ~(\text{ mod } ~\mathfrak{m })\} \end{aligned}$$

for some $\alpha _\mathfrak m \in {{\,\mathrm{SL}\,}}_2(R/\mathfrak m )$ and $\mathfrak m \subseteq R$. The surjectivity in Proposition 28.3.5 then implies that $U \cap {{\,\mathrm{SL}\,}}_2(R) \ne \emptyset$.

For the statement itself, we again argue with elementary matrices. Let $\widehat{\alpha }=(\alpha _v)_v \in {{\,\mathrm{SL}\,}}_2(\widehat{F})$; then $\alpha _v \in {{\,\mathrm{SL}\,}}_2(R_v)$ for all but finitely many v. For these finitely many v, we know ${{\,\mathrm{SL}\,}}_2(F_v)$ is generated by elementary matrices by Lemma 28.3.2, so by strong approximation in F (Lemma 28.7.2) we can approximate $\alpha _v$ by an element of ${{\,\mathrm{SL}\,}}_2(F)$ that belongs to any open neighborhood of $\alpha _v$; for the remaining places we apply the previous paragraph, and we finish using the continuity of multiplication.$\square$

4 Strong approximation and the ideal class set

In this section, we provide one more motivation for strong approximation, relating it to the ideal class set as previewed in Eichler’s theorem (see section 17.8).

We adopt the following notation for the rest of this chapter. Let R be a global ring with eligible set ${{{\texttt {\textit{S}}}}}$ and $F={{\,\mathrm{Frac}\,}}R$ its global field. Let B be a quaternion algebra over F and let $\mathcal {O}\subseteq B$ be an R-order.

28.4.1

By 27.7.1, the reduced norm map

$$\begin{aligned} {{\,\mathrm{nrd}\,}}:{{\,\mathrm{Cls}\,}}\mathcal {O}= B^\times \backslash \widehat{B}^\times \!/ \widehat{\mathcal {O}}^\times \rightarrow F_{>_{\Omega } 0}^\times \backslash \widehat{F}^\times \!/{{\,\mathrm{nrd}\,}}(\widehat{\mathcal {O}}^\times ) \end{aligned}$$

(28.4.2)

is surjective. Then by class field theory 27.7.4, the codomain $F_{>_{\Omega } 0}^\times \backslash \widehat{F}^\times \!/{{\,\mathrm{nrd}\,}}(\widehat{\mathcal {O}}^\times )={{\,\mathrm{Cl}\,}}_{G(\mathcal {O})} R$ admits a description as a class group.

The important point that we will soon see: the reduced norm map is injective, therefore bijective, when strong approximation holds for the group $B^1$. But before we get there, we have some explaining to do.

We now investigate the injectivity of the reduced norm map (28.4.2). This map is only a map of (pointed) sets, so first we show that it suffices to look at an appropriate kernel.

28.4.3

For all $\widehat{\beta }\in \widehat{B}^\times$, the map $\widehat{\alpha }\mathcal {O}\mapsto \widehat{\alpha }\mathcal {O}\widehat{\beta }$ gives a bijection

$$\begin{aligned} {{\,\mathrm{Cls}\,}}\mathcal {O}= B^\times \backslash \widehat{B}^\times / \widehat{\mathcal {O}}^\times \leftrightarrow B^\times \backslash \widehat{B}^\times / \widehat{\mathcal {O}}'^\times = {{\,\mathrm{Cls}\,}}\mathcal {O}' \end{aligned}$$

where $\mathcal {O}' :=B \cap \widehat{\beta }^{-1} \widehat{\mathcal {O}}\widehat{\beta }$ is connected (locally isomorphic) to $\mathcal {O}$. So it is sensible to consider the maps (28.4.2) for all orders $\mathcal {O}'$ connected to $\mathcal {O}$, i.e., the entire genus ${{\,\mathrm{Gen}\,}}\mathcal {O}$. We recall that the type set is finite (Corollary 27.6.25, or Main Theorem 17.7.1 in the number field case using the geometry of numbers).

Our investigations will involve the kernels of the reduced norm maps:

$$\begin{aligned} B^1 :=\{\alpha \in B^\times : {{\,\mathrm{nrd}\,}}(\alpha )=1\} \le \widehat{B}^1 :=\{\widehat{\alpha }\in \widehat{B}^\times : {{\,\mathrm{nrd}\,}}(\widehat{\alpha })=1\} \end{aligned}$$

(28.4.4)

Example 28.4.5

If $B=({a,b} \mid {F})$, then $B^1$ admits the Diophantine description

$$\begin{aligned} B^1 \simeq \{(x,y,z,w) \in F^4 : x^2 - ay^2 - bz^2 + abw^2 = 1\}, \end{aligned}$$

and the group $\widehat{B}^1$ consists of local solutions at all primes that belong to $R_v^4$ for almost all places $v \in {{\,\mathrm{Pl}\,}}F$.

Lemma 28.4.6

Let $\mathcal {O}\subseteq B$ be an R-order. Then the reduced norm map (28.4.2) is injective for all orders $\mathcal {O}' \in {{\,\mathrm{Gen}\,}}\mathcal {O}$ if and only if $\widehat{B}^1 \subseteq B^\times \widehat{\mathcal {O}}'^\times$ for all $\mathcal {O}' \in {{\,\mathrm{Gen}\,}}\mathcal {O}$.

Proof. If (28.4.2) is injective, then given $\widehat{\alpha }\in \widehat{B}^1$ we have ${{\,\mathrm{nrd}\,}}(\widehat{\alpha }\widehat{\mathcal {O}}^\times ) = {{\,\mathrm{nrd}\,}}(\widehat{\mathcal {O}}^\times )$ so $\widehat{\alpha }\widehat{\mathcal {O}}^\times = z \widehat{\mathcal {O}}^\times$ for some $z \in B^\times$ and $\widehat{\alpha }\in z \widehat{\mathcal {O}}^\times \subseteq B^\times \widehat{\mathcal {O}}^\times$.

For the converse, since ${{\,\mathrm{nrd}\,}}:B^\times \rightarrow F_{>_{\Omega } 0}^\times$ and ${{\,\mathrm{nrd}\,}}:\widehat{\mathcal {O}}^\times \rightarrow {{\,\mathrm{nrd}\,}}(\widehat{\mathcal {O}}^\times )$ are both surjective, to show ${{\,\mathrm{nrd}\,}}$ is injective for $\mathcal {O}$ we may show that if ${{\,\mathrm{nrd}\,}}(\widehat{\alpha })={{\,\mathrm{nrd}\,}}(\widehat{\beta }) \in \widehat{F}^\times$ then $\widehat{\alpha }\widehat{\mathcal {O}}^\times = z \widehat{\beta }\widehat{\mathcal {O}}^\times$ for some $z \in B^\times$. We consider $(\widehat{\alpha }\widehat{\beta }^{-1})(\widehat{\beta }\widehat{\mathcal {O}}\widehat{\beta }^{-1}) = (\widehat{\alpha }\widehat{\beta }^{-1}) \widehat{\mathcal {O}}'$ where as above $\mathcal {O}'=B \cap \widehat{\beta }\widehat{\mathcal {O}}\widehat{\beta }^{-1} \in {{\,\mathrm{Gen}\,}}\mathcal {O}$. Since $\widehat{\alpha }\widehat{\beta }^{-1} \in \widehat{B}^1$, by hypothesis $\widehat{\alpha }\widehat{\beta }^{-1} = z \widehat{\mu }' = z (\widehat{\beta }\widehat{\mu }\widehat{\beta }^{-1})$ where $z \in B^\times$ and $\widehat{\mu }\in \widehat{\mathcal {O}}^\times$, and consequently $\widehat{\alpha }\widehat{\mathcal {O}}= z \widehat{\beta }\widehat{\mu }\widehat{\mathcal {O}}= z \widehat{\beta }\widehat{\mathcal {O}}$, and hence the map is injective.$\square$

28.4.7

We have $B^\times \widehat{\mathcal {O}}^\times \cap \widehat{B}^1 = B^1 \widehat{\mathcal {O}}^1$ if and only if ${{\,\mathrm{nrd}\,}}(\mathcal {O}^\times ) = F_{>_{\Omega } 0}^\times \cap {{\,\mathrm{nrd}\,}}(\widehat{\mathcal {O}}^\times )$ (Exercise 28.10).

28.4.8

Suppose that $B^1$ is dense in $\widehat{B}^1$. Then we claim that $\widehat{B}^1 \subseteq B^1 \widehat{\mathcal {O}}^1 \subseteq B^\times \widehat{\mathcal {O}}^\times$ for all orders $\widehat{\mathcal {O}}$. Indeed, if $\widehat{\alpha }=(\alpha _\mathfrak p )_\mathfrak p \in \widehat{B}^1$ then $\widehat{\alpha }\widehat{\mathcal {O}}^1 \le \widehat{B}^1$ is open, so there exists $\alpha \in B^1$ such that $\alpha =\widehat{\alpha }\widehat{\gamma }$ with $\widehat{\gamma }\in \widehat{\mathcal {O}}^1$, and $\widehat{\alpha }=\alpha \widehat{\gamma }^{-1} \in B^1\widehat{\mathcal {O}}^1$.

We should not expect hypothesis of 28.4.8 to hold for all quaternion algebras: see Exercise 28.7.

5 Statement and first applications

In this section, we set up and state the strong approximation theorem, and then derive some applications. Throughout, we abbreviate .

Definition 28.5.1

We say B is ${{{\texttt {\textit{S}}}}}$-indefinite (or B satisfies the ${{{\texttt {\textit{S}}}}}$-Eichler condition ) if ${{{\texttt {\textit{S}}}}}$ contains a place which is unramified in B.

28.5.2

If F is a number field, then this definition agrees with Definition 17.8.1; and since a complex place is necessarily split and ${{{\texttt {\textit{S}}}}}$ contains the archimedean places, if B is ${{{\texttt {\textit{S}}}}}$-definite over a number field F then F is a totally real number field.

Main Theorem 28.5.3

(Strong approximation). Let B be a quaternion algebra over a global field and suppose B is ${{{\texttt {\textit{S}}}}}$-indefinite. Then $B^1$ is dense in $\widehat{B}^1$.

28.5.4

One can think of strong approximation from the following informal perspective: if $B_v^1$ is not compact, then there is enough room for $B^1$ to “spread out” in $B_v^1$ so that correspondingly $B^1$ is dense in the ${{{\texttt {\textit{S}}}}}$-finite part $\widehat{B}^1$.

The hypothesis that $B_{{{{\texttt {\textit{S}}}}}}^1 = \prod _{v \in {{{\texttt {\textit{S}}}}}} B_v^1$ is noncompact is certainly necessary for the conclusion that $B^1$ is dense in $\widehat{B}^1$. Indeed, if $B_{{{{\texttt {\textit{S}}}}}}^1=\prod _{v \in {{{\texttt {\textit{S}}}}}} B_v^1$ is compact, then since $B^1$ is discrete in $\underline{B}^1$, the subgroup $B^1 B_{{{{\texttt {\textit{S}}}}}}^1 \le \underline{B}^1$ is closed in $\underline{B}^1$, and $B^1 B_{{{{\texttt {\textit{S}}}}}}^1 \ne \underline{B}^1$. On the other hand, if $B^1$ is dense in $\widehat{B}^1$, then adding the components for $v \in {{{\texttt {\textit{S}}}}}$ we have $B^1 B_{{{{\texttt {\textit{S}}}}}}^1 \le \underline{B}^1$ dense. This is a contradiction.

We give two proofs of strong approximation over the next two sections. For the moment, we consider some applications.

Our main motivation for strong approximation is the following proposition. We recall the class group 27.7.4 associated to $\mathcal {O}$.

Theorem 28.5.5

If B is ${{{\texttt {\textit{S}}}}}$-indefinite, then the reduced norm map (28.4.2)

$$\begin{aligned} {{\,\mathrm{nrd}\,}}:{{\,\mathrm{Cls}\,}}\mathcal {O}= B^\times \backslash \widehat{B}^\times /\widehat{\mathcal {O}}^\times \rightarrow {{\,\mathrm{Cl}\,}}_{G(\mathcal {O})} R = F_{>_{\Omega } 0}^\times \backslash \widehat{F}^\times / {{\,\mathrm{nrd}\,}}(\mathcal {O}^\times ) \end{aligned}$$

is a bijection for all R-orders $\mathcal {O}\subseteq B$: in particular, if I is a locally principal right $\mathcal {O}$-ideal, then I is principal if and only if ${{\,\mathrm{nrd}\,}}(I)$ is principal in the class group ${{\,\mathrm{Cl}\,}}_{G(\mathcal {O})} R$.

Proof. Combine Lemma 28.4.6 and 28.4.8.$\square$

Corollary 28.5.6

If B is ${{{\texttt {\textit{S}}}}}$-indefinite and ${{\,\mathrm{Cl}\,}}_{G(\mathcal {O})} R$ is trivial, then ${{\,\mathrm{Typ}\,}}\mathcal {O}$ is trivial, i.e., every order $\mathcal {O}'$ locally isomorphic to $\mathcal {O}$ is in fact isomorphic to $\mathcal {O}$.

Proof. The class set ${{\,\mathrm{Cls}\,}}\mathcal {O}$ maps surjectively onto ${{\,\mathrm{Typ}\,}}\mathcal {O}$ by Lemma 17.4.13, and the latter is trivial by Theorem 28.5.5.$\square$

28.5.7

More generally, we can grapple with the type set of $\mathcal {O}$, measured by a different (generalized) class group. Recall (27.6.24) that

$$\begin{aligned} {{\,\mathrm{Typ}\,}}\mathcal {O}\leftrightarrow B^\times \backslash \widehat{B}^\times / N_{\widehat{B}^\times }(\widehat{\mathcal {O}}). \end{aligned}$$

Let

$$\begin{aligned} GN(\mathcal {O}) :=F_{>_{\Omega } 0}^\times {{\,\mathrm{nrd}\,}}(N_{\widehat{B}^\times }(\widehat{\mathcal {O}})) \le \widehat{F}^\times . \end{aligned}$$

(28.5.8)

Since $\widehat{\mathcal {O}}^\times \le N_{\widehat{B}^\times }(\widehat{\mathcal {O}})$, we have $GN(\mathcal {O}) \ge G(\mathcal {O})$. Define accordingly the class group

$$\begin{aligned} {{\,\mathrm{Cl}\,}}_{GN(\mathcal {O})} R = \widehat{F}^\times /GN(\mathcal {O}). \end{aligned}$$

(28.5.9)

Then there is a surjective map of abelian groups

$$\begin{aligned} {{\,\mathrm{Cl}\,}}_{G(\mathcal {O})} \rightarrow {{\,\mathrm{Cl}\,}}_{GN(\mathcal {O})} R. \end{aligned}$$

Corollary 28.5.10

If B is ${{{\texttt {\textit{S}}}}}$-indefinite, then the reduced norm map induces a bijection

$$\begin{aligned} {{\,\mathrm{nrd}\,}}:{{\,\mathrm{Typ}\,}}\mathcal {O}\xrightarrow {\smash {{\sim }}}{{\,\mathrm{Cl}\,}}_{GN(\mathcal {O})} R. \end{aligned}$$

Proof. We take the further quotient by the normalizer in the bijection in Theorem 28.5.5.$\square$

28.5.11

Returning to 17.4.16, for $B={{\,\mathrm{M}\,}}_2(F)$ and $\mathcal {O}={{\,\mathrm{M}\,}}_2(R)$ we compute that ${{\,\mathrm{nrd}\,}}(N_{\widehat{B}^\times }(\widehat{\mathcal {O}}))=\widehat{F}^2 \widehat{R}^\times$, so ${{\,\mathrm{Cl}\,}}_{GN(\mathcal {O})} R = {{\,\mathrm{Cl}\,}}R/({{\,\mathrm{Cl}\,}}R)^2$. Thus by Corollary 28.5.10, the types of maximal orders in $\mathcal {O}$ are given by $\begin{pmatrix} R &{} \mathfrak a \\ \mathfrak a ^{-1} &{} R \end{pmatrix}$ for $[\mathfrak a ]$ in a set of representatives of $({{\,\mathrm{Cl}\,}}R)/({{\,\mathrm{Cl}\,}}R)^2$.

Two other immediate applications of strong approximation that served as motivation are now apparent.

Corollary 28.5.12

Suppose B is ${{{\texttt {\textit{S}}}}}$-indefinite. Then

$$\begin{aligned} \widehat{B}^{1} = B^1 \widehat{\mathcal {O}}^1 \quad \text {and} \quad \underline{B}^1 = B^1 \underline{\mathcal {O}}^1.\end{aligned}$$

(28.5.13)

Proof. The inclusion $B^1 \widehat{\mathcal {O}}^1 \subseteq \widehat{B}^1$ holds, and the converse holds when $B^1$ is dense in $\widehat{B}^1$ by 28.4.8. For the second statement, we have $\underline{B}= \widehat{B}\times B_{{{{\texttt {\textit{S}}}}}}$ and $\underline{\mathcal {O}}= \widehat{\mathcal {O}}\times B_{{{{\texttt {\textit{S}}}}}}$, so we take norm 1 units and multiply both sides of (28.5.13) by $B_{{{{\texttt {\textit{S}}}}}}^1=\prod _{v \in {{{\texttt {\textit{S}}}}}} B_v^1$.$\square$

Corollary 28.5.14

Suppose B is ${{{\texttt {\textit{S}}}}}$-indefinite. Let $\mathfrak m \subseteq R$ be an ideal. Then the reduction map

$$\begin{aligned} \mathcal {O}^1 \rightarrow (\mathcal {O}/\mathfrak m \mathcal {O})^1 \end{aligned}$$

is surjective. Moreover, $\mathcal {O}^1$ is dense in $\widehat{\mathcal {O}}^1$.

Proof. For all $\alpha _\mathfrak m \in (\mathcal {O}/\mathfrak m \mathcal {O})^1$, by strong approximation the open set

$$\begin{aligned} \{\widehat{\beta }\in \widehat{\mathcal {O}}^1 : \widehat{\beta }\equiv \alpha _\mathfrak m ~(\text{ mod } ~\mathfrak{m \mathcal {O}})\} \subseteq \widehat{\mathcal {O}}^1 \end{aligned}$$

contains an element $\alpha \in B^1 \cap \widehat{\mathcal {O}}^1 = \mathcal {O}^1$ mapping to $\alpha _\mathfrak m$ in the reduction map. The second statement follows as above from the definition of the topology.$\square$

We now give a name to a large classes of orders where the group $G(\mathcal {O})$ governing principality is explicitly given.

Definition 28.5.15

We say that an R-order $\mathcal {O}\subseteq B$ is locally norm-maximal  if ${{\,\mathrm{nrd}\,}}(\widehat{\mathcal {O}}^\times )=\widehat{R}^\times$.

Equivalently, $\mathcal {O}$ is locally norm-maximal if and only if the reduced norm maps ${{\,\mathrm{nrd}\,}}:\mathcal {O}_\mathfrak p ^\times \rightarrow R_\mathfrak p ^\times$ are surjective for all nonzero primes $\mathfrak p$ of R.

Example 28.5.16

If $\mathcal {O}$ is maximal, then $\mathcal {O}$ is locally norm-maximal (Lemma 13.4.9); more generally if $\mathcal {O}$ is Eichler then $\mathcal {O}$ is locally norm-maximal (Exercise 23.3).

Certain special cases of Theorem 28.5.5 are important in applications. Recall that $\Omega \subseteq {{\,\mathrm{Ram}\,}}B$ is the set of real ramified places, and ${{\,\mathrm{Cl}\,}}_\Omega R$ as defined in 17.8.2 is class group associated to $\Omega$, a quotient of the narrow class group.

Corollary 28.5.17

Suppose F is a number field and let ${{{\texttt {\textit{S}}}}}$ be the set of archimedean places of F. Suppose B is ${{{\texttt {\textit{S}}}}}$-indefinite and $\mathcal {O}\subseteq B$ is locally norm-maximal R-order. Then ${{\,\mathrm{nrd}\,}}:{{\,\mathrm{Cls}\,}}\mathcal {O}\rightarrow {{\,\mathrm{Cl}\,}}_{\Omega } R$ is a bijection.

Proof. This is just a restatement of Theorem 28.5.5 once we note that ${{\,\mathrm{Cl}\,}}_{G(\mathcal {O})} R = {{\,\mathrm{Cl}\,}}_{\Omega } R$ by Example 27.7.7.$\square$

Proposition 28.5.18

Let ${{{\texttt {\textit{T}}}}}\supseteq {{{\texttt {\textit{S}}}}}$ be a set of primes of R that generate ${{\,\mathrm{Cl}\,}}_{G(\mathcal {O})} R$ and suppose B is ${{{\texttt {\textit{T}}}}}$-indefinite. Then every class in ${{\,\mathrm{Cls}\,}}\mathcal {O}$ contains an integral (invertible right) $\mathcal {O}$-ideal whose reduced norm is supported in the set ${{{\texttt {\textit{T}}}}}$.

Proof. Let $R_{({{{\texttt {\textit{T}}}}})}$ denote the (further) localization of R at the primes in ${{{\texttt {\textit{T}}}}}$. We apply Theorem 28.5.5 to the order $\mathcal {O}_{({{{\texttt {\textit{T}}}}})} :=\mathcal {O}\otimes _{R} R_{({{{\texttt {\textit{T}}}}})}$: we conclude that there is a bijection ${{\,\mathrm{Cls}\,}}\mathcal {O}_{({{{\texttt {\textit{T}}}}})} \xrightarrow {\smash {{\sim }}}{{\,\mathrm{Cl}\,}}_{G(\mathcal {O}_{({{{\texttt {\textit{T}}}}})})} R_{({{{\texttt {\textit{T}}}}})}$. But ${{\,\mathrm{Cl}\,}}_{G(\mathcal {O}_{({{{\texttt {\textit{T}}}}})})} R_{({{{\texttt {\textit{T}}}}})}$ is the quotient of ${{\,\mathrm{Cl}\,}}_{G(\mathcal {O})} R$ by the primes in ${{{\texttt {\textit{T}}}}}$, and so by hypothesis is trivial. Therefore if I is a right $\mathcal {O}$-ideal, then $I_{({{{\texttt {\textit{T}}}}})} :=I \otimes _R R_{({{{\texttt {\textit{T}}}}})}$ has $I_{({{{\texttt {\textit{T}}}}})} = \alpha \mathcal {O}_{({{{\texttt {\textit{T}}}}})}$ for some $\alpha \in B^\times$. Let $J=\alpha ^{-1} I$. Then $[J]{}_{\textsf {\tiny {R}} }=[I]{}_{\textsf {\tiny {R}} }$ and $J_\mathfrak p = \mathcal {O}_\mathfrak p$ for all primes $\mathfrak p \not \in {{{\texttt {\textit{T}}}}}$ and so J has reduced norm supported in T. Replacing J by aJ with $a \in R$ nonzero and supported in ${{{\texttt {\textit{T}}}}}$, we may suppose further that $J \subseteq \mathcal {O}$ is integral, and the result follows.$\square$

Example 28.5.19

Let B be a definite quaternion algebra over a totally real (number) field F and let ${{{\texttt {\textit{S}}}}}$ be the set of archimedean places, so $R=\mathbb Z _F$. Let $\mathcal {O}$ be a locally norm-maximal R-order in B. Suppose that ${{\,\mathrm{Cl}\,}}_{\Omega } R = \{1\}$ and let $\mathfrak p \subseteq R$ be a prime of R unramified in B. Then by Proposition 28.5.18, every ideal class in ${{\,\mathrm{Cls}\,}}\mathcal {O}$ contains an integral $\mathcal {O}$-ideal whose reduced norm is a power of $\mathfrak p$.

As a special case, we may take $F=\mathbb Q$: then ${{\,\mathrm{Cl}\,}}_{\Omega } \mathbb Z ={{\,\mathrm{Cl}\,}}^+ \mathbb Z = \{1\}$. Therefore, if B is a definite quaternion algebra of discriminant D over $\mathbb Q$, and $\mathcal {O}\subseteq B$ a locally norm-maximal order (e.g., an Eichler order), then for a prime $p \not \mid D$, every invertible right $\mathcal {O}$-ideal class is represented by an integral ideal whose reduced norm is a power of p.

6 Further applications

We continue with further applications of strong approximation.

Our next consequence of strong approximation is a refinement the Hasse–Schilling theorem on norms (Main Theorem 14.7.4) as follows.

Theorem 28.6.1

(Eichler’s theorem on norms). Suppose B is ${{{\texttt {\textit{S}}}}}$-indefinite, and let $n \in R \cap F_{>_{\Omega } 0}^\times$. Then there exists $\alpha \in B^\times$ integral over R such that ${{\,\mathrm{nrd}\,}}(\alpha )=n$.

Proof. By Main Theorem 14.7.4, there exists $\alpha \in B^\times$ such that ${{\,\mathrm{nrd}\,}}(\alpha )=n$. For each prime $\mathfrak p$, the set

$$\begin{aligned} U_\mathfrak p = \{\beta _\mathfrak p \in B_\mathfrak p ^1 : {{\,\mathrm{trd}\,}}(\beta _\mathfrak p \alpha ) \in R_\mathfrak p \} \end{aligned}$$

(28.6.2)

is (closed and) open since ${{\,\mathrm{trd}\,}}$ is continuous, and further $U_\mathfrak p$ is nonempty: if $\mathfrak p \in {{\,\mathrm{Ram}\,}}B$ then already $\alpha _\mathfrak p$ is integral over $R_\mathfrak p$ and $1 \in U_\mathfrak p$, and otherwise $B_\mathfrak p \simeq {{\,\mathrm{M}\,}}_2(F_\mathfrak p )$ and we may suppose $\alpha _\mathfrak p =\begin{pmatrix} 0 &{} -n \\ 1 &{} t \end{pmatrix}$ is in rational canonical form whereby

$$\begin{aligned} {{\,\mathrm{trd}\,}}\begin{pmatrix} 0 &{} 1 \\ -1 &{} 0 \end{pmatrix} \begin{pmatrix} 0 &{} -n \\ 1 &{} t \end{pmatrix} = {{\,\mathrm{trd}\,}}\begin{pmatrix} 1 &{} t \\ 0 &{} n \end{pmatrix} = n+1 \in R_\mathfrak p \end{aligned}$$

(28.6.3)

shows $\beta _\mathfrak p =\begin{pmatrix} 0 &{} 1 \\ -1 &{} 0 \end{pmatrix} \in U_\mathfrak p$.

Let $\widehat{U} :=\left( \prod _\mathfrak p U_\mathfrak p \right) \cap \widehat{B}^1$; then $\widehat{U}$ is open and nonempty. By strong approximation, there exists $\beta \in \widehat{U} \cap B^1$. Thus ${{\,\mathrm{trd}\,}}(\beta \alpha ) \in \bigcap _\mathfrak p R_\mathfrak p = R$ and ${{\,\mathrm{nrd}\,}}(\beta \alpha )={{\,\mathrm{nrd}\,}}(\alpha )=n$. Therefore $\beta \alpha$ is as desired.$\square$

Let $R_{>_{\Omega } 0}^{\times } :=R^\times \cap F_{>_{\Omega } 0}^{\times }$ be the subgroup of units that are positive at the places $v \in \Omega$ (the set of real, ramified places in B).

Corollary 28.6.4

Suppose B is ${{{\texttt {\textit{S}}}}}$-indefinite and that ${{\,\mathrm{Cl}\,}}_{\Omega } R$ is trivial. Let $\mathcal {O}\subseteq B$ be an Eichler R-order. Then

$$\begin{aligned} {{\,\mathrm{nrd}\,}}(\mathcal {O}^\times ) = R_{>_{\Omega } 0}^{\times }. \end{aligned}$$

Proof. Let $u \in R_{>_{\Omega } 0}^{\times }$. We repeat the argument of Theorem 28.6.1, with a slight modification. Let $\mathfrak M$ be the level of $\mathcal {O}$.

Let $\mathfrak p \mid \mathfrak M$ be a prime that divides the level $\mathfrak M$. We choose an isomorphism $B_\mathfrak p \simeq {{\,\mathrm{M}\,}}_2(F_\mathfrak p )$ such that $\alpha _\mathfrak p$ is in rational canonical form, and let $\mathcal {O}'_\mathfrak p$ be the standard Eichler order in ${{\,\mathrm{M}\,}}_2(F_\mathfrak p )$ of the same level as $\mathcal {O}$. Define

$$ U_\mathfrak p = \{ \beta _\mathfrak p \in B_\mathfrak p ^1 : \beta _\mathfrak p \alpha \in \mathcal {O}'_\mathfrak p \}. $$

This is again an open condition because multiplication is continuous, and the calculation

$$\begin{aligned} \begin{pmatrix} tu^{-1} &{} 1 \\ -1 &{} 0 \end{pmatrix} \begin{pmatrix} 0 &{} -u \\ 1 &{} t \end{pmatrix} = \begin{pmatrix} 1 &{} 0 \\ 0 &{} u \end{pmatrix} \end{aligned}$$

shows also that $U_\mathfrak p \ne \emptyset$.

For all other primes $\mathfrak p \not \mid \mathfrak M$, we define $U_\mathfrak p$ as in (28.6.2). As in the proof of Theorem 28.6.1, we find $\beta \in B^1$ such that ${{\,\mathrm{nrd}\,}}(\beta \alpha )=u$ and $\gamma '=\beta \alpha$ has $\gamma '_\mathfrak p \in \mathcal {O}'_\mathfrak p$ for all $\mathfrak p \mid \mathfrak M$. Since $\gamma '$ is integral, $\gamma '$ belongs to an R-order $\mathcal {O}'$ that is equal to $\mathcal {O}'_\mathfrak p$ at all $\mathfrak p \mid \mathfrak M$ and is maximal at all $\mathfrak p \not \mid \mathfrak M$. The order $\mathcal {O}'$ is therefore an Eichler order of level $\mathfrak M$.

Finally, since ${{\,\mathrm{Cl}\,}}_{G(\mathcal {O})} R={{\,\mathrm{Cl}\,}}_{\Omega } R$ is trivial, the type set ${{\,\mathrm{Typ}\,}}(\mathcal {O})$ is also trivial (Corollary 28.5.6), so we conclude $\mathcal {O}= \nu \mathcal {O}' \nu ^{-1}$ for some $\nu \in B^\times$, and $\gamma = \nu \gamma ' \nu ^{-1} \in \mathcal {O}$ has ${{\,\mathrm{nrd}\,}}(\gamma )=u$ as desired.$\square$

Example 28.6.5

Suppose $\mathcal {O}$ is an Eichler order in an indefinite quaternion algebra over $F=\mathbb Q$. Then by Corollary 28.6.4, ${{\,\mathrm{nrd}\,}}(\mathcal {O}^\times )=\{\pm 1\}$, in particular, there exists $\gamma \in \mathcal {O}^\times$ such that ${{\,\mathrm{nrd}\,}}(\gamma )=-1$.

Remark 28.6.6. We will prove a stronger version of Corollary 28.6.4 after we have developed the theory of selectivity: see Corollary 31.1.11.

To conclude this section, we consider a variant of principalization of right fractional ideals: we ask further that the generator has totally positive reduced norm.

28.6.7

Suppose F is a number field. Let

$$\begin{aligned} B_{>0}^\times = \{\alpha \in B^\times : v({{\,\mathrm{nrd}\,}}(\alpha ))>0\text { for all real places }v\}. \end{aligned}$$

The reduced norm gives a map $B^\times /B_{>0}^\times \rightarrow F_{>_{\Omega } 0}^\times /F_{>0}^\times$ and the quotient is a finite abelian 2-group, so in particular $B_{>0}^\times \le B^\times$ has finite index.

Let $I,J \subseteq B$ be R-lattices. We say I, J are in the same narrow right class if there exists $\alpha \in B_{>0}^\times$ such that $\alpha I = J$; accordingly, we let ${{\,\mathrm{Cls}\,}}\!{}_{\textsf {\tiny {R}} }^+\,\mathcal {O}$ be the narrow (right) class set  of $\mathcal {O}$. As in Lemma 27.6.8, there is a bijection

$$\begin{aligned} {{\,\mathrm{Cls}\,}}\!{}_{\textsf {\tiny {R}} }^+\,\mathcal {O}\leftrightarrow B_{>0}^\times \backslash \widehat{B}^\times \!/\widehat{\mathcal {O}}^\times \end{aligned}$$

choosing a local generator. The projection map ${{\,\mathrm{Cls}\,}}\!{}_{\textsf {\tiny {R}} }^+\,\mathcal {O}\rightarrow {{\,\mathrm{Cls}\,}}\!{}_{\textsf {\tiny {R}} }\,\mathcal {O}$ has finite fibers as $B_{>0}^\times \le B^\times$ has finite index, so since ${{\,\mathrm{Cls}\,}}\!{}_{\textsf {\tiny {R}} }\,\mathcal {O}$ is a finite set, so too is ${{\,\mathrm{Cls}\,}}\!{}_{\textsf {\tiny {R}} }^+\,\mathcal {O}$.

Corollary 28.6.8

Let F be a number field and suppose B is ${{{\texttt {\textit{S}}}}}$-definite. Then the map

$$\begin{aligned} {{\,\mathrm{nrd}\,}}:{{\,\mathrm{Cls}\,}}\!{}_{\textsf {\tiny {R}} }^+\,\mathcal {O}\leftrightarrow B_{>0}^\times \backslash \widehat{B}^\times \!/\widehat{\mathcal {O}}^\times \rightarrow F_{>0}^\times \backslash \widehat{F}^\times \!/{{\,\mathrm{nrd}\,}}(\widehat{\mathcal {O}}^\times ) =:{{\,\mathrm{Cl}\,}}^+_{G(\mathcal {O})} R \end{aligned}$$

(28.6.9)

induced by the reduced norm is a bijection.

Proof. Repeating the argument in the proof of Lemma 28.4.6, we see that the map (28.6.9) is injective for all orders $\mathcal {O}' \in {{\,\mathrm{Gen}\,}}\mathcal {O}$ if and only if $\widehat{B}^1 \subseteq B_{>0}^\times \widehat{\mathcal {O}}'^\times$ for all $\mathcal {O}' \in {{\,\mathrm{Gen}\,}}\mathcal {O}$. And the latter holds by strong approximation (Corollary 28.5.12):

$$\begin{aligned} \widehat{B}^1 \subseteq B^1 \widehat{\mathcal {O}}'^1 \subseteq B_{>0}^\times \widehat{\mathcal {O}}'^\times \end{aligned}$$

for all R-orders $\mathcal {O}'$.$\square$

Proposition 28.6.10

Let F be a number field. Suppose that B is ${{{\texttt {\textit{S}}}}}$-indefinite and that ${{\,\mathrm{Cl}\,}}_{G(\mathcal {O})}^+ R = {{\,\mathrm{Cl}\,}}_{G(\mathcal {O})} R$. For each real place v not in $\Omega$, let $\epsilon _v \in \{\pm 1\}$. Then there exists $\gamma \in \mathcal {O}^\times$ such that ${{\,\mathrm{sgn}\,}}(v({{\,\mathrm{nrd}\,}}(\gamma )))=\epsilon _v$ for all v real not in $\Omega$.

In particular, if $F=\mathbb Q$ and $\mathcal {O}$ is a locally norm-maximal $\mathbb Z$-order in an indefinite quaternion algebra B, then there exists $\gamma \in \mathcal {O}^\times$ with ${{\,\mathrm{nrd}\,}}(\gamma )=-1$.

Proof. Let $a \in F_{>_{\Omega } 0}^\times$ be such that $v(a)=\epsilon _v$ for all v real not in $\Omega$ and the class of aR is trivial in ${{\,\mathrm{Cl}\,}}_{G(\mathcal {O})} R$: these constraints together impose congruence conditions on elements in a real cone. By the Hasse–Schilling norm theorem, there exists $\alpha \in B^\times$ such that ${{\,\mathrm{nrd}\,}}(\alpha )=a$. Thus the class of ${{\,\mathrm{nrd}\,}}(\alpha \mathcal {O})$ in ${{\,\mathrm{Cl}\,}}_{G(\mathcal {O})}^+ R = {{\,\mathrm{Cl}\,}}_{G(\mathcal {O})} R$ is trivial.

But then by (28.6.9) (a consequence of strong approximation), there exists $\beta \in B_{>0}^\times$ such that $\beta \mathcal {O}= \alpha \mathcal {O}$, and therefore $\beta =\alpha \gamma$ with $\gamma \in \mathcal {O}^\times$. Since $\beta$ is totally positive, for all real places $v \not \in \Omega$ we have ${{\,\mathrm{sgn}\,}}(v({{\,\mathrm{nrd}\,}}(\gamma )))={{\,\mathrm{sgn}\,}}(v({{\,\mathrm{nrd}\,}}(\alpha )))=\epsilon _v$, completing the proof.

For the second statement, we only need to note that ${{\,\mathrm{Cl}\,}}^+ \mathbb Z ={{\,\mathrm{Cl}\,}}\mathbb Z = \{1\}$ and recall that ${{\,\mathrm{nrd}\,}}(\mathcal {O}^\times ) \le \{\pm 1\}$.$\square$

Remark 28.6.11. More generally, let B be a central simple algebra over the global field F. We say B satisfies the ${{{\texttt {\textit{S}}}}}$ -Eichler condition if there exists a place $v \in {{{\texttt {\textit{S}}}}}$ such that $B_v$ is not a division algebra. (In this text, for quaternion algebras we prefer to use the term ${{{\texttt {\textit{S}}}}}$-indefinite because it readily conveys the notion, but both terms are common.) If F is a number field and ${{{\texttt {\textit{S}}}}}$ is the set of archimedean places of F, then B satisfies the ${{{\texttt {\textit{S}}}}}$-Eichler condition if and only if B is not a totally definite quaternion algebra (Exercise 28.5). So the condition is a mild condition, and the quaternion algebra case requires special effort.

When B satisfies the ${{{\texttt {\textit{S}}}}}$-Eichler condition, then $B^1 \hookrightarrow \widehat{B}^1$ is dense, and for $\mathcal {O}\subseteq B$ a maximal R-order, a locally principal right fractional $\mathcal {O}$-ideal $I \subseteq B$ is principal if and only if its reduced norm ${{\,\mathrm{nrd}\,}}(I) \subseteq R$ is trivial in ${{\,\mathrm{Cl}\,}}_\Omega R$, where $\Omega$ is the set of real places $v \in {{\,\mathrm{Pl}\,}}F$ such that $B_v \simeq {{\,\mathrm{M}\,}}_n(\mathbb H )$, generalizing the quaternion case.

Eichler proved Theorem 28.5.5 and the more general statement of the previous paragraph [Eic37, Satz 2], also providing several reformulations and applications [Eic38a, Eic38c]. For a full exposition, see Reiner [Rei2003, §34].

7 First proof

Now we proceed with the proof of strong approximation in Theorem 28.5.3; we follow roughly the same lines as in the proof of Eichler’s theorem on norms, but here instead we will be concerned with traces. In other words, we replace strong approximation of elements by strong approximation of traces, and then we just have to chase conjugacy classes.

We start with a statement of weak and strong approximation for the global field F.

Lemma 28.7.1

(Weak approximation for F). Let ${{{\texttt {\textit{S}}}}}\subseteq {{\,\mathrm{Pl}\,}}F$ be a finite nonempty set of places. Then the images of the maps

$$\begin{aligned}F \hookrightarrow F_{{{{\texttt {\textit{S}}}}}} :=\prod _{v \in {{{\texttt {\textit{S}}}}}} F_v \quad \text {and}\quad F^\times \hookrightarrow F_{{{{\texttt {\textit{S}}}}}}^\times :=\prod _{v \in {{{\texttt {\textit{S}}}}}} F_v^\times \end{aligned}$$

are dense.

Proof. See e.g. Neukirch [Neu99, Theorem II.3.4] or O’Meara [O’Me73, §11E].$\square$

Lemma 28.7.2

(Strong approximation for F). Let ${{{\texttt {\textit{S}}}}}\subseteq {{\,\mathrm{Pl}\,}}F$ be a finite nonempty set of places. Then the image of is dense.

Proof. See e.g. Neukirch [Neu99, Exercise III.1.1] or O’Meara [O’Me73, §33G].$\square$

We recall that B is a quaternion algebra over F and we write $B_{{{{\texttt {\textit{S}}}}}} :=\prod _{v \in {{{\texttt {\textit{S}}}}}} B_v$.

Proposition 28.7.3

(Weak approximation for B). Let ${{{\texttt {\textit{S}}}}}\subseteq {{\,\mathrm{Pl}\,}}F$ be a finite nonempty set of places. Then the images

$$\begin{aligned}B \hookrightarrow B_{{{{\texttt {\textit{S}}}}}} \quad \text {and}\quad B^\times \hookrightarrow B_{{{{\texttt {\textit{S}}}}}}^\times \quad \text {and}\quad B^1 \hookrightarrow B_{{{{\texttt {\textit{S}}}}}}^1\end{aligned}$$

are dense.

Proof. By weak approximation for F (Lemma 28.7.1), we have F dense in $\prod _{v\in {{{\texttt {\textit{S}}}}}} F_v$. Choosing an F-basis for B, we have $B \simeq F^4$ as topological F-vector spaces, and so by approximating in each coordinate, we conclude that B is dense in $\prod _{v\in {{{\texttt {\textit{S}}}}}} B_v$. The multiplicative case follows a fortiori, restricting open neighborhoods.

Finally we treat $B^1$. By Exercise 7.31, we know that $B^1=[B^\times ,B^\times ]$ is the commutator. Let $(\gamma _v)_v \in \prod _{v \in {{{\texttt {\textit{S}}}}}} B_v^1$. Then for each $v \in {{{\texttt {\textit{S}}}}}$, we can write $\gamma _v=\alpha _v\beta _v\alpha _v^{-1}\beta _v^{-1}$ with $\alpha _v,\beta _v \in B^\times$. Then by weak approximation for $B^\times$, we can find a sequence $\alpha _n \in B^\times$ such that $\alpha _n \rightarrow (\alpha _v)_v \in B_{{{{\texttt {\textit{S}}}}}}^\times$, and similarly with $\beta _n \rightarrow (\beta _v)_v$. Then since multiplication is continuous, we conclude that $\gamma _n=\alpha _n\beta _n\alpha _n^{-1}\beta _n^{-1} \rightarrow (\gamma _v)_v \in B^1$.$\square$

Next, we need to approximate polynomials: this kind of lemma was first performed in section 14.7 to prove the Hasse–Schilling theorem of norms, and here we need another variant.

Lemma 28.7.4

Let ${{{\texttt {\textit{S}}}}}\subseteq {{\,\mathrm{Pl}\,}}F$ be a finite nonempty set of places and $\Sigma \subseteq {{\,\mathrm{Pl}\,}}F$ a finite set of noncomplex places disjoint from ${{{\texttt {\textit{S}}}}}$. Let $\widehat{t} \in \widehat{B}$ be such that $x^2-t_vx+1 \in F_v[x]$ is irreducible for $v \in \Sigma$.

Let $\epsilon >0$. Then there exists $t \in F$ such that:

• $f(x)=x^2-tx+1$ is irreducible and separable over F;

• $|t-t_v \,|_v<\epsilon$ for all $v \in \Sigma$;

• f(x) is irreducible over $F_v$ for all $v \in \Sigma$; and

• $|t-t_v \,|_v \le 1$ for all $v \not \in {{{\texttt {\textit{S}}}}}\cup \Sigma$.

Proof. We argue as in Lemma 14.7.6 (and Corollary 14.7.8), but instead of weak approximation we now use strong approximation (Lemma 28.7.2). Our job is a bit easier because we are only asking for irreducibility, not separability.

Since ${{{\texttt {\textit{S}}}}}$ is nonempty, by strong approximation for F we can find t arbitrarily close to $\widehat{t}$, thus ensuring that the desired inequalities hold and that f(x) is irreducible over $F_v$ for $v \in \Sigma$; to ensure that f(x) is separable, we need only avoid the locus $t^2=4$, and similarly we may ensure f(x) is irreducible.$\square$

We now embark on the proof of strong approximation.

Proof of Main Theorem 28.5.3

We follow Vignéras [Vig80a, Théorème III.4.3]; see also Miyake [Miy2006, Theorems 5.2.9–5.2.10] for the case $F=\mathbb Q$. We show that the closure of $B^1$ is equal to $\widehat{B}^1$. Let $\widehat{\gamma }=(\gamma _v)_v \in \widehat{B}^1$; we will find a sequence of elements of $B^1$ converging to $\widehat{\gamma }$.

Step 1: Setup. We claim it is enough to consider the case where $\gamma _v=1$ for all but finitely many v, by a Cantor-style diagonalization argument. Indeed, for a finite set ${{{\texttt {\textit{T}}}}}\subseteq {{\,\mathrm{Pl}\,}}F$ disjoint from ${{{\texttt {\textit{S}}}}}$, we let $\widehat{\gamma }_{[{{{\texttt {\textit{T}}}}}]}$ be the idele obtained from $\widehat{\gamma }$ replacing $\gamma _v=1$ for $v \not \in {{{\texttt {\textit{T}}}}}$. Then for a sequence of subsets ${{{\texttt {\textit{T}}}}}$ eventually containing each place v, we have $\widehat{\gamma }_{[{{{\texttt {\textit{T}}}}}]} \rightarrow \widehat{\gamma }$. Thus, if we can find sequences in $B^1$ converging to each $\widehat{\gamma }_{[{{{\texttt {\textit{T}}}}}]}$ we can diagonalize to find a sequence converging to $\widehat{\gamma }$, since ${{\,\mathrm{Pl}\,}}F$ is countable.

So we may suppose without loss of generality that $\gamma _v=1$ for all but finitely many v. To find a sequence in $B^1$ converging to $\widehat{\gamma }$, our strategy in the proof is as follows: in shrinking open neighborhoods of $\widehat{\gamma }$ we first find an element in $B^1$ whose reduced characteristic polynomial is close to an element in the open neighborhood, and then we conjugate by $B^\times$ to get the limits themselves to match.

To this end, let $\mathcal {O}\subset B$ be a reference R-order, let ${{{\texttt {\textit{T}}}}}\subseteq {{\,\mathrm{Pl}\,}}F$ be a finite set of places disjoint from ${{{\texttt {\textit{S}}}}}$ containing the primes v where $\gamma _v \ne 1$ and the ramified places of B not in ${{{\texttt {\textit{S}}}}}$. We consider open neighborhoods of the form

$$\begin{aligned} U = \prod _{v \in {{{\texttt {\textit{T}}}}}} \gamma _v U_v \times \prod _{v \not \in {{{\texttt {\textit{S}}}}}\cup {{{\texttt {\textit{T}}}}}} \mathcal {O}_v^1 \end{aligned}$$

where $U_v \subseteq B_v^1$ is an open neighborhood of 1.

Step 2: Polynomial approximation. Now comes the polynomial approximation step: we will show that there exists $\widehat{\alpha }\in \widehat{B}^\times$ such that $B^1 \cap \widehat{\alpha }^{-1} U \widehat{\alpha }\ne \emptyset$. (This is about as good as could be expected at this stage: if we argue by approximating polynomials, we should only be able to expect to get something up to conjugation.)

We define the idele $\widehat{t}$ in each component, as follows.

$\bullet$:

If $v \not \in {{{\texttt {\textit{S}}}}}\cup {{{\texttt {\textit{T}}}}}$, we take $t_v={{\,\mathrm{trd}\,}}(\gamma _v)=2$.

$\bullet$:

If $v \in {{{\texttt {\textit{T}}}}}$ is unramified in B, we take $t_v={{\,\mathrm{trd}\,}}(\gamma _v)$.

$\bullet$:

If $v \in {{{\texttt {\textit{T}}}}}$ is ramified in B, we choose $\mu _v \in \gamma _v U_v$ such that $\mu _v \not \in F_v$ is separable over $F_v$, and take $t_v={{\,\mathrm{trd}\,}}(\mu _v)$. Since $B_v$ is a division algebra, we have $F_v[\mu _v]$ is a field, and so its reduced characteristic polynomial is irreducible.

$\bullet$:

If $v \in {{{\texttt {\textit{S}}}}}$ is ramified in B, we choose $t_v \in F_v$ such that $x^2-t_v x + 1$ is irreducible; such an element $t_v$ exists by Lemma 14.7.5).

The only places that remain are those $v \in {{{\texttt {\textit{S}}}}}$ such that v is unramified in B. By hypothesis that B is ${{{\texttt {\textit{S}}}}}$-indefinite, we know that there is at least one such split place $v_{\text {spl}} \in {{{\texttt {\textit{S}}}}}$ remaining (and in particular, $v_{\text {spl}} \not \in {{\,\mathrm{Ram}\,}}B \cup {{{\texttt {\textit{T}}}}}$).

We now apply our polynomial approximation Lemma 28.7.4, we conclude that there exists $t \in F$ such that $f(x)=x^2-tx+1$ is separable and irreducible over F, and irreducible over $F_v$ for all $v \in {{\,\mathrm{Ram}\,}}B$, and such that t is arbitrarily close to $\widehat{t}$. Let $K=F[x]/(f(x))$. Then either ${{\,\mathrm{Ram}\,}}B = \emptyset$ and $K \hookrightarrow B$ automatically, or ${{\,\mathrm{Ram}\,}}B \ne \emptyset$ so f(x) is irreducible and then $K \hookrightarrow B$ by the local-global principle for embeddings (Proposition 14.6.7). Let $\beta \in B^1$ have minimal polynomial f(x). Since $\widehat{t} \in {{\,\mathrm{trd}\,}}(U)$ and the reduced trace is an open (linear) map, with a closer approximation we may suppose ${{\,\mathrm{trd}\,}}(\beta ) \in {{\,\mathrm{trd}\,}}(U)$, and therefore there exists $\widehat{\gamma }' \in U$ such that ${{\,\mathrm{trd}\,}}(\beta )={{\,\mathrm{trd}\,}}(\widehat{\gamma }')$ so that $\beta ,\widehat{\gamma }'$ have the same irreducible minimal polynomial. By the Skolem–Noether theorem (Corollary 7.7.3), there exists $\widehat{\alpha }\in \widehat{B}^\times$ such that

$$\begin{aligned} \beta = \widehat{\alpha }^{-1} \widehat{\gamma }' \widehat{\alpha }. \end{aligned}$$

(28.7.5)

Step 3: Finding a sequence. We then repeat the above argument with a sequence of open neighborhoods $U_n \ni \widehat{\alpha }$ such that $\bigcap _n U_n = \{\widehat{\gamma }\}$; we obtain a sequence

$$\begin{aligned} B^1 \ni \beta _n = \widehat{\alpha }_n^{-1} \widehat{\gamma }_n' \widehat{\alpha }_n \in \widehat{\alpha }_n^{-1} U_n \widehat{\alpha }_n. \end{aligned}$$

(28.7.6)

Since $\widehat{\gamma }_n' \in U_n$, we have $\widehat{\gamma }_n' \rightarrow \widehat{\gamma }$, and in particular for $v \in {{\,\mathrm{Pl}\,}}F \smallsetminus {{{\texttt {\textit{S}}}}}\cup {{{\texttt {\textit{T}}}}}$, we have $\gamma _{n,v}' \rightarrow \gamma _v=1$.

Step 4: Harmonizing the sequence. By ‘harmonizing’ the conjugating elements $\widehat{\alpha }_n$, we will realize a sequence in $B^1$ tending to $\widehat{\gamma }$ as desired, in two (subset)steps. First, by Main Theorem 27.6.14, $\widehat{B}^\times \!/B^\times$ is compact. So restricting to a subsequence, we have $\widehat{\alpha }_n = \widehat{\delta }_n \mu _n$ with $\mu _n \in B^\times$ and $\widehat{\delta }_n \rightarrow \widehat{\delta }= (\delta _v)_v \in \widehat{B}^\times$. Second, by weak approximation for B (Proposition 28.7.3), $B^\times$ is dense in $\prod _{v \in {{{\texttt {\textit{T}}}}}} B_v^\times$, so there is a sequence $\nu _n$ from $B^\times$ such that $\nu _n \rightarrow (\delta _v)_{v \in {{{\texttt {\textit{T}}}}}}$.

Step 5: Conclusion. To conclude, we consider the sequence

$$\begin{aligned} (\nu _n\mu _n)\beta _n(\nu _n\mu _n)^{-1} = (\nu _n\widehat{\delta }_n^{-1}) \widehat{\gamma }_n' (\widehat{\delta }_n \nu _n^{-1}). \end{aligned}$$

(28.7.7)

We claim that this sequence tends to $\widehat{\gamma }$. For $v \in {{{\texttt {\textit{T}}}}}$, we have $\nu _{n,v} \widehat{\delta }_{n,v}^{-1} \rightarrow \delta _v \delta _v^{-1} = 1$ so

$$\begin{aligned} (\nu _{n,v}\delta _{n,v}^{-1}) \gamma _{n,v}' (\delta _{n,v} \nu _{n,v}^{-1}) \rightarrow \gamma _v. \end{aligned}$$

(28.7.8)

On the other hand, for $v \in {{\,\mathrm{Pl}\,}}F \smallsetminus ({{{\texttt {\textit{S}}}}}\cup {{{\texttt {\textit{T}}}}})$, we have $\gamma _{n,v}' \rightarrow 1$, so

$$\begin{aligned} (\nu _{n,v}\delta _{n,v}^{-1}) \gamma _{n,v}' (\delta _{n,v} \nu _{n,v}^{-1}) \rightarrow 1=\gamma _v. \end{aligned}$$

(28.7.9)

Putting together these cases, we conclude that $(\nu _n\mu _n)\beta _n(\nu _n\mu _n)^{-1} \rightarrow \widehat{\gamma }$, and therefore $\widehat{\gamma }$ is in the closure of $B^1$. $\square$

Remark 28.7.10. Strong approximation has a more general formulation, as follows. Let G be a semisimple, simply-connected algebraic group over the global field F. Let ${{{\texttt {\textit{S}}}}}$ be an eligible set containing a place v such that $G(F_v)$ is not compact. Then G(F) is dense in , and we say G satisfies strong approximation (relative to ${{{\texttt {\textit{S}}}}}$). Over number fields, strong approximation was established by Kneser [Kne66a, Kne66b] and Platonov [Pla69, Pla69-70], and over function fields by Margulis and Prasad [Pra77]; see also Platonov–Rapinchuk [PR94, Theorem 7.12]. For a survey with discussion and bibliography, see Rapinchuk [Rap2014] and the description of Kneser’s work by Scharlau [Scha2009, §2.1].

8 Second proof

Because of its importance, we now give a second proof of strong approximation; this has the same essential elements, but uses some facts from group theory to simplify away the final steps of the previous proof. We follow Swan [Swa80, §14], who references Kneser [Kne66a, Kne66b], Platonov [Pla69-70], and Prasad [Pra77]; see also the exposition by Kleinert [Klt2000, §4.2].

Let $Z :={{\,\mathrm{cl}\,}}(B^1) \le \widehat{B}^1$ be the closure of $B^1$ in $\widehat{B}^1$. We embed $B_v^1 \hookrightarrow \widehat{B}^1$ by $(\dots ,1,\alpha _v,1,\dots )$ in the vth component, for $v \not \in {{{\texttt {\textit{S}}}}}$.

Lemma 28.8.1

If $B_v^1 \subseteq Z$ for almost all $v \not \in {{{\texttt {\textit{S}}}}}$, then $Z=\widehat{B}^1$.

Proof. Suppose that $B_v^1 \subseteq Z$ for all $v \not \in {{{\texttt {\textit{T}}}}}$ where ${{{\texttt {\textit{T}}}}}$ is a finite set. Let $\widehat{\gamma }\in \widehat{B}^1$. If $\gamma _v=1$ for all $v \in {{{\texttt {\textit{T}}}}}$, then $\widehat{\gamma }$ is a limit of elements in Z (approximating at a finite level), so $\widehat{\gamma }\in Z$. Otherwise, by weak approximation for B (Proposition 28.7.3), there exists $\gamma \in B^1$ such that $\gamma$ is near $\gamma _v$ for all $v \in {{{\texttt {\textit{T}}}}}$. Let $\widehat{\beta }$ have $\beta _v=1$ for $v \in {{{\texttt {\textit{T}}}}}$ and $\beta _v=\gamma ^{-1}\gamma _v$ for $v \not \in {{{\texttt {\textit{T}}}}}$; then $\widehat{\beta }\in Z$, and $\widehat{\gamma }$ is the limit of the $\gamma \widehat{\beta }$.$\square$

Now we consider

$$\begin{aligned} Z_1 :=\{\widehat{\gamma }\in Z : \gamma _v=1\text { for all but finitely many }v\}. \end{aligned}$$

(28.8.2)

Lemma 28.8.3

$Z_1 \trianglelefteq \widehat{B}^1$ is a normal subgroup.

Proof. Let $\widehat{\gamma }\in \widehat{B}^1$ and let $\widehat{\alpha }\in Z_1$ with $\alpha _v=1$ for $v \not \in {{{\texttt {\textit{T}}}}}$ with ${{{\texttt {\textit{T}}}}}$ a finite set. By weak approximation for B (Proposition 28.7.3), there exists $\gamma \in B^1$ with $\gamma$ close to $\gamma _v$ for all $v \in {{{\texttt {\textit{T}}}}}$. Therefore $\gamma ^{-1} \widehat{\alpha }\gamma$ is near $\widehat{\gamma }^{-1} \widehat{\alpha }\widehat{\gamma }$ for $v \in {{{\texttt {\textit{T}}}}}$ and $\gamma ^{-1} \alpha _v \gamma = \gamma _v^{-1} \alpha _v \gamma _v=1$ for $v \not \in {{{\texttt {\textit{T}}}}}$, so is the limit of such in Z. Therefore $\widehat{\gamma }^{-1}\widehat{\alpha }\widehat{\gamma }\in Z_1$, thus $\widehat{\gamma }^{-1} Z_1 \widehat{\gamma }\subseteq Z_1$ and $Z_1 \trianglelefteq \widehat{B}^1$.$\square$

Lemma 28.8.4

Let F be an infinite field. Then ${{\,\mathrm{PSL}\,}}_2(F)$ is a simple group.

Proof. See e.g. Grove [Grov2002, Theorem 1.13]. Briefly, the result can be proven using Iwasawa’s criterion, since ${{\,\mathrm{SL}\,}}_2(F)$ acts doubly transitively on the linear subspaces of $F^2$: the kernel of the action is the center $\{\pm 1\}$, and the stabilizer subgroup of a standard basis element is the subgroup of upper triangular matrices, whose conjugates generate ${{\,\mathrm{SL}\,}}_2(F)$.$\square$

Proof of Main Theorem 28.5.3 (Strong approximation)

We show that $Z=\widehat{B}^1$. By Lemma 28.8.1, it is enough to show that $B_v^1 \subseteq Z$ for almost all v. By Lemma 28.8.3, each $Z_1 \cap B_v^1 \trianglelefteq B_v^1$ is a normal subgroup; by Lemma 28.8.4, either this normal subgroup is either scalar, or we have $Z_1 \cap B_v^1=B_v^1 \le Z_1 \le Z$ and we are done. So it suffices to show that for almost all v, we have $Z_1 \cap B_v^1$ nonscalar, which is to say, $Z_1 \cap B_v^1 \ne \{\pm 1\}$.

So let $w \in {{\,\mathrm{Pl}\,}}F$ be unramified. We perform polynomial approximation, as in Step 2 of the first proof. For $v \in {{\,\mathrm{Ram}\,}}B$, let $x^2-t_vx+1$ be a separable irreducible polynomial (which exists by Lemma 14.7.5) with $t_v \in R_v$, and do the same for w; let $\widehat{t}$ be the corresponding idele, with $t_v=1$ for the remaining places v. By Lemma 28.7.4, there exists $t \in F$ such that $f(x)=x^2-tx+1$ is:

• irreducible and separable over F,

• irreducible over $F_v$ for all $v \in {{\,\mathrm{Ram}\,}}B$, and

• such that t arbitrarily well-approximates $\widehat{t}\in \widehat{R}$, so we may suppose $t \in R$.

By the local-global principle for embeddings (Proposition 14.6.7), there exists $\beta \in B^1$ with $f(\beta )=0$; but moreover, $\beta$ is integral and so belongs to a maximal order. In this way, we manufacture a sequence $\beta _n$ with ${{\,\mathrm{trd}\,}}(\beta _n) \rightarrow \widehat{t}$. Repeating this with $t_v \rightarrow 1$ for $v \in {{\,\mathrm{Ram}\,}}B$ and diagonalizing, we may suppose $t_v=1$ for all $v \ne w$.

Let $\mathcal {O}$ be a maximal R-order. The type set ${{\,\mathrm{Typ}\,}}\mathcal {O}$ is finite by Corollary 27.6.25; choose representatives $\mathcal {O}_i$ for ${{\,\mathrm{Typ}\,}}\mathcal {O}$. After conjugating the elements $\beta _n$, we may suppose without loss of generality that each $\beta _n$ belongs to one of the orders $\mathcal {O}_i$. By the pigeonhole principle, there is an order containing infinitely many, so restricting to a subsequence we may suppose $\beta _n \in \mathcal {O}^1$ for all n.

But now the kicker: $\widehat{\mathcal {O}}^1$ is compact, so we may restrict to a convergent subsequence $\beta _n \rightarrow \widehat{\beta }\in \widehat{\mathcal {O}}^1$. By construction, each $\beta _n$ has separable reduced characteristic polynomial, and ${{\,\mathrm{trd}\,}}(\beta _{n,v}) \rightarrow 1$ for all $v \ne w$, so $\beta _{n,v} \rightarrow \beta _v=1$ for all $v \ne w$. But ${{\,\mathrm{trd}\,}}(\beta _{n,w}) \rightarrow t_w$, and $x^2-t_wx+1$ is irreducible, so $\beta _w \not \in F_w^\times$, as desired. $\square$

9 $*$ Normalizer groups

In this section, we apply strong approximation to the normalizer of an order, and we compare the normalizers for locally isomorphic orders. We recall the notation from section 18.5. Let ${{\,\mathrm{Idl}\,}}(\mathcal {O})$ be the group of invertible two-sided fractional $\mathcal {O}$-ideals, let ${{\,\mathrm{PIdl}\,}}(\mathcal {O}) \le {{\,\mathrm{Idl}\,}}(\mathcal {O})$ the subgroup of principal fractional $\mathcal {O}$-ideals, and let ${{\,\mathrm{PIdl}\,}}(R) \le {{\,\mathrm{PIdl}\,}}(\mathcal {O})$ be the image of the group of principal fractional R-ideals.

We suppose throughout this section that ${{\,\mathrm{PIdl}\,}}(\mathcal {O}) \trianglelefteq {{\,\mathrm{Idl}\,}}(\mathcal {O})$ is a normal subgroup. This is true whenever ${{\,\mathrm{Idl}\,}}\mathcal {O}$ is abelian, which holds when $\mathcal {O}$ is Eichler order (using Lemma 23.3.13 for the primes where $\mathfrak p$ is maximal and Proposition 23.4.14 the remaining primes).

28.9.1

Recall there is a natural exact sequence

$$\begin{aligned} 1 \rightarrow N_{B^\times }(\mathcal {O})/(F^\times \mathcal {O}^\times ) \rightarrow {{\,\mathrm{Pic}\,}}_R(\mathcal {O}) \rightarrow {{\,\mathrm{Idl}\,}}(\mathcal {O})/{{\,\mathrm{PIdl}\,}}(\mathcal {O}) \rightarrow 1 \end{aligned}$$

(18.5.5)

obtained by considering the class of a bimodule as a two-sided ideal modulo principal ideals. The idelic dictionary (27.6.26) gives another proof of exactness: there is a canonical bijection

$$\begin{aligned} {{\,\mathrm{Idl}\,}}(\mathcal {O}) \xrightarrow {\smash {{\sim }}}N_{\widehat{B}^\times }(\widehat{\mathcal {O}})/\widehat{\mathcal {O}}^\times , \end{aligned}$$

to obtain ${{\,\mathrm{Idl}\,}}(\mathcal {O})/{{\,\mathrm{PIdl}\,}}(\mathcal {O})$ we take the quotient by $N(\mathcal {O})$ and to obtain ${{\,\mathrm{Idl}\,}}(\mathcal {O})/{{\,\mathrm{PIdl}\,}}(R)$ we take the quotient by $F^\times$; therefore the exact sequence (18.5.5) can be rewritten

$$\begin{aligned} 1 \rightarrow F^\times \backslash N_{B^\times }(\mathcal {O}) / \mathcal {O}^\times \rightarrow F^\times \backslash N_{\widehat{B}^\times }(\widehat{\mathcal {O}})/\widehat{\mathcal {O}}^\times \rightarrow N_{B^\times }(\mathcal {O}) \backslash N_{\widehat{B}^\times }(\widehat{\mathcal {O}})/\widehat{\mathcal {O}}^\times \rightarrow 1 \end{aligned}$$

(28.9.2)

and its exactness is now visible.

28.9.3

We have a map of pointed sets

$$\begin{aligned} {{\,\mathrm{Idl}\,}}(\mathcal {O})&\rightarrow {{\,\mathrm{Cls}\,}}\mathcal {O}\\ I&\mapsto [I] \end{aligned}$$

and ${{\,\mathrm{PIdl}\,}}(\mathcal {O})$ is the kernel of this map, the preimage of the trivial class in ${{\,\mathrm{Cls}\,}}\mathcal {O}$. The composition of this map with the reduced norm gives a group homomorphism:

$$\begin{aligned} c :{{\,\mathrm{Idl}\,}}(\mathcal {O})&\rightarrow {{\,\mathrm{Cl}\,}}_{G(\mathcal {O})}(R)=F_{>_{\Omega } 0}^\times \backslash \widehat{F}^\times \!/{{\,\mathrm{nrd}\,}}(\widehat{\mathcal {O}}) \\ I&\mapsto [{{\,\mathrm{nrd}\,}}(I)] \end{aligned}$$

Lemma 28.9.4

Suppose that B is ${{{\texttt {\textit{S}}}}}$-indefinite. Then ${{\,\mathrm{PIdl}\,}}(\mathcal {O})=\ker c$, i.e.,

$$\begin{aligned} {{\,\mathrm{PIdl}\,}}(\mathcal {O})=\{ I \in {{\,\mathrm{Idl}\,}}(\mathcal {O}) : [{{\,\mathrm{nrd}\,}}(I)]\text { is trivial in }{{\,\mathrm{Cl}\,}}_{G(\mathcal {O})}(R)\}. \end{aligned}$$

Proof. By strong approximation (Theorem 28.5.5), the reduced norm gives a bijection ${{\,\mathrm{nrd}\,}}:{{\,\mathrm{Cls}\,}}\mathcal {O}\rightarrow {{\,\mathrm{Cl}\,}}_{G(\mathcal {O})}(R)$; thus $I \in {{\,\mathrm{Idl}\,}}(\mathcal {O})$ is principal, belonging to ${{\,\mathrm{PIdl}\,}}(\mathcal {O})$, if and only if $[{{\,\mathrm{nrd}\,}}(I)]$ is trivial in ${{\,\mathrm{Cl}\,}}_{G(\mathcal {O})}(R)$.$\square$

Now let $\mathcal {O},\mathcal {O}'$ be locally isomorphic orders (in the same genus) with connecting $\mathcal {O},\mathcal {O}'$-ideal J.

28.9.5

By 18.4.7, there is an isomorphism of groups

$$\begin{aligned} \begin{aligned} {{\,\mathrm{Idl}\,}}(\mathcal {O})&\xrightarrow {\smash {{\sim }}}{{\,\mathrm{Idl}\,}}(\mathcal {O}') \\ I&\mapsto J^{-1}IJ. \end{aligned} \end{aligned}$$

(28.9.6)

which induces an isomorphism ${{\,\mathrm{Pic}\,}}_R(\mathcal {O}) \simeq {{\,\mathrm{Pic}\,}}_R(\mathcal {O}')$.

We now come to the first major result of this section.

Proposition 28.9.7

Suppose that B is ${{{\texttt {\textit{S}}}}}$-indefinite. Then the map (28.9.6) induces a commutative diagram

with vertical maps isomorphisms.

Proof. We verify that $J^{-1}{{\,\mathrm{PIdl}\,}}(\mathcal {O})J = {{\,\mathrm{PIdl}\,}}(\mathcal {O}')$, from which both statements follow; and this verification comes from Lemma 28.9.4, as

$$\begin{aligned}{}[{{\,\mathrm{nrd}\,}}(J^{-1}IJ)]=[{{\,\mathrm{nrd}\,}}(I)] \in {{\,\mathrm{Cl}\,}}_{G(\mathcal {O})} R = {{\,\mathrm{Cl}\,}}_{G(\mathcal {O}')} R \end{aligned}$$

(recall 27.7.4), so $I \in {{\,\mathrm{PIdl}\,}}(\mathcal {O})$ if and only if $J^{-1}IJ \in {{\,\mathrm{PIdl}\,}}(\mathcal {O}')$.$\square$

Proposition 28.9.7 says that when B is ${{{\texttt {\textit{S}}}}}$-indefinite, then the structure of the normalizer group, the Picard group, and group of ideals modulo principal ideals are all isomorphic for all orders in a genus. The same is not true when B is ${{{\texttt {\textit{S}}}}}$-definite; we always have an isomorphism in the middle, but for locally isomorphic orders, the Picard group may be distributed differently between the normalizer and the ideal group.

By chasing a few diagrams, we can be more specific about the structure of ${{\,\mathrm{Idl}\,}}(\mathcal {O})$ by seeking out primitive ideals.

Definition 28.9.8

The Atkin–Lehner group of $\mathcal {O}$ is

$$\begin{aligned} {{\,\mathrm{AL}\,}}(\mathcal {O}) :=\bigl \{J \in {{\,\mathrm{Idl}\,}}(\mathcal {O}) : [{{\,\mathrm{nrd}\,}}(J)] \in ({{\,\mathrm{Cl}\,}}_{G(\mathcal {O})} R)^2 \bigr \}/{{\,\mathrm{Idl}\,}}(R). \end{aligned}$$

(28.9.9)

The definition (28.9.9) makes sense because ${{\,\mathrm{Idl}\,}}(R)$ is indeed a subgroup: since if $\mathfrak a \in {{\,\mathrm{Idl}\,}}(R)$ then $[{{\,\mathrm{nrd}\,}}(\mathfrak a \mathcal {O})]=[\mathfrak a ]^2 \in ({{\,\mathrm{Cl}\,}}_{G(\mathcal {O})} R)^2$.

Example 28.9.10

Suppose that $\mathcal {O}$ is an Eichler order with ${{\,\mathrm{discrd}\,}}\mathcal {O}=\mathfrak N$. Then (23.4.21) gives an isomorphism

$$\begin{aligned} {{\,\mathrm{Idl}\,}}(\mathcal {O})/{{\,\mathrm{Idl}\,}}(R) \simeq \prod _\mathfrak{p \mid \mathfrak N } \mathbb Z /2\mathbb Z . \end{aligned}$$

The group ${{\,\mathrm{AL}\,}}(\mathcal {O})$ is therefore an abelian 2-group, isomorphic to $\prod _\mathfrak{p \mid \mathfrak N } \mathbb Z /2\mathbb Z$ when $({{\,\mathrm{Cl}\,}}_{G(\mathcal {O})} R)^2$ is trivial. For example, for $\mathcal {O}={{\,\mathrm{M}\,}}_2(R) \subset B={{\,\mathrm{M}\,}}_2(F)$, we have ${{\,\mathrm{AL}\,}}(\mathcal {O})$ the trivial group.

28.9.11

There is a group homomorphism

$$\begin{aligned} \begin{aligned} {{\,\mathrm{Idl}\,}}(R)&\rightarrow {{\,\mathrm{Idl}\,}}(\mathcal {O}) \\ \mathfrak a&\mapsto \mathfrak a \mathcal {O}; \end{aligned} \end{aligned}$$

(28.9.12)

this map is injective, since $\mathfrak a \mathcal {O}=\mathcal {O}$ implies $\mathfrak a =R$. We obtain an exact sequence

$$\begin{aligned} 1 \rightarrow {{\,\mathrm{Cl}\,}}R \rightarrow {{\,\mathrm{Pic}\,}}_R(\mathcal {O}) \rightarrow {{\,\mathrm{Idl}\,}}(\mathcal {O})/{{\,\mathrm{Idl}\,}}(R) \rightarrow 1 \end{aligned}$$

(28.9.13)

(compare to (18.5.5)). From Lemma 28.9.4 and the fact that ${{\,\mathrm{PIdl}\,}}(R) \subseteq \ker (c)$, we obtain an exact sequence

$$\begin{aligned} 1 \rightarrow N_{B^\times }(\mathcal {O})/(F^\times \mathcal {O}^\times ) \rightarrow {{\,\mathrm{Pic}\,}}_R(\mathcal {O}) \xrightarrow {c} {{\,\mathrm{Cl}\,}}_{G(\mathcal {O})}(R). \end{aligned}$$

(28.9.14)

From (28.9.14), we see that $c({{\,\mathrm{Idl}\,}}(R)/{{\,\mathrm{PIdl}\,}}(R)) = ({{\,\mathrm{Cl}\,}}_{G(\mathcal {O})} R)^2$; further, we have $c({{\,\mathrm{Cl}\,}}R)=({{\,\mathrm{Cl}\,}}_{G(\mathcal {O})} R)^2$ with

$$\begin{aligned} \ker c|_{{{\,\mathrm{Cl}\,}}R} = ({{\,\mathrm{Cl}\,}}R)[2]_{\uparrow \mathcal {O}} :={{\,\mathrm{img}\,}}\bigl ({{\,\mathrm{Cl}\,}}_{G(\mathcal {O})}(R)[2] \rightarrow {{\,\mathrm{Cl}\,}}(R)\bigr ) \le ({{\,\mathrm{Cl}\,}}R)[2]. \end{aligned}$$

(28.9.15)

We write $({{\,\mathrm{Cl}\,}}R)[2]_{\uparrow \mathcal {O}}$ because this is the subgroup of ideal classes that lift to the group ${{\,\mathrm{Cl}\,}}_{G(\mathcal {O})}(R)[2]$ (having order dividing 2). Therefore the following diagram commutes, with exact rows and columns:

(28.9.16)

The second main result of this section is the following.

Proposition 28.9.17

Suppose B is ${{{\texttt {\textit{S}}}}}$-indefinite. Then there is a (non-canonically) split exact sequence

$$\begin{aligned} 1 \rightarrow ({{\,\mathrm{Cl}\,}}R)[2]_{\uparrow \mathcal {O}} \rightarrow N_{B^\times }(\mathcal {O})/(F^\times \mathcal {O}^\times ) \rightarrow {{\,\mathrm{AL}\,}}(\mathcal {O}) \rightarrow 1. \end{aligned}$$

(28.9.18)

Proof. The snake lemma implies that the top row of (28.9.16) is exact; the sequence is split by choosing for each class in ${{\,\mathrm{AL}\,}}(\mathcal {O})$ a generator of a representative ideal.

Here is second, self-contained proof which captures the above discussion. Say that a two-sided (integral) $\mathcal {O}$-ideal I is R-primitive if I is not divisible by any integral ideal of the form $\mathfrak a \mathcal {O}$ with $\mathfrak a \subsetneq R$. (If I is integral but not R-primitive, with $I \subseteq \mathfrak a \mathcal {O}$ and $\mathfrak a$ as small as possible, then $\mathfrak a ^{-1} I \subseteq \mathcal {O}$ is integral and now R-primitive.) Let $\alpha \in N_{B^\times }(\mathcal {O})$. Then $\mathcal {O}\alpha \mathcal {O}=I \in {{\,\mathrm{Idl}\,}}(\mathcal {O})$, so we can factor $I=\mathfrak c J$ uniquely with $\mathfrak c$ a fractional ideal of R and J an R-primitive ideal. We have

$$\begin{aligned} aR={{\,\mathrm{nrd}\,}}(\alpha )R = {{\,\mathrm{nrd}\,}}(I)=\mathfrak c ^2 {{\,\mathrm{nrd}\,}}(J)=\mathfrak c ^2 \mathfrak a \end{aligned}$$

(28.9.19)

and so

$$\begin{aligned} 1=[(a)]=[\mathfrak c ]^2[\mathfrak a ] \in {{\,\mathrm{Cl}\,}}_{G(\mathcal {O})}(R) \end{aligned}$$

and in particular $[{{\,\mathrm{nrd}\,}}(J)] \in ({{\,\mathrm{Cl}\,}}_{G(\mathcal {O})} R)^2$. Therefore there is a map

$$\begin{aligned} N_{B^\times }(\mathcal {O}) \rightarrow {{\,\mathrm{AL}\,}}(\mathcal {O}). \end{aligned}$$

(28.9.20)

This map is surjective by strong approximation (see Lemma 28.9.4): if $J \in {{\,\mathrm{Idl}\,}}(\mathcal {O})$ has ${{\,\mathrm{nrd}\,}}(J)=\mathfrak a$ and $[\mathfrak a ]=[\mathfrak c ^{-1}]^2 \in ({{\,\mathrm{Cl}\,}}_{G(\mathcal {O})} R)^2$, then $[{{\,\mathrm{nrd}\,}}(\mathfrak c J)]=1 \in {{\,\mathrm{Cl}\,}}_{G(\mathcal {O})}(R)$ so by Theorem 28.5.5, there exists $\alpha \in B^\times$ such that $\mathcal {O}\alpha \mathcal {O}=\mathfrak c J$ and since $\mathfrak c J \in {{\,\mathrm{Idl}\,}}(\mathcal {O})$ we have $\alpha \in N_{B^\times }(\mathcal {O})$. The map is split by this construction, with a choice of J up to ${{\,\mathrm{Idl}\,}}R$. The kernel of the map in (28.9.20) consists of $\alpha \in N_{B^\times }(\mathcal {O})$ such that $\mathcal {O}\alpha \mathcal {O}=\mathfrak c \mathcal {O}$ with $\mathfrak c \in {{\,\mathrm{Idl}\,}}(R)$, and from the preceding paragraph $[\mathfrak c ^2]=1 \in {{\,\mathrm{Cl}\,}}_{G(\mathcal {O})}(R)$ so $[\mathfrak c ] \in {{\,\mathrm{Cl}\,}}_{G(\mathcal {O})}(R)[2]$; however, the kernel also contains $F^\times \mathcal {O}^\times$, whence the class of $\mathfrak c$ is well-defined only in ${{\,\mathrm{Cl}\,}}(R)$. Therefore the kernel is canonically identified with $({{\,\mathrm{Cl}\,}}R)[2]_{\uparrow \mathcal {O}}$.$\square$

Corollary 28.9.21

Suppose that $\mathcal {O}$ is an Eichler order with ${{\,\mathrm{discrd}\,}}\mathcal {O}= \mathfrak N$ and that B is ${{{\texttt {\textit{S}}}}}$-indefinite. Then

$$\begin{aligned} N_{B^\times }(\mathcal {O})/(F^\times \mathcal {O}^\times ) \simeq {{\,\mathrm{AL}\,}}(\mathcal {O}) \times ({{\,\mathrm{Cl}\,}}R)[2]_{\uparrow \mathcal {O}} \end{aligned}$$

is an abelian 2-group with rank at most $\omega (\mathfrak N )+h_2(R)$, where $\omega (\mathfrak N )$ is the number of prime divisors of $\mathfrak N$ and $h_2(R)=\dim _\mathbb{F _2} ({{\,\mathrm{Cl}\,}}R)[2]$.

Proof. Combine Proposition 28.9.17 and Example 28.9.10.$\square$

Corollary 28.9.22

We have

$$\begin{aligned} N_{{{\,\mathrm{GL}\,}}_2(F)}({{\,\mathrm{M}\,}}_2(R))/(F^\times {{\,\mathrm{GL}\,}}_2(R)) \simeq ({{\,\mathrm{Cl}\,}}R)[2]. \end{aligned}$$

Proof. Apply Corollary 28.9.21 with ${{\,\mathrm{Cl}\,}}_{G(\mathcal {O})}(R) = {{\,\mathrm{Cl}\,}}(R)$ so $({{\,\mathrm{Cl}\,}}R)[2]_{\uparrow \mathcal {O}} = ({{\,\mathrm{Cl}\,}}R)[2]$ and ${{\,\mathrm{AL}\,}}(\mathcal {O})$ the trivial group by Example 28.9.10.$\square$

Remark 28.9.23. Corollary 28.9.21 corrects Vignéras [Vig80a, Exercise III.5.4] to account for possible class group factors.

10 $*$ Stable class group

We conclude this epic chapter with a final result on the stable class group, restoring some generality; we announced a special case of this theorem as Theorem 20.7.16.

Theorem 28.10.1

(Fröhlich–Swan). Let R be a global ring with $F={{\,\mathrm{Frac}\,}}R$, let B be a central simple F-algebra, and let $\mathcal {O}\subset B$ be an R-order. Let $\Omega \subseteq {{\,\mathrm{Pl}\,}}F$ be the set of real places v of F such that $B_v \simeq {{\,\mathrm{M}\,}}_m(\mathbb H )$ for some $m \ge 1$. Then the reduced norm induces an isomorphism

$$\begin{aligned} \begin{aligned} {{\,\mathrm{nrd}\,}}:{{\,\mathrm{StCl}\,}}\mathcal {O}&\xrightarrow {\smash {{\sim }}}F_{>_{\Omega } 0}^\times \backslash \widehat{F}^\times \!/ {{\,\mathrm{nrd}\,}}(\widehat{\mathcal {O}}^\times ) \end{aligned} \end{aligned}$$

(28.10.2)

of finite abelian groups. In particular, if B is a quaternion algebra and $\mathcal {O}$ is locally norm-maximal order, then

$$ {{\,\mathrm{StCl}\,}}\mathcal {O}\simeq {{\,\mathrm{Cl}\,}}_\Omega R $$

where $\Omega \subseteq {{\,\mathrm{Ram}\,}}B$ is the set of real ramified places in B.

Proof. We give only a sketch of the proof. For further details, see the references given for the proof of Theorem 20.7.16.

We first show that the map (28.10.2) is well-defined. Suppose that $I \oplus \mathcal {O}^r \simeq I' \oplus \mathcal {O}^r$ with $r \ge 0$. If $r=0$, we are done; so suppose $r \ge 1$. Extending scalars, we find an isomorphism $\phi :B^{r+1} \rightarrow B^{r+1}$ of left B-modules, represented by an element $\gamma \in {{\,\mathrm{GL}\,}}_{r+1}(B)$ acting on the left. In a similar way, associated to $I \oplus \mathcal {O}^r$ we obtain a class

$$\begin{aligned} {{\,\mathrm{GL}\,}}_{r+1}(B) \widehat{\alpha }{{\,\mathrm{GL}\,}}_{r+1}(\widehat{\mathcal {O}}) \in {{\,\mathrm{GL}\,}}_{r+1}(B) \backslash {{\,\mathrm{GL}\,}}_{r+1}(\widehat{B}) / {{\,\mathrm{GL}\,}}_{r+1}(\widehat{\mathcal {O}}) \end{aligned}$$

by choosing in each completion an isomorphism with $\mathcal {O}_\mathfrak p ^{r+1}$ represented by a matrix, well-defined up to a change of basis on the right (and on the left by a global isomorphism). Now by strong approximation in the advanced version announced in Remark 28.7.10, the reduced norm induces a bijection

$$\begin{aligned} {{\,\mathrm{nrd}\,}}:{{\,\mathrm{GL}\,}}_{r+1}(B) \backslash {{\,\mathrm{GL}\,}}_{r+1}(\widehat{B}) / {{\,\mathrm{GL}\,}}_{r+1}(\widehat{\mathcal {O}}) \rightarrow F_{>_{\Omega } 0}^\times \backslash \widehat{F}^\times \!/{{\,\mathrm{nrd}\,}}(\widehat{\mathcal {O}}^\times ) \end{aligned}$$

after we check that the codomain is indeed the image of the reduced norm. Then

$$\begin{aligned} {{\,\mathrm{nrd}\,}}(\widehat{\alpha }')={{\,\mathrm{nrd}\,}}(\gamma \widehat{\alpha })={{\,\mathrm{nrd}\,}}(\gamma ){{\,\mathrm{nrd}\,}}(\widehat{\alpha })={{\,\mathrm{nrd}\,}}(\widehat{\alpha }) \in F_{>_{\Omega } 0}^\times \backslash \widehat{F}^\times \!/{{\,\mathrm{nrd}\,}}(\widehat{\mathcal {O}}^\times ) \end{aligned}$$

so the map is well-defined.

Similarly, the map (28.10.2) is a group homomorphism: an isomorphism $I \oplus I' \simeq J \oplus \mathcal {O}$ gives ${{\,\mathrm{nrd}\,}}(\gamma \beta )={{\,\mathrm{nrd}\,}}(\beta )={{\,\mathrm{nrd}\,}}(\widehat{\alpha }\widehat{\alpha }')$. The map is surjective. If $[I]{}_{\textsf {\tiny {St}} }$ is in the kernel and ${{\,\mathrm{nrd}\,}}(I)$ is trivial, then so too is ${{\,\mathrm{nrd}\,}}(\widehat{\alpha }_1)$ where $\widehat{\alpha }_1$ corresponds to $I \oplus \mathcal {O}$; by strong approximation, this means that there exists $\beta \in {{\,\mathrm{GL}\,}}_2(B)$ and $\widehat{\mu }\in {{\,\mathrm{GL}\,}}_2(\widehat{\mathcal {O}})$ such that $\widehat{\alpha }_1 = \beta \widehat{\mu }$ and so via $\beta$ we have $I \oplus \mathcal {O}\simeq \mathcal {O}^{\oplus 2}$, which means $[I]{}_{\textsf {\tiny {St}} }=[\mathcal {O}]$.$\square$

Exercises

Unless otherwise specified, let R be a global ring with eligible set ${{{\texttt {\textit{S}}}}}$ and $F={{\,\mathrm{Frac}\,}}R$, and let B be a quaternion algebra over F, and let $\mathcal {O}\subset B$ be an R-order.

1.:

Give another proof weak approximation for $\mathbb Q$, as follows.

(a):

Let ${{{\texttt {\textit{S}}}}}\subseteq {{\,\mathrm{Pl}\,}}\mathbb Q$ be a finite, nonempty set of places. Show that, for each $v \in {{{\texttt {\textit{S}}}}}$, there exists $y_v \in \mathbb Q ^\times$ such that $|y_v \,|_v < 1$ and $|y_v \,|_{v'} > 1$ for all $v' \in {{{\texttt {\textit{S}}}}}\smallsetminus \{v\}$. [Hint: let $y_\infty :=\prod _{p \in {{{\texttt {\textit{S}}}}}} 1/p$ and to get $y_p$, multiply $y_\infty$ by a power of p.]

(b):

For the elements $y_v$ constructed in (a), show that for all $v \in {{{\texttt {\textit{S}}}}}$ we have $1/(1+y_v^n) \rightarrow 1 \in \mathbb Q _v$ and $1/(1+y_v^n) \rightarrow 0\in \mathbb Q _{v'}$ for $v' \ne v$. [How does this relate to the proof of the CRT?]

(c):

Prove weak approximation. [Hint: given $x_v \in \mathbb Q _v$ for each $v \in {{{\texttt {\textit{S}}}}}$, show that

$$\begin{aligned} z_n :=\sum _{v \in {{{\texttt {\textit{S}}}}}} \frac{x_v}{1+y_v^n} \rightarrow x_v \in \mathbb Q _v \end{aligned}$$

as $n \rightarrow \infty$.]

2.:

Show that for all $N \ge 1$, the group ${{\,\mathrm{SL}\,}}_2(\mathbb Z /N\mathbb Z )$ is generated by two elements of order N.

3.:

Let F be a field and let $n \in \mathbb Z _{\ge 2}$. Show that the elementary matrices (which differ from the identity matrix in exactly one off-diagonal place) generate ${{\,\mathrm{SL}\,}}_n(F)$ as a group. [Hint: Argue by induction, and reduce a matrix to the identity by elementary row and column operations.]

4.:

Let $n \ge 2$.

(a):

Let R be a Euclidean domain. Show that the elementary matrices generate ${{\,\mathrm{SL}\,}}_n(R)$. [Hint: In view of Lemma 28.3.3, argue by induction.]

(b):

Using (a), show that Proposition 28.3.5 and Corollary 28.3.6 hold for ${{\,\mathrm{SL}\,}}_n$.

5.:

Let B be a central simple algebra over the global field F. We say B satisfies the ${{{\texttt {\textit{S}}}}}$ -Eichler condition if there exists a place $v \in {{{\texttt {\textit{S}}}}}$ such that $B_v$ is not a division algebra. Show that if F is a number field and ${{{\texttt {\textit{S}}}}}$ is the set of archimedean places of F, then B satisfies the ${{{\texttt {\textit{S}}}}}$-Eichler condition if and only if B is not a totally definite quaternion algebra.

6.:

In this exercise, we provide details in the proof of Proposition 28.3.5, considering elementary matrices of rings that are not necessarily domains.

Let A be a commutative ring (with 1), not necessarily a domain. We say A isEuclidean if there exists a function $N:A \rightarrow \mathbb Z _{\ge 0}$ such that for all $a,b \in A$ with $b \ne 0$, there exists $q,r \in A$ such that $a=qb+r$ and $N(r)<N(b)$.

In the first parts, we suppose A is Euclidean with respect to N and show that the situation is quite analogous to the case when A is a domain.

(a):

Show that for all $b \in A$ we have $N(b) \ge N(0)$ with equality if and only if $b=0$.

(b):

Let $m = \min \{ N(b) : b \ne 0\}$. Show that if $N(b)=m$ then $b \in A^\times$.

(c):

Show that every ideal of A is principal. [We call A a principal ideal ring.]

We now consider examples.

(d):

Let A be an Artinian local principal ideal ring with a unique maximal ideal $\mathfrak m =\pi A$. (For example, we may take $A=R/\mathfrak p ^e$ where R is a Dedekind domain, $\mathfrak p$ is a nonzero prime ideal, and $e \in \mathbb Z _{\ge 0}$.) Show that for all $x \in A$ with $x \ne 0$, there exists $u \in A^\times$ and a unique $n \in \mathbb Z _{\ge 0}$ such that $x=u\pi ^n$. Conclude that A is Euclidean with $N(x)=n$.

(e):

If $A \simeq \prod _{i=1}^r A_i$ with each $A_i$ an Artinian local principal ideal ring, show that A is Euclidean under $N(x) = \sum _{i=1}^r N_i(\pi _i(x))$ where $N_i$ is as given in (a) for $A_i$ and $\pi _i :A \rightarrow A_i$ is the projection.

We conclude with the application.

(f):

Let A be a Euclidean ring. Show that ${{\,\mathrm{SL}\,}}_2(A)$ is generated by elementary matrices. [Hint: Show that the proof in Lemma 28.3.3 carries over.]

7.:

Consider the quaternion algebra $B :=({-11,-17} \mid \mathbb{Q })$ and let $\mathcal {O}=\mathbb Z \oplus \mathbb Z i \oplus \mathbb Z j \oplus \mathbb Z ij$. Then B is definite, so strong approximation does not apply. Indeed, show that $\widehat{B}^1 \not \subseteq B^\times \widehat{\mathcal {O}}^\times$ as follows.

(a):

Find $a,b,c,d,m \in \mathbb Z$ with $3 \not \mid m$ such that

$$\begin{aligned} a^2+11b^2=c^2+17d^2=3m. \end{aligned}$$

(b):

Now let $\widehat{\alpha }= (\alpha _p)_p \in \widehat{B}^\times$ be such that

$$\begin{aligned} \alpha _3=(a+bi)(c+dj)^{-1}=\frac{(a+bi)(c-dj)}{3m} \end{aligned}$$

and $\alpha _p=1$ if $p \ne 3$. Show that $\widehat{\alpha }\in \widehat{B}^1$ and $\widehat{\alpha }\not \in B^\times \widehat{\mathcal {O}}^\times$. [Hint: Observe that ${{\,\mathrm{nrd}\,}}|_{\mathcal {O}}$ only represents 9 by $\pm 3$.]

(c):

Prove that the right $\mathcal {O}$-ideals $I_1 :=3\mathcal {O}+ (a+bi)\mathcal {O}$ and $I_2 :=3\mathcal {O}+ (c+di)\mathcal {O}$ are not principal. How does this relate to (b)?

8.:

Let F be a number field with ring of integers R. Show that there is a finite set ${{{\texttt {\textit{S}}}}}$ of (rational) primes such that every totally positive element of R can be written as a sum of four squares of elements of F whose denominator is a product of primes in ${{{\texttt {\textit{S}}}}}$. (We may not be able to write every such element as sum of four squares from R, but we only need denominators in ${{{\texttt {\textit{S}}}}}$.)

9.:

Suppose that B is ${{{\texttt {\textit{S}}}}}$-indefinite. Suppose $\mathcal {O}$ is locally norm-maximal. Give a direct proof using strong approximation that if $\mathcal {O}\subseteq B$ is an R-order and I is an invertible right fractional $\mathcal {O}$-ideal, then I is principal if and only if $[{{\,\mathrm{nrd}\,}}(I)]$ is trivial in ${{\,\mathrm{Cl}\,}}_{\Omega } R$. [Hint: If $\alpha \in B^\times$ satisfies ${{\,\mathrm{nrd}\,}}(\alpha )R = {{\,\mathrm{nrd}\,}}(I)$, consider $\alpha ^{-1} I$.]

$\triangleright$ 10.:

Show that $B^\times \widehat{\mathcal {O}}^\times \cap \widehat{B}^1 = B^1 \widehat{\mathcal {O}}^1$ if and only if ${{\,\mathrm{nrd}\,}}(\mathcal {O}^\times ) = F_{>_{\Omega } 0}^\times \cap {{\,\mathrm{nrd}\,}}(\widehat{\mathcal {O}}^\times )$ (see 28.4.7).

11.:

Suppose that B is ${{{\texttt {\textit{S}}}}}$-indefinite. Show that $\#{{\,\mathrm{Typ}\,}}\mathcal {O}$ is a power of 2.

12.:

Suppose ${{\,\mathrm{disc}\,}}B=\mathfrak D$ and $\mathcal {O}$ is an Eichler order of level $\mathfrak M$ and reduced discriminant $\mathfrak N =\mathfrak D \mathfrak M$. Show that ${{\,\mathrm{Cl}\,}}_{GN(\mathcal {O})} R$ is the quotient of $({{\,\mathrm{Cl}\,}}R)/({{\,\mathrm{Cl}\,}}R)^2$ by the subgroup generated by the classes of ideals $\mathfrak p \mid \mathfrak D$ together with $\mathfrak q \mid \mathfrak N$ such that ${{\,\mathrm{ord}\,}}_\mathfrak q (\mathfrak N )$ is odd.

Conclude that for $B={{\,\mathrm{M}\,}}_2(F)$ and $\mathcal {O}=\begin{pmatrix} R &{} R \\ \mathfrak N &{} R \end{pmatrix}$, the type set ${{\,\mathrm{Typ}\,}}\mathcal {O}$ is represented by the isomorphism classes of the orders $\begin{pmatrix} R &{} \mathfrak b \\ \mathfrak N \mathfrak b ^{-1} &{} R \end{pmatrix}$ for $[\mathfrak b ] \in {{\,\mathrm{Cl}\,}}_{GN(\mathcal {O})} R$.