Quaternion orders

Abstract

In the previous chapter, we gave a rather general classification of quaternion orders in terms of ternary quadratic modules. In this chapter, we take a guided tour of the most important animals in the zoo of quaternion orders, identifying those with good local properties. We continue in the next chapter with a second visit to the zoo.

In the previous chapter, we gave a rather general classification of quaternion orders in terms of ternary quadratic modules. In this chapter, we take a guided tour of the most important animals in the zoo of quaternion orders, identifying those with good local properties. We continue in the next chapter with a second visit to the zoo.

1 $\triangleright$ Highlights of quaternion orders

We begin in this section by providing some highlights of this tour. Let B be a quaternion algebra over $\mathbb Q$ of discriminant $D :={{\,\mathrm{disc}\,}}B$ and let $\mathcal {O}\subset B$ be an order with reduced discriminant $N :={{\,\mathrm{discrd}\,}}(\mathcal {O})$. Then $N=DM$ with $M \in \mathbb {Z}_{\ge 1}$, and $\mathcal {O}$ is maximal if and only if $N=D$.

23.1.1

(Maximal orders). The nicest orders are undoubtedly the maximal orders, those not properly contained in another order. An order is maximal if and only if it is locally maximal (Lemma 10.4.3), i.e. p-maximal for all primes p; globally, an order $\mathcal {O}$ is maximal if and only if $N=D$ (i.e., $M=1$).

We have either $B \simeq {{\,\mathrm{M}\,}}_2(\mathbb Q _p)$ or B is a division algebra over $\mathbb Q _p$ (unique up to isomorphism). If B is split, then a maximal order is isomorphic (conjugate) to ${{\,\mathrm{M}\,}}_2(\mathbb {Z}_p)$, and the corresponding ternary quadratic form is the determinant $xy-z^2$ (see Example 22.3.12). If instead B is division, then the unique maximal order is the valuation ring, with corresponding anisotropic form $x^2-ey^2+pz^2$ for $p \ne 2$, where $e \in \mathbb {Z}$ is a quadratic nonresidue modulo p (and for $p=2$, the associated form is $x^2+xy+y^2+2z^2$).

Maximal orders have modules with good structural properties: all lattices $I \subset B$ with left or right order equal to a maximal order $\mathcal {O}$ are invertible (Theorem 18.1.2).

There is a combinatorial structure, called the Bruhat–Tits tree, that classifies maximal orders in ${{\,\mathrm{M}\,}}_2(\mathbb Q _p)$ (as endomorphism rings of lattices, up to scaling): the Bruhat–Tits tree is a $p+1$-regular tree (see section 23.5).

Examining orders beyond maximal orders is important for the development of the theory: already the Lipschitz order—an order which arises when considering if a positive integer is the sum of four squares—is properly contained inside the Hurwitz order (Chapter 11).

23.1.2

(Hereditary orders). More generally, we say that $\mathcal {O}$ is hereditary if every left or right fractional $\mathcal {O}$-ideal (i.e., lattice $I \subseteq B$ with left or right order containing $\mathcal {O}$) is invertible. Maximal orders are hereditary, and being hereditary is a local property. A hereditary $\mathbb {Z}_p$-order $\mathcal {O}_p \subseteq B_p$ is either maximal or

$$ \mathcal {O}_p \simeq \begin{pmatrix} \mathbb {Z}_p &{} \mathbb {Z}_p \\ p\mathbb {Z}_p &{} \mathbb {Z}_p \end{pmatrix} = \left\{ \begin{pmatrix} a &{} b \\ pc &{} d \end{pmatrix} : a,b,c,d \in \mathbb {Z}_p\right\} \subseteq {{\,\mathrm{M}\,}}_2(\mathbb Q _p) \simeq B_p $$

with associated ternary quadratic form $xy-pz^2$. Thus $\mathcal {O}\subset B$ is hereditary if and only if ${{\,\mathrm{discrd}\,}}(\mathcal {O})=D M$ is squarefree, so in particular $\gcd (D,M)=1$.

Hereditary orders share the nice structural property of maximal orders: all lattices $I \subset B$ with hereditary left or right order are invertible. The different ideal ${{\,\mathrm{diff}\,}}\mathcal {O}_p$ is generated by any element $\mu \in \mathcal {O}_p$ such that $\mu ^2 \in p\mathbb {Z}_p$.

23.1.3

(Eichler orders). More generally, we can consider orders that are “upper triangular modulo M” with $\gcd (D,M)=1$ (i.e., avoiding primes that ramify in B). The order

$$\begin{aligned} \begin{pmatrix} \mathbb {Z}_p &{} \mathbb {Z}_p \\ p^e\mathbb {Z}_p &{} \mathbb {Z}_p \end{pmatrix} \subseteq {{\,\mathrm{M}\,}}_2(\mathbb {Z}_p) \end{aligned}$$

is called the standard Eichler order of level $p^e$ in ${{\,\mathrm{M}\,}}_2(\mathbb Q _p)$. A $\mathbb {Z}_p$-order $\mathcal {O}_p \subseteq {{\,\mathrm{M}\,}}_2(\mathbb Q _p)$ is an Eichler order if $\mathcal {O}_p$ is isomorphic to a standard Eichler order. The ternary quadratic form associated to an Eichler order of level $p^e$ is $xy-p^ez^2$.

Globally, we say $\mathcal {O}\subset B$ is a Eichler order of level M if ${{\,\mathrm{discrd}\,}}(\mathcal {O})=N=DM$ with $\gcd (D,M)=1$ and $\mathcal {O}_p$ is an Eichler order of level $p^e$ for all $p^e \parallel M$. In particular, $\mathcal {O}_p$ is maximal at all primes $p \mid D$. Every hereditary order is Eichler, and an Eichler order is hereditary if and only if its level M (or reduced discriminant N) is squarefree. A maximal $\mathbb {Z}_p$-order $\mathcal {O}_p \subseteq {{\,\mathrm{M}\,}}_2(\mathbb {Z}_p)$ is an Eichler order of level $1=p^0$. Eichler orders play a crucial role in the context of modular forms, as we will see in the final part of this monograph.

This local description of Eichler orders also admits a global description. The standard Eichler order $\mathcal {O}_p$ can be written

$$\begin{aligned} \mathcal {O}_p=\begin{pmatrix} \mathbb {Z}_p &{} \mathbb {Z}_p \\ p^e\mathbb {Z}_p &{} \mathbb {Z}_p \end{pmatrix} = {{\,\mathrm{M}\,}}_2(\mathbb {Z}_p) \cap \begin{pmatrix} \mathbb {Z}_p &{} p^{-e} \mathbb {Z}_p \\ p^e \mathbb {Z}_p &{} \mathbb {Z}_p \end{pmatrix}&= {{\,\mathrm{M}\,}}_2(\mathbb {Z}_p) \cap \varpi ^{-1} {{\,\mathrm{M}\,}}_2(\mathbb {Z}_p) \varpi \end{aligned}$$

as the intersection of a (unique) pair of maximal orders, with

$$\begin{aligned} \varpi :=\begin{pmatrix} 0 &{} 1 \\ p^e &{} 0 \end{pmatrix} \in N_{{{\,\mathrm{GL}\,}}_2(\mathbb Q _p)}(\mathcal {O}_p) \end{aligned}$$

(23.1.4)

a generator of the group $N_{{{\,\mathrm{GL}\,}}_2(\mathbb Q _p)}(\mathcal {O}_p)/\mathbb Q _p^\times \mathcal {O}_p^\times \simeq \mathbb {Z}/2\mathbb {Z}$, and $\varpi ^2=p^e$. The different ${{\,\mathrm{diff}\,}}\mathcal {O}_p$ is the two-sided ideal generated by $\varpi$.

From the local-global dictionary, it follows that $\mathcal {O}\subset B$ is Eichler if and only if $\mathcal {O}$ is the intersection of two (not necessarily distinct) maximal orders.

2 Maximal orders

Throughout this chapter, we impose the following notation: let R be a Dedekind domain with field of fractions $F={{\,\mathrm{Frac}\,}}R$, let B be a quaternion algebra over F, and let $\mathcal {O}\subseteq B$ an R-order.

23.2.1

We make the following convention. When we say “R is local”, we mean that R is a complete DVR, and in this setting we let $\mathfrak p = \pi R$ be its maximal ideal, and $k=R/\mathfrak p$ the residue field. When we want to return to the general context, we will say “R is Dedekind”.

Recall that an R-order is maximal if it is not properly contained in another order. We begin in this section by summarizing the properties of maximal orders for convenience.

23.2.2

Being maximal is a local property (Lemma 10.4.3), so the following are equivalent:

• (i) $\mathcal {O}$ is a maximal R-order;

• (ii) $\mathcal {O}_{(\mathfrak p )}$ is a maximal $R_{(\mathfrak p )}$-order for all $\mathfrak p \subseteq R$; and

• (iii) $\mathcal {O}_\mathfrak p$ is a maximal $R_\mathfrak p$-order for all $\mathfrak p \subseteq R$.

We recall (Theorem 15.5.5) that an order over a global ring R is maximal if and only if ${{\,\mathrm{discrd}\,}}(\mathcal {O})={{\,\mathrm{disc}\,}}_R(B)$. Local maximal R-orders have the nice local description.

23.2.3

Suppose R is local and that $B \simeq {{\,\mathrm{M}\,}}_2(F)$ is split. Then by Corollary 10.5.5, every maximal R-order in ${{\,\mathrm{M}\,}}_2(F)$ is conjugate to ${{\,\mathrm{M}\,}}_2(R)$ by an element of ${{\,\mathrm{GL}\,}}_2(R)$, i.e. $\mathcal {O}\simeq {{\,\mathrm{M}\,}}_2(R)$. We have ${{\,\mathrm{discrd}\,}}(\mathcal {O})=R$. All two-sided ideals of $\mathcal {O}$ are powers of ${{\,\mathrm{rad}\,}}(\mathcal {O})=\mathfrak p \mathcal {O}$, and

$$\begin{aligned} \mathcal {O}/{{\,\mathrm{rad}\,}}(\mathcal {O}) \simeq {{\,\mathrm{M}\,}}_2(k). \end{aligned}$$

The associated ternary quadratic form is similar to $Q(x,y,z)=xy-z^2$, by Example 22.3.12 and the classification theorem (Main Theorem 22.1.1). Finally,

$$\begin{aligned} N_{B^\times }(\mathcal {O}) = N_{{{\,\mathrm{GL}\,}}_2(F)}({{\,\mathrm{M}\,}}_2(R)) = F^\times \mathcal {O}^\times . \end{aligned}$$

(23.2.4)

23.2.5

Suppose R is local but now that B is a division algebra. Then the valuation ring $\mathcal {O}\subset B$ is the unique maximal R-order by Proposition 13.3.4.

Suppose further that the residue field k is finite (equivalently, that F is a local field). Then Theorem 13.3.11 applies, and we have

$$\begin{aligned} \mathcal {O}\simeq S \oplus S j \subseteq \displaystyle {\biggl (\frac{K,\pi }{F}\biggr )} \end{aligned}$$

where $K \supseteq F$ is the unique quadratic unramified extension of F and S its valuation ring. We computed in 15.2.12 that ${{\,\mathrm{discrd}\,}}(\mathcal {O})=\mathfrak p$. All two-sided ideals of $\mathcal {O}$ are powers of the unique maximal ideal ${{\,\mathrm{rad}\,}}(\mathcal {O})=P=\mathcal {O}j \mathcal {O}$, and $\ell :=\mathcal {O}/{{\,\mathrm{rad}\,}}(\mathcal {O})$ is a quadratic field extension of k. By Exercise 13.7, we have $P=[\mathcal {O},\mathcal {O}]$ equal to the commutator. We also have $P={{\,\mathrm{diff}\,}}\mathcal {O}$; this can be computed directly, or it follows from the condition that ${{\,\mathrm{nrd}\,}}({{\,\mathrm{diff}\,}}\mathcal {O})={{\,\mathrm{discrd}\,}}\mathcal {O}=\mathfrak p$.

Write $S=R[i]$ with $i^2=ui-b$, and $u,b \in R$. Then 1, i, j, k where $k=-ij$ is an R-basis for $\mathcal {O}$. We have

$$\begin{aligned} k^2=i(ji)j=i(\overline{i}j)j={{\,\mathrm{nrd}\,}}(i)\pi , \end{aligned}$$

so ${{\,\mathrm{trd}\,}}(k)=0$, and $\overline{k}=-k=ij$. This gives multiplication table

$$\begin{aligned} \begin{aligned} i^2&= ui-b \quad&jk&= -\pi \overline{i} \\ j^2&= \pi&ki&= b\overline{j} \\ k^2&= b\pi&ij&= \overline{k} \end{aligned} \end{aligned}$$

(23.2.6)

realizing the basis as a good basis in the sense of 22.4.7; the associated ternary quadratic form is

$$\begin{aligned} {{\,\mathrm{nrd}\,}}^\sharp (\mathcal {O})(x,y,z)=-\pi x^2+by^2+uyz+z^2= -\pi x^2 + {{\,\mathrm{Nm}\,}}_{K|F}(z+yi). \end{aligned}$$

(23.2.7)

Finally, since the valuation ring is the unique maximal order and conjugation respects integrality, we have

$$\begin{aligned} N_{B^\times }(\mathcal {O}) = B^\times . \end{aligned}$$

(23.2.8)

Finally, lattices over maximal orders are necessarily invertibility, as follows.

23.2.9

All lattices $I \subset B$ with left or right order equal to a maximal order $\mathcal {O}$ are invertible, by Theorem 18.1.2, proven in Proposition 18.3.2. (We also gave a different proof of this fact in Proposition 16.6.15(b).)

The classification of two-sided ideals and their classes follows from that of hereditary orders: see 23.3.19.

3 Hereditary orders

Hereditary orders were investigated in section 21.4 in general; here, we provide a quick development specific to quaternion algebras. As mentioned before, a good general reference for (maximal and) hereditary orders is Reiner [Rei2003, Chapters 3–6, 9].

We recall that $\mathcal {O}$ is hereditary if every left (or right) ideal $I \subseteq \mathcal {O}$ is projective as a left (or right) $\mathcal {O}$-module. Being hereditary is a local property 21.4.4 because projectivity is.

23.3.1

Suppose R is local. By Main Theorem 21.1.4 and Corollary 21.1.5, the following are equivalent:

• (i) $\mathcal {O}$ is hereditary;

• (ii) ${{\,\mathrm{rad}\,}}\mathcal {O}$ is projective as a left (or right) $\mathcal {O}$-module;

• (iii) $\mathcal {O}{}_{\textsf {\tiny {L}} }({{\,\mathrm{rad}\,}}\mathcal {O})=\mathcal {O}{}_{\textsf {\tiny {R}} }({{\,\mathrm{rad}\,}}\mathcal {O})=\mathcal {O}$;

• (iv) ${{\,\mathrm{rad}\,}}\mathcal {O}$ is invertible as a (sated) two-sided $\mathcal {O}$-ideal; and

• (v) either $\mathcal {O}$ is maximal or

  $$\begin{aligned} \mathcal {O}\simeq \begin{pmatrix} R &{} R \\ \mathfrak p &{} R \end{pmatrix} \subseteq {{\,\mathrm{M}\,}}_2(F) \simeq B. \end{aligned}$$

We now spend some time investigating ‘the’ local hereditary order that is not maximal. So until further notice, suppose R is local and let

$$\begin{aligned} \mathcal {O}_0(\mathfrak p ) :=\begin{pmatrix} R &{} R \\ \mathfrak p &{} R \end{pmatrix} \subseteq {{\,\mathrm{M}\,}}_2(R). \end{aligned}$$

To avoid clutter, we will just write $\mathcal {O}= \mathcal {O}_0(\mathfrak p )$. We have

$$\begin{aligned} {{\,\mathrm{discrd}\,}}(\mathcal {O})=[{{\,\mathrm{M}\,}}_2(R):\mathcal {O}]_{R} {{\,\mathrm{disc}\,}}({{\,\mathrm{M}\,}}_2(R))=\mathfrak p . \end{aligned}$$

(23.3.2)

23.3.3

A multiplication table for $\mathcal {O}$ is obtained from the one for ${{\,\mathrm{M}\,}}_2(R)$ in Example 22.3.12, scaling j by $\pi$ in (22.3.12), which gives the same multiplication laws as (22.3.13) except now $ij=-\pi \overline{k}$ and c is scaled by $\pi$. Therefore, the similarity class of ternary quadratic forms associated to $\mathcal {O}$ is represented by

$$\begin{aligned} Q(x,y,z)=xy-\pi z^2. \end{aligned}$$

(23.3.4)

23.3.5

Let $J :={{\,\mathrm{rad}\,}}(\mathcal {O})$. Then

$$\begin{aligned} J = \begin{pmatrix} \mathfrak p &{} R \\ \mathfrak p &{} \mathfrak p \end{pmatrix} \end{aligned}$$

(23.3.6)

by (21.3.6), and we find

$$\begin{aligned} \mathcal {O}/J \simeq \begin{pmatrix} k &{} 0 \\ 0 &{} k \end{pmatrix} \simeq k \times k \end{aligned}$$

(23.3.7)

as k-algebras. Now let

$$\begin{aligned} \varpi = \begin{pmatrix} 0 &{} 1 \\ \pi &{} 0 \end{pmatrix}. \end{aligned}$$

(23.3.8)

Then a direct calculation yields

$$\begin{aligned} J=\mathcal {O}\varpi = \varpi \mathcal {O}\end{aligned}$$

(23.3.9)

in agreement with 23.3.1(ii)–23.3.1(iii), and J is an invertible $\mathcal {O}$-ideal. Since $\varpi ^2=\pi$, we have

$$\begin{aligned} J^2=\mathfrak p \mathcal {O}. \end{aligned}$$

(23.3.10)

In particular, $J^{-1} = \pi ^{-1} J$, and the powers of J give a filtration

$$\begin{aligned} \mathcal {O}\supsetneq J \supsetneq \mathfrak p \mathcal {O}\supsetneq J^3 \supsetneq \dots . \end{aligned}$$

(23.3.11)

23.3.12

We compute that

$$ \varpi {{\,\mathrm{M}\,}}_2(R) \varpi ^{-1} = \begin{pmatrix} R &{} \mathfrak p ^{-1}R \\ \mathfrak p &{} R \end{pmatrix} $$

and hence

$$\begin{aligned} \mathcal {O}= {{\,\mathrm{M}\,}}_2(R) \cap \varpi {{\,\mathrm{M}\,}}_2(R) \varpi ^{-1} \end{aligned}$$

is the intersection of two maximal orders.

Lemma 23.3.13

The group ${{\,\mathrm{Idl}\,}}(\mathcal {O})={{\,\mathrm{PIdl}\,}}(\mathcal {O})$ is generated by $J={{\,\mathrm{rad}\,}}\mathcal {O}$, with $J^2=\mathfrak p \mathcal {O}$.

Proof. We have ${{\,\mathrm{Idl}\,}}(\mathcal {O})={{\,\mathrm{PIdl}\,}}(\mathcal {O})$ since R is local: invertible is equivalent to principal (Main Theorem 16.6.1).

Let $I \subseteq \mathcal {O}$ be an invertible two-sided $\mathcal {O}$-ideal. Then by (23.3.11), we can replace I by a power of $J^{-1}$ and suppose that $\mathcal {O}\subsetneq I \subseteq J$. Invertible means locally principal, so $I = \mathcal {O}\alpha =\alpha \mathcal {O}$ and

$$\begin{aligned}{}[\mathcal {O}:I]_{R}={{\,\mathrm{nrd}\,}}(I)^2={{\,\mathrm{nrd}\,}}(\alpha )^2 R \mid [\mathcal {O}:J]_{R} = \mathfrak p ^2. \end{aligned}$$

Thus $[\mathcal {O}:I]_{R}=\mathfrak p ^2$, so $[I:J]_R=R$ and $I=J$. (This also follows directly from Proposition 16.4.3.)

Here is a second computational proof. The image $I/J \subseteq \mathcal {O}/J \simeq k \times k$ (23.3.7) is a two-sided ideal, therefore

$$\begin{aligned} I=\begin{pmatrix} R &{} \mathfrak p \\ \mathfrak p &{} \mathfrak p \end{pmatrix} \quad \text {or} \quad I=\begin{pmatrix} \mathfrak p &{} \mathfrak p \\ \mathfrak p &{} R \end{pmatrix}. \end{aligned}$$

But

$$ \begin{pmatrix} R &{} \mathfrak p \\ \mathfrak p &{} \mathfrak p \end{pmatrix} \begin{pmatrix} 0 &{} 1 \\ 0 &{} 0 \end{pmatrix} = \begin{pmatrix} 0 &{} R \\ 0 &{} \mathfrak p \end{pmatrix} \not \subseteq \begin{pmatrix} R &{} \mathfrak p \\ \mathfrak p &{} \mathfrak p \end{pmatrix}; $$

we get a contradiction with the other possibility by multiplying instead on the left. A third “pure matrix multiplication” proof is also requested in Exercise 23.1.

The second statement was already proven in (23.3.10).

$\square$

Corollary 23.3.14

We have $N_{B^\times }(\mathcal {O})/(F^\times \mathcal {O}^\times ) \simeq \mathbb {Z}/2\mathbb {Z}$ generated by $\varpi$, and

$$ {{\,\mathrm{nrd}\,}}(N_{B^\times }(\mathcal {O})) = {\left\{ \begin{array}{ll} F^{\times 2} R_\mathfrak p ^\times , &{} \text { if}\, e\, \text {is even;} \\ F^\times , &{} \text { if}\, e\,\text {is odd.} \end{array}\right. } $$

Proof. By (18.5.4), we have an isomorphism

$$\begin{aligned} N_{B^\times }(\mathcal {O})/(F^\times \mathcal {O}^\times ) \simeq {{\,\mathrm{PIdl}\,}}(\mathcal {O})/{{\,\mathrm{PIdl}\,}}(R); \end{aligned}$$

by Lemma 23.3.13, the latter is generated by $J=\varpi \mathcal {O}$ with $J^2=\pi \mathcal {O}$. The computation of reduced norms is immediate. $\square$

23.3.15

Lemma 23.3.13 also implies the description

$$\begin{aligned} J=[\mathcal {O},\mathcal {O}] \end{aligned}$$

(23.3.16)

as the commutator. Since $\mathcal {O}/J$ is commutative, we know $[\mathcal {O},\mathcal {O}] \subseteq J$; but $[\mathcal {O},\mathcal {O}] \subsetneq J^2=\mathfrak p \mathcal {O}$ since $\mathcal {O}/\mathfrak p \mathcal {O}$ is noncommutative. We also have $J={{\,\mathrm{diff}\,}}\mathcal {O}$ for the same reason, since ${{\,\mathrm{nrd}\,}}({{\,\mathrm{diff}\,}}\mathcal {O})={{\,\mathrm{discrd}\,}}\mathcal {O}=\mathfrak p$. (A matrix proof of these facts are requested in Exercise 23.8.)

23.3.17

We now classify the left $\mathcal {O}$-lattices, up to isomorphism. Each such $\mathcal {O}$-lattice is projective, since $\mathcal {O}$ is hereditary.

By the Krull-Schmidt theorem (Theorem 20.6.2), every $\mathcal {O}$-lattice can be written as the direct sum of indecomposables, so it is enough to classify the indecomposables; and we did just this in 21.5.3. Explicitly, we have $V=\begin{pmatrix} F \\ F \end{pmatrix} \simeq F^2$ the simple $B={{\,\mathrm{M}\,}}_2(F)$-module, and we take the $\mathcal {O}$-lattice $M=\begin{pmatrix} R \\ R \end{pmatrix}\subset V$. We have $JM=\begin{pmatrix} R \\ \mathfrak p \end{pmatrix}$ and $J^2 M = \begin{pmatrix} \mathfrak p \\ \mathfrak p \end{pmatrix} = \pi M$, and M, JM give a complete set of indecomposable left $\mathcal {O}$-modules. As expected, $\mathcal {O}= JM \oplus M$ is a decomposition of $\mathcal {O}$ into projective indecomposable left $\mathcal {O}$-modules.

The preceding local results combine to determine global structure. Now let R be a global ring.

Lemma 23.3.18

$\mathcal {O}$ is hereditary if and only if ${{\,\mathrm{discrd}\,}}(\mathcal {O})$ is squarefree.

Proof. We argue locally; and then we use the characterization (iv), the computation of the reduced discriminant (23.3.2), and the same argument as in Theorem 15.5.5 to finish. $\square$

23.3.19

Let $\mathcal {O}$ be a hereditary (possibly maximal) R-order. By Theorem 21.4.9, we know that the group ${{\,\mathrm{Idl}\,}}(\mathcal {O})$ is an abelian group generated by the prime (equivalently, maximal) invertible two-sided ideals. We claim that the map

$$\begin{aligned} \begin{aligned} \left\{ \text {Prime two-sided invertible}\, \mathcal {O}-\,\text {ideals} \right\}&\leftrightarrow \left\{ \text {Prime ideals of}\, R\, \right\} \\ P&\mapsto P \cap R \end{aligned} \end{aligned}$$

(23.3.20)

is a bijection, generalizing Theorem 18.3.6. If $\mathfrak p \not \mid \mathfrak N$ then we have $P=\mathfrak p \mathcal {O}$; and if $\mathfrak p \mid \mathfrak D$ then we have a prime two-sided ideal $P=\mathcal {O}\cap {{\,\mathrm{rad}\,}}(\mathcal {O}_\mathfrak p )$ with $P^2=\mathfrak p \mathcal {O}$. Otherwise, $\mathfrak p \mid \mathfrak N$ but $\mathfrak p \not \mid \mathfrak D$, so $\mathcal {O}_\mathfrak p$ is hereditary but not maximal; from the local description in Lemma 23.3.13, we get a prime ideal $P=\mathcal {O}\cap {{\,\mathrm{rad}\,}}(\mathcal {O}_\mathfrak p )$ with $P^2=\mathfrak p \mathcal {O}$ as in the ramified case. This proves (23.3.20), and that the sequence

$$\begin{aligned} 0 \rightarrow {{\,\mathrm{Idl}\,}}(R) \rightarrow {{\,\mathrm{Idl}\,}}(\mathcal {O}) \rightarrow \prod _\mathfrak{p \mid \mathfrak N } \mathbb {Z}/2\mathbb {Z}\rightarrow 0 \end{aligned}$$

(23.3.21)

is exact.

Taking the quotient by ${{\,\mathrm{PIdl}\,}}(R)$, we obtain the exact sequence

$$\begin{aligned} 0 \rightarrow {{\,\mathrm{Cl}\,}}R \rightarrow {{\,\mathrm{Pic}\,}}_R(\mathcal {O}) \rightarrow \prod _\mathfrak{p \mid \mathfrak N } \mathbb {Z}/2\mathbb {Z}\rightarrow 0. \end{aligned}$$

(23.3.22)

In particular, if $\mathfrak D =(1)$, then ${{\,\mathrm{Pic}\,}}_R(\mathcal {O}) \simeq {{\,\mathrm{Cl}\,}}R$. Finally, the group of two-sided ideals modulo principal two-sided ideals is related to the Picard group by the exact sequence (18.5.5):

$$\begin{aligned} 0 \rightarrow N_{B^\times }(\mathcal {O})/(F^\times \mathcal {O}^\times )&\rightarrow {{\,\mathrm{Pic}\,}}_R(\mathcal {O}) \rightarrow {{\,\mathrm{Idl}\,}}(\mathcal {O})/{{\,\mathrm{PIdl}\,}}(\mathcal {O}) \rightarrow 0 \\ \alpha F^\times \mathcal {O}^\times&\mapsto [\alpha \mathcal {O}]=[\mathcal {O}\alpha ] \end{aligned}$$

(This exact sequence is sensitive to $\mathcal {O}$ even within its genus: see Remark 18.5.9.)

4 Eichler orders

We now consider a more general class of orders inspired by the hereditary orders.

Definition 23.4.1

An Eichler order $\mathcal {O}\subseteq B$ is the intersection of two (not necessarily distinct) maximal orders.

23.4.2

By the local-global dictionary for lattices (and orders), the property of being an Eichler order is local. Moreover, from 23.3.12, it follows that a hereditary order is Eichler.

Proposition 23.4.3

Suppose R is local and $\mathcal {O}\subseteq B={{\,\mathrm{M}\,}}_2(F)$. Then the following are equivalent:

• (i) $\mathcal {O}$ is Eichler;

• (ii) $\mathcal {O}\simeq \begin{pmatrix} R &{} R \\ \mathfrak p ^e &{} R \end{pmatrix}$;

• (iii) $\mathcal {O}$ contains an R-subalgebra that is $B^\times$-conjugate to $\begin{pmatrix} R &{} 0 \\ 0 &{} R \end{pmatrix}$; and

• (iv) $\mathcal {O}$ is the intersection of a uniquely determined pair of maximal orders (not necessarily distinct).

Proof. We follow Hijikata [Hij74, 2.2(i)]. Apologies in advance for all of the explicit matrix multiplication!

We prove (i) $\Rightarrow$ (ii) $\Leftrightarrow$ (iii) and then (ii) $\Rightarrow$ (iv) $\Rightarrow$ (i). The implications (ii) $\Rightarrow$ (iii) and (iv) $\Rightarrow$ (i) are immediate.

So first (i) $\Rightarrow$ (ii). Suppose $\mathcal {O}= \mathcal {O}_1 \cap \mathcal {O}_2$. All maximal orders in B are $B^\times$-conjugate to ${{\,\mathrm{M}\,}}_2(R)$, so there exist $\alpha _1,\alpha _2 \in B^\times$ such that $\mathcal {O}_i=\alpha _i^{-1} {{\,\mathrm{M}\,}}_2(R) \alpha _i$ for $i=1,2$. Conjugating by $\alpha _1$, we may suppose $\alpha _1=1$ and we write $\alpha =\alpha _2^{-1}$ for convenience, so $\mathcal {O}\simeq {{\,\mathrm{M}\,}}_2(R) \cap \alpha {{\,\mathrm{M}\,}}_2(R) \alpha ^{-1}$. Scaling by $\pi$, we may suppose $\alpha \in {{\,\mathrm{M}\,}}_2(R) \smallsetminus \pi {{\,\mathrm{M}\,}}_2(R)$. By row and column operations (Smith normal form, proven as part of the structure theorem for finitely generated modules over a PID), there exist $\beta ,\gamma \in {{\,\mathrm{GL}\,}}_2(R)$ such that

$$\begin{aligned} \beta \alpha \gamma = \begin{pmatrix} 1 &{} 0 \\ 0 &{} \pi ^e \end{pmatrix} \end{aligned}$$

is in standard invariant form with $e \ge 0$. Then

$$\begin{aligned} \mathcal {O}\simeq \beta \mathcal {O}\beta ^{-1} = {{\,\mathrm{M}\,}}_2(R) \cap \beta \alpha {{\,\mathrm{M}\,}}_2(R) \alpha ^{-1}\beta ^{-1} \end{aligned}$$

(23.4.4)

since $\beta \in {{\,\mathrm{GL}\,}}_2(R)$, and

$$\begin{aligned} \beta \alpha {{\,\mathrm{M}\,}}_2(R) \alpha ^{-1}\beta ^{-1} = \begin{pmatrix} 1 &{} 0 \\ 0 &{} \pi ^{e} \end{pmatrix} \begin{pmatrix} R &{} R \\ R &{} R \end{pmatrix} \begin{pmatrix} 1 &{} 0 \\ 0 &{} \pi ^{-e} \end{pmatrix} =\begin{pmatrix} R &{} \mathfrak p ^{-e} \\ \mathfrak p ^e &{} R \end{pmatrix} \end{aligned}$$

(23.4.5)

so

$$\begin{aligned} \mathcal {O}\simeq \begin{pmatrix} R &{} R \\ R &{} R \end{pmatrix} \cap \begin{pmatrix} R &{} \mathfrak p ^{-e} \\ \mathfrak p ^e &{} R \end{pmatrix} = \begin{pmatrix} R &{} R \\ \mathfrak p ^e &{} R \end{pmatrix}. \end{aligned}$$

(23.4.6)

To show (iii) $\Rightarrow$ (ii), we may suppose $\mathcal {O}\supseteq \begin{pmatrix} R &{} 0 \\ 0 &{} R \end{pmatrix}=R e_{11} + Re_{22}$ with $e_{11}=\begin{pmatrix} 1 &{} 0 \\ 0 &{} 0 \end{pmatrix}, e_{22}=\begin{pmatrix} 0 &{} 0 \\ 0 &{} 1 \end{pmatrix}$. Then $\mathcal {O}e_{11} \subseteq \begin{pmatrix} F &{} 0 \\ F &{} 0 \end{pmatrix}$. Let $\pi _{ij}$ be the projection onto the ij-coordinate. Then

$$\begin{aligned} \pi _{11}(\mathcal {O}e_{11})={{\,\mathrm{trd}\,}}(\mathcal {O}e_{11})e_{11} \subseteq R=\begin{pmatrix} R &{} 0 \\ 0 &{} 0 \end{pmatrix} \end{aligned}$$

and so equality holds. Therefore

$$\begin{aligned} \mathcal {O}\supseteq \pi _{21}(\mathcal {O})=\begin{pmatrix} 0 &{} 0 \\ \mathfrak p ^a &{} 0 \end{pmatrix} \end{aligned}$$

for some $a \in \mathbb {Z}$. Arguing again with the other matrix unit $e_{22}$ we conclude that

$$\begin{aligned} \mathcal {O}= \begin{pmatrix} R &{} \mathfrak p ^b \\ \mathfrak p ^a &{} R \end{pmatrix} \end{aligned}$$

(23.4.7)

with $a,b \in \mathbb {Z}$. Multiplying

$$ \begin{pmatrix} R &{} \mathfrak p ^b \\ \mathfrak p ^a &{} R \end{pmatrix} \begin{pmatrix} R &{} \mathfrak p ^b \\ \mathfrak p ^a &{} R \end{pmatrix} = \begin{pmatrix} R + \mathfrak p ^{a+b} &{} \mathfrak p ^b \\ \mathfrak p ^a &{} R + \mathfrak p ^{a+b} \end{pmatrix} $$

we conclude that $e=a+b \ge 0$. Such an order is maximal if and only if $a+b=0$: if $a\ge 0$, then $\begin{pmatrix} R &{} \mathfrak p ^b \\ \mathfrak p ^a &{} R \end{pmatrix} \subseteq \begin{pmatrix} R &{} \mathfrak p ^{-a} \\ \mathfrak p ^a &{} R \end{pmatrix}$ and similarly if $a \le 0$. The element $\alpha =\begin{pmatrix} 0 &{} 1 \\ \pi ^a &{} 0 \end{pmatrix}$ has

$$\begin{aligned} \alpha ^{-1} \mathcal {O}\alpha = \begin{pmatrix} 0 &{} 1 \\ \pi ^a &{} 0 \end{pmatrix} \begin{pmatrix} R &{} \mathfrak p ^b \\ \mathfrak p ^a &{} R \end{pmatrix} \begin{pmatrix} 0 &{} \pi ^{-a} \\ 1 &{} 0 \end{pmatrix} = \begin{pmatrix} R &{} R \\ \mathfrak p ^e &{} R \end{pmatrix} \end{aligned}$$

(23.4.8)

(and normalizes the given subalgebra) so the result is proven.

To conclude, we show (ii) $\Rightarrow$ (iv). Let $\mathcal {O}' \supseteq \mathcal {O}=\begin{pmatrix} R &{} R \\ \mathfrak p ^e &{} R \end{pmatrix}$ be a maximal R-order. Since $\begin{pmatrix} R &{} 0 \\ 0 &{} R \end{pmatrix} \subseteq \mathcal {O}'$, the argument of the previous paragraph applies, and $\mathcal {O}'=\begin{pmatrix} R &{} \mathfrak p ^{-c} \\ \mathfrak p ^{c} &{} R \end{pmatrix}$ with $c \in \mathbb {Z}$ satisfying $0 \le c \le e$. The intersection of another such maximal orders with the parameter d is the order $\begin{pmatrix} R &{} \mathfrak p ^{a} \\ \mathfrak p ^{b} &{} R \end{pmatrix}$ where $a=\max (c,d)$ and $b=-\min (c,d)$ so is equal to $\begin{pmatrix} R &{} R \\ \mathfrak p ^e &{} R \end{pmatrix}$ if and only if $e=a=\max (c,d)$ and $0=b=\min (c,d)$, which uniquely determine c, d up to swapping.

$\square$

Remark 23.4.9. There is a further important equivalent characterization of Eichler orders as being maximal or residually split: see Lemma 24.3.6.

Corollary 23.4.10

Every superorder of an Eichler order is Eichler.

Proof. The corollary is local, so we may apply Proposition 23.4.3(iii) to every superorder.

$\square$

Definition 23.4.11

Suppose R is local. The standard Eichler order of level $\mathfrak p ^e$  in ${{\,\mathrm{M}\,}}_2(F)$ is the order

$$\begin{aligned} \mathcal {O}_0(\mathfrak p ^e) :=\begin{pmatrix} R &{} R \\ \mathfrak p ^e &{} R \end{pmatrix}. \end{aligned}$$

By Proposition 23.4.3, if R is local then an order $\mathcal {O}\subseteq {{\,\mathrm{M}\,}}_2(F)$ is Eichler if and only if $\mathcal {O}$ is conjugate to a standard Eichler order.

Suppose until further notice that R is local, and let $\mathcal {O}=\mathcal {O}_0(\mathfrak p ^e)$ be the standard Eichler order of level $\mathfrak p ^e$ with $e \ge 0$.

23.4.12

First two basic facts about the Eichler order $\mathcal {O}$ of level $\mathfrak p ^e$: We have

$$\begin{aligned} {{\,\mathrm{discrd}\,}}(\mathcal {O})=[{{\,\mathrm{M}\,}}_2(R):\mathcal {O}]_R = \mathfrak p ^e \end{aligned}$$

and its associated ternary quadratic form $Q(x,y,z)=xy-\pi ^e z^2$ as in 23.3.3.

23.4.13

Let

$$\begin{aligned} \varpi =\begin{pmatrix} 0 &{} 1 \\ \pi ^e &{} 0 \end{pmatrix} \in \mathcal {O}. \end{aligned}$$

Then

$$\begin{aligned} \mathcal {O}= {{\,\mathrm{M}\,}}_2(R) \cap \varpi ^{-1} {{\,\mathrm{M}\,}}_2(R) \varpi \end{aligned}$$

as in (23.4.6); by Proposition 23.4.3 these two orders are the uniquely determined pair of maximal orders containing $\mathcal {O}$. We have $\varpi ^2=\pi ^e$, and so $\varpi \in N_{B^\times }(\mathcal {O})$. It follows (and can be checked directly) that $I=\mathcal {O}\varpi = \varpi \mathcal {O}$ is a two-sided $\mathcal {O}$-ideal. If $e=0$, we have $I=\mathcal {O}$.

Proposition 23.4.14

Suppose that $e \ge 1$. Then we have $N_{B^\times }(\mathcal {O})/(F^\times \mathcal {O}^\times ) = \langle \varpi \rangle \simeq \mathbb {Z}/2\mathbb {Z}$. Moreover, the group ${{\,\mathrm{Idl}\,}}(\mathcal {O})={{\,\mathrm{PIdl}\,}}(\mathcal {O})$ is abelian, generated by I and $\mathfrak p \mathcal {O}$ with the single relation $I^2=\mathfrak p ^e\mathcal {O}$.

Proof. Let $\alpha \in N_{B^\times }(\mathcal {O})$. Then by uniqueness of the intersection in 23.4.13, conjugation by $\alpha$ permutes these two orders, so we have a homomorphism $N_{B^\times }(\mathcal {O})$ to a cyclic group of order 2. This homomorphism is surjective, since $\varpi$ transposes the orders. If $\alpha$ is in the kernel, then $\alpha \in N_{B^\times }({{\,\mathrm{M}\,}}_2(R))=F^\times {{\,\mathrm{GL}\,}}_2(R)$ and unconjugating the second factor we similarly get $\varpi \alpha \varpi ^{-1} \in F^\times {{\,\mathrm{GL}\,}}_2(R)$, so

$$\begin{aligned} \alpha \in F^\times ( {{\,\mathrm{GL}\,}}_2(R) \cap \varpi ^{-1} {{\,\mathrm{GL}\,}}_2(R) \varpi ) = F^\times \mathcal {O}^\times . \end{aligned}$$

Again since R is local, we have ${{\,\mathrm{Idl}\,}}(\mathcal {O})={{\,\mathrm{PIdl}\,}}(\mathcal {O})$, and by (18.5.4), we have an isomorphism

$$\begin{aligned} N_{B^\times }(\mathcal {O})/(F^\times \mathcal {O}^\times ) \simeq {{\,\mathrm{PIdl}\,}}(\mathcal {O})/{{\,\mathrm{PIdl}\,}}(R) \end{aligned}$$

so ${{\,\mathrm{Idl}\,}}(\mathcal {O})$ is generated by I and the generator $\mathfrak p$ for ${{\,\mathrm{PIdl}\,}}(R)$. $\square$

23.4.15

It is helpful to consider the Jacobson radical of an Eichler order, to compare to the hereditary case.

$$\begin{aligned} J=\begin{pmatrix} \mathfrak p &{} R \\ \mathfrak p ^e &{} \mathfrak p \end{pmatrix}. \end{aligned}$$

We claim that $J={{\,\mathrm{rad}\,}}\mathcal {O}$. We verify directly that $J \subseteq \mathcal {O}$ is a two-sided ideal and

$$\begin{aligned} J^2= \begin{pmatrix} \mathfrak p ^f &{} \mathfrak p \\ \mathfrak p ^{e+1} &{} \mathfrak p ^f \end{pmatrix} \subseteq \mathfrak p \mathcal {O}, \end{aligned}$$

(23.4.16)

where $f=\min (e,2)$, so by Corollary 20.5.5, $J \subseteq {{\,\mathrm{rad}\,}}\mathcal {O}$; on the other hand, the quotient

$$\begin{aligned} \mathcal {O}/J \simeq k \times k \end{aligned}$$

(23.4.17)

is semisimple, so ${{\,\mathrm{rad}\,}}\mathcal {O}\subseteq J$ by Corollary 20.4.11(a).

However, the radical is not an invertible (sated) two-sided $\mathcal {O}$-ideal unless $\mathcal {O}$ is hereditary ($e=1$), by 23.3.1. Indeed, we verify that

$$\begin{aligned} \mathcal {O}{}_{\textsf {\tiny {L}} }(J) = \begin{pmatrix} R &{} \mathfrak p ^{-1} \\ \mathfrak p ^{e-1} &{} R \end{pmatrix} = \mathcal {O}{}_{\textsf {\tiny {R}} }(J) \end{aligned}$$

(23.4.18)

(Exercise 23.6); this recovers $\mathcal {O}$ if and only if $e=1$, and if $e \ge 2$ then it is an Eichler order of level $\mathfrak p ^{e-2}$ (conjugating as in (23.4.8)). By (23.4.16), if $e \ge 2$ then $J^2=\mathfrak p J$, and so we certainly could not have J invertible!

We now repackage these local efforts into a global characterization.

23.4.19

Suppose that R is a global ring. Let ${{\,\mathrm{disc}\,}}_R B = \mathfrak D$ and let $\mathcal {O}$ be an Eichler order with ${{\,\mathrm{discrd}\,}}\mathcal {O}=\mathfrak N$. If $\mathfrak p \mid \mathfrak D$, then $B_\mathfrak p$ has a unique maximal order, so (as an ‘intersection’) $\mathcal {O}_\mathfrak p$ is necessarily the maximal order. If $\mathfrak p \not \mid \mathfrak D$, and ${{\,\mathrm{ord}\,}}_\mathfrak p \mathfrak N = e \ge 0$, then $\mathcal {O}_\mathfrak p$ is isomorphic to the standard Eichler order of level $\mathfrak p ^e$.

We have $\mathfrak N =\mathfrak D \mathfrak M$ with $\mathfrak M \subseteq R$ and we just showed that $\mathfrak M$ is coprime to $\mathfrak D$. We call $\mathfrak M$ the level  of the Eichler order $\mathcal {O}$. The pair $\mathfrak D ,\mathfrak M$ (or $\mathfrak D ,\mathfrak N$) determines a unique genus of Eichler R-orders, i.e., this data uniquely determines the isomorphism class of $\mathcal {O}_\mathfrak p$ for each $\mathfrak p$.

Putting together Proposition 23.4.14 together with 23.3.19 for the remaining primes where the order is maximal, we have an exact sequence

$$\begin{aligned} 0 \rightarrow {{\,\mathrm{Idl}\,}}(R) \rightarrow {{\,\mathrm{Idl}\,}}(\mathcal {O}) \rightarrow \prod _\mathfrak{p \mid \mathfrak N } \mathbb {Z}/2\mathbb {Z}\rightarrow 0 \end{aligned}$$

(23.4.20)

and we may take the quotient by ${{\,\mathrm{PIdl}\,}}R$ to get

$$\begin{aligned} 0 \rightarrow {{\,\mathrm{Cl}\,}}R \rightarrow {{\,\mathrm{Pic}\,}}_R(\mathcal {O}) \rightarrow \prod _\mathfrak{p \mid \mathfrak N } \mathbb {Z}/2\mathbb {Z}\rightarrow 0 \end{aligned}$$

(23.4.21)

Remark 23.4.22. Eichler [Eic56a] developed his orders in detail for prime level, and employed them in the study of modular correspondences [Eic56c] and the trace formula [Eic73] in the case of squarefree level. (As mentioned in Remark 24.1.5, Eichler’s earlier work [Eic36, §6] over $\mathbb Q$ included some more general investigations, but his later work seemed mostly confined to the hereditary orders.) Hijikata [Hij74, §2.2] later studied these orders in the attempt to generalize Eichler’s result beyond the squarefree case; calling the orders split (like our name, residually split). Pizer [Piz73, p. 77] may be the first who explicitly called them Eichler orders.

Remark 23.4.23. An R-order $\mathcal {O}\subseteq {{\,\mathrm{M}\,}}_n(F)$ that contains a ${{\,\mathrm{GL}\,}}_n(F)$-conjugate of the diagonal matrices ${{\,\mathrm{diag}\,}}(R,\dots ,R)$ (equivalently, containing n orthogonal idempotents) is said to be tiled. By 23.4.3, an order $\mathcal {O}\subseteq {{\,\mathrm{M}\,}}_2(F)$ is tiled if and only if it is Eichler. Tiled orders also go by other names (including graduated orders) and they arise naturally in many contexts, including representation theory [Pl83] and modular forms.

5 Bruhat–Tits tree

In the previous section, we examined Eichler orders as the intersection of two maximal orders. There is a beautiful and useful combinatorial construction—a tree—which keeps track of the containments among maximal orders in aggregate, as follows. For further reading, see Serre [Ser2003, §II.1].

Let F be a nonarchimedean local field with valuation ring R, maximal ideal $\mathfrak p =\pi R$, and residue field $k :=R/\mathfrak p$, and let $q :=\#k$. Let $B :={{\,\mathrm{M}\,}}_2(F)$, and let $V :=F^2$ as column vectors, so that $B :={{\,\mathrm{End}\,}}_F(V)$ acts on the left.

Recalling again section 10.5, every maximal order $\mathcal {O}\subset B$ has $\mathcal {O}={{\,\mathrm{End}\,}}_R(M)$ where $M \subset V$ is an R-lattice. So to understand maximal orders, it is equivalent to understand lattices (and their containments) and the positioning of one lattice inside another.

Lemma 23.5.1

Let $L,M \subset V$ be R-lattices. Then there exists an R-basis $x_1,x_2$ of L such that $\pi ^{f_1}x_1,\pi ^{f_2}x_2$ is an R-basis of M with $f_1,f_2 \in \mathbb {Z}$ and $f_1 \le f_2$.

Proof. Exercise 23.7. $\square$

Lemma 23.5.2

We have ${{\,\mathrm{End}\,}}_R(L)={{\,\mathrm{End}\,}}_R(M)$ if and only if there exists $a \in F^\times$ such that $M=aL$.

Proof. If $M=aL$ for $a \in F^\times$, then ${{\,\mathrm{End}\,}}_R(L)={{\,\mathrm{End}\,}}_R(M)$. Conversely, suppose that ${{\,\mathrm{End}\,}}_R(L)={{\,\mathrm{End}\,}}_R(M)$. Replacing M by aM with $a \in F^\times$, we may suppose without loss of generality that $L \subseteq M$. By Lemma 23.5.1, we may identify $L=R^2$ with the standard basis and $M=\pi ^{f_1} e_1 \oplus \pi ^{f_2} e_2$ with $f_1,f_2 \in \mathbb {Z}_{\ge 0}$; rescaling again, and interchanging the basis elements if necessary, we may suppose $e_1=0$. Then ${{\,\mathrm{End}\,}}_R(M)={{\,\mathrm{End}\,}}_R(L) \simeq {{\,\mathrm{M}\,}}_2(R)$ implies $f_2=0$ and $L=M$. $\square$

With this lemma in mind, we make the following definition (recalling this definition made earlier in the context of algebras).

Definition 23.5.3

Two R-lattices $L,L' \subset V$ are homothetic if there exists $a \in F^\times$ such that $L'=aL$.

The relation of homothety is an equivalence relation on the set of R-lattices in V, and we write [L] for the homothety class of L.

23.5.4

Let $L \subset V$ be an R-lattice. In a homothety class of lattices, there is a unique lattice $L' \subseteq L$ in this homothety class satisfying any of the following equivalent conditions:

• (i) $L' \subseteq L$ is maximal;

• (ii) $L' \not \subseteq \pi L$; and

• (iii) $L/L'$ is cyclic as an R-module (has one generator).

These equivalences follow from Lemma 23.5.1: they are equivalent to $f_1=0$, and correspond to a maximal scaling of $L'$ by a power of $\pi$ within L. For such an $L'$, we have $L/L' \simeq R/\pi ^f R$ for a unique $f \ge 0$.

Definition 23.5.5

Let $\mathcal T$ be the graph whose vertices are homothety classes of R-lattices in V and where an undirected edge joins two vertices (exactly) when there exist representative lattices $L,L'$ for these vertices such that

$$\begin{aligned} \pi L \subsetneq L' \subsetneq L. \end{aligned}$$

(23.5.6)

Equivalently, by Lemma 23.5.2, the vertices of $\mathcal T$ are in bijection with maximal orders in $B={{\,\mathrm{M}\,}}_2(F)$ by $[L] \mapsto {{\,\mathrm{End}\,}}_R(L)$ for every choice of $L \in [L]$.

23.5.7

The adjacency relation (23.5.6) implies $L' \subsetneq \pi L \subsetneq \pi L'$, so it is sensible to have undirected edges.

A class $[L']$ has an edge to L if and only if the representative $L'$ in 23.5.4 has $f=1$.

Proposition 23.5.8

The graph $\mathcal T$ is a connected tree such that each vertex has degree $q+1$.

Proof. We have $L/\pi L \simeq k^2$, and so the lattices $L'$ satisfying (23.5.6) are in bijection with k-subspaces of dimension 1 in $L/\pi L$; such a subspace is given by a choice of generator up to scaling, so there are exactly $(q^2-1)/(q-1)=q+1$ such, and each vertex has $q+1$ adjacent vertices. The graph is connected: given two vertices, we may choose representative lattices $L,L'$ such that $L' \subseteq L$ as in 23.5.4. The quotient $L/L'$ is cyclic, so by induction the lattices $L_i=\pi ^i L + L'$ for $i=0,\dots ,f$ have $L_i$ adjacent to $L_{i+1}$, and $L_0=L$ and $L_f=L'$, giving a path from [L] to $[L']$.

The following argument comes from Dasgupta–Teitelbaum [DT2008, Proposition 1.3.2]. Suppose there is a nontrivial cycle in $\mathcal T$

$$\begin{aligned} \pi ^v L=L_s \subsetneq L_{s-1} \subsetneq \dots \subsetneq L_1 \subsetneq L_0=L \end{aligned}$$

(23.5.9)

so that $v \ge 1$. We may suppose this cycle is minimal, meaning that no intermediate lattices are equivalent. The quotient $L/L_s=L/\pi ^v L \simeq (R/\mathfrak p ^v)^2$ is not cyclic; let i be the largest index such that $L/L_i$ is cyclic but $L/L_{i+1}$ is not. Thus $L/L_{i+1} \simeq R/\mathfrak p ^i \oplus R/\mathfrak p$, and so $\pi ^i$ annihilates $L/L_{i+1}$ and $\pi ^i L \subseteq L_{i+1}$. Since $L/L_i$ is cyclic, just as in the previous paragraph, we conclude $L_{i-1}=\pi ^{i-1} L + L_i$. Putting these together, we find that $\pi L_{i-1} = \pi ^i L + \pi L_i \subseteq L_{i+1}$, the latter by definition of the adjacency between $L_i$ and $L_{i+1}$. By adjacency, $L_i \subsetneq \pi L_{i_1} \subseteq L_{i+1}$. And again by adjacency, $L_{i+1}$ is maximal inside $L_i$, so $\pi L_{i-1}=L_{i+1}$. This contradicts the minimality of the cycle; we conclude that $\mathcal T$ has no cycles. $\square$

We call $\mathcal T$ the Bruhat–Tits tree for ${{\,\mathrm{GL}\,}}_2(F)$. The Bruhat–Tits tree for $F=\mathbb Q _2$ is sketched in Figure 23.5.10. We write ${{\,\mathrm{Ver}\,}}(\mathcal T )$ and ${{\,\mathrm{Edg}\,}}(\mathcal T )$ for the set of vertices and edges of $\mathcal T$.

Figure 23.5.10:

Bruhat–Tits tree for ${{\,\mathrm{GL}\,}}_2(\mathbb Q _2)$

23.5.11

We define a transitive action of ${{\,\mathrm{GL}\,}}_2(F)$ on $\mathcal T$ as follows.

Let $L \subseteq V$ be a lattice. Choose a R-basis for L and put the vectors in the columns of a matrix $\beta \in {{\,\mathrm{M}\,}}_2(F)$. Since these columns span V over F, we have $\beta \in {{\,\mathrm{GL}\,}}_2(F)$, and the matrix $\beta$ is well-defined up to a change of basis over R; therefore the coset $\beta {{\,\mathrm{GL}\,}}_2(R) \in {{\,\mathrm{GL}\,}}_2(F)/{{\,\mathrm{GL}\,}}_2(R)$ is well-defined. (Check that the action of change of basis on columns is given by matrix multiplication the right.) Therefore a homothety class [L] gives a well-defined element of ${{\,\mathrm{GL}\,}}_2(F)/(F^\times {{\,\mathrm{GL}\,}}_2(R))$. Conversely, given such a class we can consider the R-lattice spanned by its columns, and its homothety class is well-defined. We have shown there is a bijection

$$\begin{aligned} {{\,\mathrm{Ver}\,}}(\mathcal T ) \leftrightarrow {{\,\mathrm{GL}\,}}_2(F)/(F^\times {{\,\mathrm{GL}\,}}_2(R)). \end{aligned}$$

(23.5.12)

The group ${{\,\mathrm{GL}\,}}_2(F)$ acts transitively on the left on the cosets ${{\,\mathrm{GL}\,}}_2(F)/(F^\times {{\,\mathrm{GL}\,}}_2(R))$ and we transport via the bijection (23.5.12) to an action on ${{\,\mathrm{Ver}\,}}(\mathcal T )$.

We claim this action preserves the adjacency relation on $\mathcal T$: if $L \supseteq L'$ are adjacent, then by invariant factors we can choose a basis $x_1,x_2$ for L such that $x_1,\pi x_2$ is a basis for $L'$, i.e.,

$$\begin{aligned} \beta '=\beta \begin{pmatrix} 1 &{} 0 \\ 0 &{} \pi \end{pmatrix}. \end{aligned}$$

(23.5.13)

If $\alpha \in {{\,\mathrm{GL}\,}}_2(F)$, then multiplying (23.5.13) on the left by $\alpha$ shows that $\alpha L,\alpha L'$ are adjacent.

23.5.14

The tree $\mathcal T$ has a natural notion of distance d between two vertices, given by the length of the shortest path between them, giving each edge of $\mathcal T$ length 1. Consequently, we have a notion of distance $d(\mathcal {O},\mathcal {O}')$ between every two maximal orders $\mathcal {O},\mathcal {O}' \subseteq B$.

Lemma 23.5.15

Let $L,L'$ be lattices with bases $x_1,x_2$ and $\pi ^f x_1,\pi ^{e+f} x_2$, respectively. Then in the basis $x_1,x_2$, we have $\mathcal {O}= {{\,\mathrm{End}\,}}_R(L) \simeq {{\,\mathrm{M}\,}}_2(R)$ and $\mathcal {O}' = {{\,\mathrm{End}\,}}_R(L') \simeq \begin{pmatrix} R &{} \mathfrak p ^{-e} \\ \mathfrak p ^e &{} R \end{pmatrix}$, and $d(\mathcal {O},\mathcal {O}')=e$.

Proof. The statement on endomorphism rings comes from Example 10.5.2; we may suppose up to homothety that $L'$ has basis $x_1, \pi ^e x_2$; the maximal lattices as in 23.5.4 are given by $L_i=Rx_1 + \mathfrak p ^i R x_2$ with $i=0,\dots ,e$, so the distance is $d([L],[L'])=e$. $\square$

23.5.16

Importantly, now, we turn to Eichler orders: they are the intersection of two unique maximal orders, and so correspond to a pair of vertices in $\mathcal T$, or equivalently a path. By Lemma 23.5.15, the standard Eichler order of level $\mathfrak p ^e$ corresponds to a path of length e, and by transitivity the same is true of every Eichler order. The normalizer $\varpi$ of an Eichler order 23.4.13 acts by swapping the two vertices. Each vertex of the path corresponds to the $e+1$ possible maximal superorders.

In this way, the Bruhat–Tits tree provides a visual way to keep track of many calculations with Eichler orders.

Remark 23.5.17. The theory of Bruhat–Tits trees beautifully generalizes to become the theory of buildings, pioneered by Tits; see the survey by Tits [Tit79], as well as introductions by Abramenko–Brown [AB2008].

6 Exercises

Unless otherwise specified, let R be a Dedekind domain with $F={{\,\mathrm{Frac}\,}}R$ and let $\mathcal {O}\subseteq B$ be an R-order in a quaternion algebra B.

1.:

Let R be a DVR with maximal ideal $\mathfrak p =(\pi )$, and let

$$\begin{aligned} \mathcal {O}=\begin{pmatrix} R &{} R \\ \mathfrak p &{} R \end{pmatrix}. \end{aligned}$$

(a):

Suppose that $\alpha =\begin{pmatrix} x &{} y \\ \pi z &{} w \end{pmatrix} \in N_{B^\times }(\mathcal {O})$. After scaling, we may suppose that $x,y,z,w \in R$. By determinants, show that if $x,w \in R^\times$, then $\alpha \in \mathcal {O}^\times$.

(b):

Compute

$$\begin{aligned} \alpha \varpi \overline{\alpha } = \begin{pmatrix} \pi (wy-xz) &{} x^2-\pi y^2 \\ \pi (w^2-\pi z^2) &{} -\pi (wy-xz) \end{pmatrix}. \end{aligned}$$

Show that $\pi \mid x$, and then $\pi \mid w$, whence $\alpha \in \varpi \mathcal {O}$.

(c):

Conclude that $N_{B^\times }(\mathcal {O})/(F^\times \mathcal {O}^\times )$ is generated by $\varpi =\begin{pmatrix} 0 &{} 1 \\ \pi &{} 0 \end{pmatrix}$.

[See also the matrix proof by Eichler [Eic56a, Satz 5], which uses a normal form for one-sided ideals simplifying the above computation.]

$\triangleright$ 2.:

Let $K \supseteq F$ be a field extension, let S be the integral closure of R in K, and let $\mathfrak a ,\mathfrak b \subset F$ be fractional ideals of R. Show that $\mathfrak a = \mathfrak b$ if and only if $\mathfrak a S = \mathfrak b S$.

$\triangleright$ 3.:

Extend Lemma 13.4.9, and show that if R is a DVR and $\mathcal {O}$ is an Eichler R-order then ${{\,\mathrm{nrd}\,}}(\mathcal {O}^\times )=R^\times$.

$\triangleright$ 4.:

Suppose R is local and $\mathcal {O}$ is a hereditary order. Show that if $\mu \in \mathcal {O}$ has $\mu ^2=\pi$, then $\mu$ generates $N_{B^\times }(\mathcal {O})/(F^\times \mathcal {O}^\times )$.

5.:

Suppose R is local and $\mathcal {O}\subseteq B={{\,\mathrm{M}\,}}_2(F)$ is the intersection of two maximal orders. Give another (independent) proof that $\mathcal {O}$ is isomorphic to a standard Eichler order which replaces matrix calculations in Proposition 23.4.3 with some representation theory as follows.

(a):

Write $\mathcal {O}\simeq {{\,\mathrm{M}\,}}_2(R) \cap \mathcal {O}'$. Let $e_{11}$ be the top-left matrix unit and let $I={{\,\mathrm{M}\,}}_2(R)e_{11}$. Show $I'=\mathcal {O}' e_{11}$ is an R-lattice in $V={{\,\mathrm{M}\,}}_2(F)e_{11} \simeq F^2$.

(b):

Use elementary divisors to show that there exists an R-basis $x_1,x_2$ of I such that $x_1,\pi ^e x_2$ is an R-basis for $I'$.

(c):

Show that the corresponding change of basis matrix $\alpha =\begin{pmatrix} 1 &{} 0 \\ 0 &{} \pi ^e \end{pmatrix}$ has $I'=\alpha I$, and use this to identify $\mathcal {O}$ with the standard Eichler order of level $\mathfrak p ^e$.

[See Brzezinski [Brz83a, Proposition 2.1].]

$\triangleright$ 6.:

Let R be local and let $\mathcal {O}$ be the standard Eichler order of level $\mathfrak p ^e$ for $e \ge 1$. Let $J={{\,\mathrm{rad}\,}}\mathcal {O}$. Show that

$$\begin{aligned} \mathcal {O}{}_{\textsf {\tiny {L}} }(J) = \begin{pmatrix} R &{} \mathfrak p ^{-1} \\ \mathfrak p ^{e-1} &{} R \end{pmatrix} = \mathcal {O}{}_{\textsf {\tiny {R}} }(J). \end{aligned}$$

$\triangleright$ 7.:

Prove Lemma 23.5.1. [Hint: use direct matrix methods or the theory of invariant factors.]

8.:

Let R be local and let $\mathcal {O}$ be a hereditary quaternion R-order. Show that ${{\,\mathrm{rad}\,}}\mathcal {O}=[\mathcal {O},\mathcal {O}]$ is the commutator (cf. Exercise 13.7) and that ${{\,\mathrm{diff}\,}}\mathcal {O}= {{\,\mathrm{rad}\,}}\mathcal {O}$.

9.:

Let R be local. Let $\mathcal {O},\mathcal {O}' \subseteq B$ be maximal R-orders. Recall that $\mathcal {O},\mathcal {O}'$ are vertices in the Bruhat–Tits tree. Define the distance ${{\,\mathrm{dist}\,}}(\mathcal {O},\mathcal {O}')$ to be the distance in the Bruhat–Tits tree between the respective vertices. Show that

$$\begin{aligned}{}[\mathcal {O}:\mathcal {O}\cap \mathcal {O}']={{\,\mathrm{dist}\,}}(\mathcal {O},\mathcal {O}')=[\mathcal {O}':\mathcal {O}\cap \mathcal {O}']. \end{aligned}$$

10.:

Let R be local, and let $\mathcal {O}$ be an Eichler order of level $\mathfrak p ^e$. Consider the graph whose vertices are R-superorders $\mathcal {O}' \supseteq \mathcal {O}$ in B and with a directed edge whenever the containment $\mathcal {O}' \supsetneq \mathcal {O}$ is proper and minimal. What does this graph look like? [Hint: use the Bruhat–Tits tree; it helps to draw the Eichler orders of the same level at the same height.]