Classical zeta functions

Abstract

In this chapter, we prove the Eichler mass number for a definite quaternion order over a totally real field using classical analytic methods.

In this chapter, we prove the Eichler mass number for a definite quaternion order over a totally real field using classical analytic methods.

1 $\triangleright$ Eichler mass formula

In the previous chapter, we saw a sketch of how analytic methods with quaternionic zeta functions provide a weighted class number formula for a quaternion order in a definite quaternion algebra over $\mathbb Q$, analogous to the analytic class number formula of Dirichlet for a quadratic field. The main result of this section is then the generalization of the Eichler mass formula to a definite quaternion order over a totally real number field. In this section, we give the statement of this result.

26.1.1

Let F be a totally real number field of degree $n=[F:\mathbb Q ]$, absolute discriminant $d_F$, and ring of integers $R :=\mathbb Z _F$. Let $h_F$ be the class number of F. Let

$$\begin{aligned} \zeta _F(s) :=\sum _\mathfrak{a \subseteq R} \frac{1}{\mathsf{N }(\mathfrak a )^s} \end{aligned}$$

be the Dedekind zeta function of F, where $\mathsf{N }(\mathfrak a )=[R:\mathfrak a ] \in \mathbb Z _{>0}$. Let B be a totally definite quaternion algebra over F of discriminant $\mathfrak D$. Let $\mathcal {O}\subset B$ be an R-order with reduced discriminant ${{\,\mathrm{discrd}\,}}(\mathcal {O})=\mathfrak N$.

For a prime $\mathfrak p \mid \mathfrak N$ with $\mathsf{N }(\mathfrak p )=q$, let $\biggl (\displaystyle {\frac{\mathcal {O}}{\mathfrak{p }}}\biggr ) \in \{-1,0,1\}$ be the Eichler symbol (Definition 24.3.2), and let

$$\begin{aligned} \lambda (\mathcal {O},\mathfrak p ) :=\frac{1-\mathsf{N }(\mathfrak p )^{-2}}{1-\biggl (\displaystyle {\frac{\mathcal {O}}{\mathfrak{p }}}\biggr )\mathsf{N }(\mathfrak p )^{-1}}= {\left\{ \begin{array}{ll} 1+1/q, &{} \text {if }({\mathcal {O}} \mid {p})=1; \\ 1-1/q, &{} \text {if }({\mathcal {O}} \mid {p})=-1; \\ 1-1/q^2, &{} \text {if }({\mathcal {O}} \mid {p})=0. \end{array}\right. } \end{aligned}$$

(26.1.2)

26.1.3

We saw in Lemma 17.7.13 that for each definite order $\mathcal {O}$, the group $\mathcal {O}^1$ of units of reduced norm 1 is a finite group; we will see in Lemma 26.5.1 that further the group $\mathcal {O}^\times /R^\times$ is finite. For a right $\mathcal {O}$-ideal J, the automorphism group of J (as a right $\mathcal {O}$-module) consists of right multiplication maps by elements $\mu \in B^\times$ with $\mu J = J$, i.e., $\mu \in \mathcal {O}{}_{\textsf {\tiny {L}} }(J)^\times$.

It was already evident in the Eichler mass formula (and remains a general principle in mathematics) that one often gets a better count of objects when they are weighted by the inverse size of the automorphism group, so we weight a right ideal class [J] by $[\mathcal {O}{}_{\textsf {\tiny {L}} }(J)^\times :R^\times ]^{-1}$ and make the following definition of a weighted class number.

Definition 26.1.4

Define the mass of ${{\,\mathrm{Cls}\,}}\mathcal {O}$ to be

$$\begin{aligned} {{\,\mathrm{mass}\,}}({{\,\mathrm{Cls}\,}}\mathcal {O})=\sum _{[J] \in {{\,\mathrm{Cls}\,}}\mathcal {O}} [\mathcal {O}{}_{\textsf {\tiny {L}} }(J)^\times :R^\times ]^{-1}. \end{aligned}$$

Main Theorem 26.1.5

(Eichler mass formula). With notation as in 26.1.1, we have

$$\begin{aligned} {{\,\mathrm{mass}\,}}({{\,\mathrm{Cls}\,}}\mathcal {O}) = \frac{2\zeta _F(2)}{(2\pi )^{2n}}d_F^{3/2} h_F \mathsf{N }(\mathfrak N ) \prod _\mathfrak{p \mid \mathfrak N } \lambda (\mathcal {O},\mathfrak p ). \end{aligned}$$

(26.1.6)

26.1.7

The functional equation for the Dedekind zeta function relates s to $1-s$, giving an alternative way of writing (26.1.6) as

$$\begin{aligned} \frac{2\zeta _F(2)}{(2\pi )^{2n}}d_F^{3/2} = \frac{|\zeta _F(-1) \,|}{2^{n-1}}. \end{aligned}$$

(26.1.8)

We notice that the Eichler mass formula then implies that $\zeta _F(-1) \in \mathbb Q$.

Remark 26.1.9. More generally, the rationality of the values $\zeta _F(-n)$ with $n \in \mathbb Z _{>0}$ is a theorem of Siegel [Sie69] and Deligne–Ribet [DR80].

Remark 26.1.10. The weighting in the mass is what makes Main Theorem 26.1.5 so simple. In the (unlikely) situation where $w_J = w_\mathcal {O}$ is independent of J, we would have a formula for the class number, but more generally we will need to take account of unit groups by computing embedding numbers of cyclotomic quadratic orders: we will do this in Chapter 30.

We now make the formula (26.1.6) a bit more explicit for the case of Eichler orders.

26.1.11

Let $\mathcal {O}$ be an Eichler order of level $\mathfrak M$, so that $\mathfrak N =\mathfrak D \mathfrak M$ with $\mathfrak D ,\mathfrak M$ coprime. Then

$$ \biggl (\displaystyle {\frac{\mathcal {O}}{\mathfrak{p }}}\biggr )= {\left\{ \begin{array}{ll} -1, &{} \text {if }\mathfrak p \mid \mathfrak D ; \\ 1, &{} \text {if }\mathfrak p \mid \mathfrak M ; \\ *, &{} \text {if }\mathfrak p \not \mid \mathfrak N . \end{array}\right. } $$

Accordingly, we define the generalized Euler $\varphi$-function and Dedekind $\psi$-function by

$$\begin{aligned} \varphi (\mathfrak D )&:=\prod _\mathfrak{p \mid \mathfrak D } (\mathsf{N }(\mathfrak p )-1) = \mathsf{N }(\mathfrak D ) \prod _\mathfrak{p \mid \mathfrak D }\left( 1-\frac{1}{\mathsf{N }(\mathfrak p )}\right) \\ \psi (\mathfrak M )&:=\prod _\mathfrak{p ^e \parallel \mathfrak M } \mathsf{N }(\mathfrak p )^{e-1}(\mathsf{N }(\mathfrak p )+1) = \mathsf{N }(\mathfrak M ) \prod _\mathfrak{p \mid \mathfrak M } \left( 1+\frac{1}{\mathsf{N }(\mathfrak p )}\right) \end{aligned}$$

(recalling $\mathfrak D$ is squarefree, with the natural extension $\varphi (\mathfrak D )=\#(R/\mathfrak D )^\times$ for all $\mathfrak D$). The $\psi$-function computes a unit index: see Lemma 26.6.7.

The Eichler mass formula (Main Theorem 26.1.5) for Eichler orders then reads as follows.

Theorem 26.1.12

(Eichler mass formula, Eichler orders). With notation as in 26.1.11, we have

$$\begin{aligned} {{\,\mathrm{mass}\,}}({{\,\mathrm{Cls}\,}}\mathcal {O}) = \frac{2\zeta _F(2)}{(2\pi )^{2n}}d_F^{3/2} h_F \varphi (\mathfrak D )\psi (\mathfrak M ). \end{aligned}$$

(26.1.13)

Remark 26.1.14. The Eichler mass formula in the form (26.1.13) for maximal orders was proven by Eichler (working over a general totally real field) using the techniques in this chapter [Eic38b, Satz 1], and was extended to squarefree level $\mathfrak N$ (i.e., hereditary orders) again by Eichler [Eic56a, §4]. This was extended by Brzezinski [Brz90, (4.6)] to a general formula over $\mathbb Q$ and by Körner [Kör87, Theorem 1], using idelic methods.

The classical method to prove Main Theorem 26.1.5 is similar to the one we sketched over $\mathbb Q$ in chapter 25, with some added technicalities of working over a number field. We follow this approach, first proving the formula when $\mathcal {O}$ is a maximal order, and then deducing the general case. We will return in chapter 29 and reconsider the Eichler mass formula from an idelic point of view, thinking of it as a special case of a volume formula (for a finite set of “quotient points”). It is hoped that this chapter will serve to show both the power and limits of classical methods before we build upon them using idelic methods.

2 Analytic class number formula

In this section, in preparation for the quaternionic case we briefly review what we need from the analytic class number formula for a number field F. References for this material include Borevich–Shafarevich [BS66, Chapter 5], Lang [Lang94, Chapter VI], and Neukirch [Neu99, Chapter VII].

We begin by setting some notation that will be used throughout the rest of this chapter. Let F be a number field of degree $n :=[F:\mathbb Q ]$ with r real places and c complex places, so that $n=r+2c$. Let $R :=\mathbb Z _F$ be the ring of integers in F, and let $d_F$ be the discriminant of F. Let $w_F$ be the number of roots of unity in F, let $h_F :=\# {{\,\mathrm{Cl}\,}}R$ be the class number of F, and let ${Reg }_F$ be the regulator of F (the covolume of $R^\times$ under the Minkowski embedding).

Define the Dedekind zeta function for $s \in \mathbb C$ with ${{\,\mathrm{Re}\,}}(s)>1$ by

$$\begin{aligned} \zeta _F(s) :=\sum _\mathfrak{a \subseteq R} \frac{1}{\mathsf{N }(\mathfrak a )^s} \end{aligned}$$

where the sum is over all nonzero ideals of R and $\mathsf{N }(\mathfrak a )=\#(R/\mathfrak a )=[R:\mathfrak a ]$ is the absolute norm; we have $\mathsf{N }(\mathfrak a )={{\,\mathrm{Nm}\,}}(\mathfrak a )$ with norm taken from F to $\mathbb Q$ and positive generator chosen.

26.2.1

The Dedekind zeta function converges for ${{\,\mathrm{Re}\,}}s>1$ and has an Euler product

$$\begin{aligned} \zeta _F(s) = \prod _\mathfrak{p } \left( 1-\frac{1}{\mathsf{N }(\mathfrak p )^s}\right) ^{-1} \end{aligned}$$

(26.2.2)

where the product is over all nonzero primes of R—this follows formally from unique factorization of ideals after one shows that the pruned product converges.

The Dedekind zeta function has properties analogous to the Riemann zeta function, which is the special case $F=\mathbb Q$. In particular, we can extend $\zeta _F(s)$ in a manner analogous to 25.2.1 to ${{\,\mathrm{Re}\,}}s>0$. For $a \in \mathbb C$ we write $\zeta _F^*(a)$ for the leading coefficient in the Laurent series expansion for $\zeta _F$ at $s=a$.

Theorem 26.2.3

(Analytic class number formula). $\zeta _F(s)$ has analytic continuation to ${{\,\mathrm{Re}\,}}s>0$, with a simple pole at $s=1$ having residue

$$\begin{aligned} \zeta _F^*(1) = \lim _{s \rightarrow 1}(s-1) \zeta _F(s) = \frac{2^{r} (2\pi )^{c}}{w_F \sqrt{|d_F \,|}} h_F {Reg }_F. \end{aligned}$$

(26.2.4)

Remark 26.2.5. The formula (26.2.4) is known as Dirichlet’s analytic class number formula (even though the original form of Dirichlet’s theorem concerned quadratic forms rather than classes of ideals, so is closer to Theorem 25.2.12).

Example 26.2.6

When F is an imaginary quadratic field ($r=0$ and $c=1$) we have ${Reg }_F=1$ and Theorem 26.2.3 is Theorem 25.2.12.

Before we finish this section, we review a few ingredients from the proof of the analytic class number formula (26.2.4) to set up the Eichler mass formula.

26.2.7

We first write the Dedekind zeta function as a sum over ideals in a given ideal class $[\mathfrak b ] \in {{\,\mathrm{Cl}\,}}(R)$: we define the partial zeta function

(26.2.8)

convergent for ${{\,\mathrm{Re}\,}}s>1$ by comparison to the harmonic series, so that

$$\begin{aligned} \zeta _F(s) = \sum _{[\mathfrak b ] \in {{\,\mathrm{Cl}\,}}R} \zeta _{F,[\mathfrak b ]}(s). \end{aligned}$$

(26.2.9)

Now note that $[\mathfrak a ]=[\mathfrak b ]$ if and only if $\mathfrak a = a \mathfrak b$ for some nonzero

$$\begin{aligned} a \in \mathfrak b ^{-1} = \{x \in F : x\mathfrak b \subseteq R\}, \end{aligned}$$

so there is a bijection between nonzero ideals $\mathfrak a \subseteq R$ such that $[\mathfrak a ]=[\mathfrak b ]$ and the set of nonzero elements in $\mathfrak b ^{-1} / R^\times$. So

$$\begin{aligned} \zeta _{F,[\mathfrak b ]}(s) = \frac{1}{\mathsf{N }(\mathfrak b )^s} \sum _{0 \ne a \in \mathfrak b ^{-1} / R^\times } \frac{1}{{{\,\mathrm{Nm}\,}}(a)^s}. \end{aligned}$$

(26.2.10)

One now reduces to a problem concerning lattice points in a fundamental domain for the action of $R^\times$, and examining the residue of the pole at $s=1$ fits into a more general framework (invoked again below).

Definition 26.2.11

A cone $X \subseteq \mathbb R ^n$ is a subset closed under multiplication by positive scalars, so $t X = X$ for all $t \in \mathbb R _{>0}$.

Theorem 26.2.12

Let $X \subseteq \mathbb R ^n$ be a cone. Let $N:X \rightarrow \mathbb R _{>0}$ be a function satisfying

$$\begin{aligned} N(tx)=t^n N(x) \quad \text {for all }x \in X, t \in \mathbb R _{>0}. \end{aligned}$$

Suppose that

$$\begin{aligned} X_{\le 1} :=\{x \in X : N(x) \le 1\} \subseteq \mathbb R ^n \end{aligned}$$

(26.2.13)

is a bounded subset with volume ${{\,\mathrm{vol}\,}}(X_{\le 1})$. Let $\Lambda \subseteq \mathbb R ^n$ be a (full) $\mathbb Z$-lattice in $\mathbb R ^n$, and let

$$\begin{aligned} \zeta _{\Lambda ,X}(s) :=\sum _{\lambda \in X \cap \Lambda } \frac{1}{N(\lambda )^s}. \end{aligned}$$

Then $\zeta _{\Lambda ,X}(s)$ converges for ${{\,\mathrm{Re}\,}}s>1$ and has a simple pole at $s=1$ with residue

$$\begin{aligned} \zeta _{\Lambda ,X}^*(1) = \lim _{s \searrow 1} (s-1) \zeta _{\Lambda ,X}(s) = \frac{{{\,\mathrm{vol}\,}}(X_{\le 1})}{{{\,\mathrm{covol}\,}}(\Lambda )}. \end{aligned}$$

Proof. See Borevich–Shafarevich [BS66, Chapter 5, Section 1.1, Theorem 1]. $\square$

26.2.14

To apply Theorem 26.2.12 for $\zeta _{F,[\mathfrak b ]}(s)$, we embed $F \hookrightarrow F_\mathbb R \simeq \mathbb R ^r \times \mathbb C ^c$ and we equip $F_\mathbb R$ with the inner product

$$\begin{aligned} \langle x,y \rangle = \sum _{i=1}^r x_iy_i + \sum _{j=1}^c 2{{\,\mathrm{Re}\,}}(x_{r+j}\overline{y_{r+j}}) \end{aligned}$$

(26.2.15)

for $x=(x_i)_i,y=(y_i)_i \in F_\mathbb R$. This inner product modifies the usual one by rescaling complex coordinates, and the volume form ${{\,\mathrm{vol}\,}}$ induced by $\langle \,,\,\rangle$ is $2^c$ times the standard Lebesgue volume on $\mathbb R ^r \times \mathbb C ^c$. With this convention, we have $\langle x, 1 \rangle = {{\,\mathrm{Tr}\,}}_{F|\mathbb Q }(x)$ and ${{\,\mathrm{covol}\,}}(R)=\sqrt{|d_F \,|}$.

We then take $\Lambda$ to be the image of $\mathfrak b ^{-1}$, and take X to be a cone fundamental domain for the action of the unit group $R^\times$. The absolute norm $\mathsf{N }(x)=|{{\,\mathrm{Nm}\,}}_{F|\mathbb Q }(x) \,|$ then satisfies the required homogeneity property, and $X_{\le 1}$ is bounded, so by Theorem 26.2.12,

$$\begin{aligned} \zeta _{F,[\mathfrak b ]}^*(1) = \frac{1}{\mathsf{N }(\mathfrak b )}\frac{{{\,\mathrm{vol}\,}}(X_{\le 1})}{{{\,\mathrm{covol}\,}}(\Lambda )}. \end{aligned}$$

(26.2.16)

We have

$$\begin{aligned} {{\,\mathrm{covol}\,}}(\Lambda )=\frac{{{\,\mathrm{covol}\,}}(R)}{\mathsf{N }(\mathfrak b )} = \frac{\sqrt{|d_F \,|}}{\mathsf{N }(\mathfrak b )}. \end{aligned}$$

(26.2.17)

It requires a bit more work to compute ${{\,\mathrm{vol}\,}}(X_{\le 1})$.

Proposition 26.2.18

We have

$$\begin{aligned} {{\,\mathrm{vol}\,}}(X_{\le 1})= \frac{2^r (2\pi )^c {Reg }_F}{w_F} \end{aligned}$$

(26.2.19)

Proof. See Exercise 26.3: the proof is well-summarized as a “change of variables”, but the reader may prefer the idelic point of view (Chapter 29) instead, where the integrals are ‘easier’. A detailed proof can be found in Borevich–Shafarevich [BS66, §5.1.3], Lang [Lang94, §VI.3, Theorem 3], and Neukirch [Neu99, §VII.5]. $\square$

Plugging (26.2.19) and (26.2.17) into (26.2.16),

$$\begin{aligned} \zeta _{F,[\mathfrak b ]}^*(1) = \frac{2^{r} (2\pi )^{c}}{w_F \sqrt{|d_F \,|}} {Reg }_F; \end{aligned}$$

(26.2.20)

note in particular that this does not depend on the class $[\mathfrak b ]$! The analytic class number formula (Theorem 26.2.3) then follows as

$$\begin{aligned} \zeta _{F}^*(1) = \sum _{[\mathfrak b ]} \zeta _{F,[\mathfrak b ]}^*(1) = \frac{2^{r} (2\pi )^{c}}{w_F \sqrt{|d_F \,|}} {Reg }_F h_F. \end{aligned}$$

(26.2.21)

3 Classical zeta functions of quaternion algebras

We now embark on a proof in our quaternionic setting, mimicking the above. We retain our notation on the number field F. We further let throughout B be a quaternion algebra over F of discriminant $\mathfrak D$ and let $\mathcal {O}\subseteq B$ be an R-order. (Our emphasis will be on the case $\mathcal {O}$ a maximal order, but many definitions carry through.)

To begin, in this section we define the classical zeta function and show it has an Euler product.

26.3.1

Let I be an invertible, integral right $\mathcal {O}$-ideal, so that $I \subseteq \mathcal {O}$, and by definition I is sated so $\mathcal {O}{}_{\textsf {\tiny {R}} }(I)=\mathcal {O}$. Recall we have defined $\mathsf{N }(I)=\#(\mathcal {O}/I)$; we have $\mathsf{N }(I)=\mathsf N ({{\,\mathrm{nrd}\,}}(I))^2$ (Paragraph 16.4.10).

For example, if $\mathfrak a \subseteq R$ is a nonzero ideal then $\mathsf{N }(\mathfrak a \mathcal {O})=\mathsf{N }(\mathfrak a )^4$.

We then define the (classical) zeta function  of $\mathcal {O}$ to be

$$\begin{aligned} \zeta _\mathcal {O}(s) :=\sum _{I \subseteq \mathcal {O}} \frac{1}{\mathsf{N }(I)^s} = \sum _\mathfrak{n } \frac{a_\mathfrak n (\mathcal {O})}{\mathsf{N }(\mathfrak n )^{2s}} \end{aligned}$$

(26.3.2)

where the first sum is over all (nonzero) integral, invertible right $\mathcal {O}$-ideals I and in the second sum we define

$$\begin{aligned} a_\mathfrak n (\mathcal {O}) :=\#\{I \subseteq \mathcal {O}: {{\,\mathrm{nrd}\,}}(I)=\mathfrak n \} \end{aligned}$$

(26.3.3)

(and $a_\mathfrak n (\mathcal {O})$ is finite by Lemma 17.7.26).

Lemma 26.3.4

If $\mathcal {O},\mathcal {O}'$ are locally isomorphic, then $a_\mathfrak n (\mathcal {O})=a_\mathfrak n (\mathcal {O}')$ for all $\mathfrak n$.

Proof. We use the local-global dictionary for lattices (Theorem 9.4.9). To ease parentheses in the notation, we work in the completion, but one can also work just in the localization. For all $\mathfrak p$, we have $\mathcal {O}'_\mathfrak{p }=\nu _\mathfrak p ^{-1} \mathcal {O}_\mathfrak{p } \nu _\mathfrak p$ for some $\nu _\mathfrak p \in B_\mathfrak p ^\times$, and we may take $\nu _\mathfrak p =1$ for all but finitely many $\mathfrak p$; the element $\nu _\mathfrak p$ is well-defined up to left multiplication by $\mathcal {O}_\mathfrak p ^\times$ and right multiplication by $\mathcal {O}_\mathfrak p '^\times$.

Then to an integral, invertible right $\mathcal {O}$-ideal I, we associate the unique lattice $I'$ such that $I'_\mathfrak p = \nu _\mathfrak p ^{-1} I_\mathfrak p \nu _\mathfrak p$; such a lattice is well-defined independent of the choice of $\nu _\mathfrak p$. By construction, $\mathcal {O}{}_{\textsf {\tiny {R}} }(I'_\mathfrak p )=\mathcal {O}_\mathfrak p '$ so $\mathcal {O}{}_{\textsf {\tiny {R}} }(I')=\mathcal {O}'$. And since I is integral, $I_\mathfrak p \subseteq \mathcal {O}_\mathfrak p$ whence $I_\mathfrak p ' \subseteq \nu _\mathfrak p ^{-1} I_\mathfrak p \nu _\mathfrak p \subseteq \mathcal {O}_\mathfrak p '$ and I is locally integral hence integral. Since I is invertible, I is locally principal, so $I'$ is also locally principal, hence invertible. Finally, again checking locally, we have ${{\,\mathrm{nrd}\,}}(I')={{\,\mathrm{nrd}\,}}(I)$.

Repeating this argument going from $I'$ to I, we see that the corresponding sets of ideals are in bijection, as claimed. $\square$

26.3.5

From Lemma 26.3.4, we see that $\zeta _\mathcal {O}(s)$ only depends on the genus of $\mathcal {O}$. Since there is a unique genus of maximal orders in B, following the number field case we will write $\zeta _B(s)=\zeta _\mathcal {O}(s)$ where $\mathcal {O}$ is any maximal order.

Our next order of business is to establish an Euler product for $\zeta _\mathcal {O}(s)$. We prove a more general result on the factorization of invertible lattices.

Lemma 26.3.6

Let I be an invertible, integral lattice and suppose that ${{\,\mathrm{nrd}\,}}(I)=\mathfrak m \mathfrak n$ with $\mathfrak m ,\mathfrak n \subseteq R$ coprime ideals. Then there exists a unique invertible, integral lattice J such that I is compatible with $J^{-1}$ with $IJ^{-1}$ integral and ${{\,\mathrm{nrd}\,}}(J)=\mathfrak m$.

Proof. We use the local-global dictionary for lattices, and we define $J \subseteq B$ to be the unique lattice such that

$$\begin{aligned} J_{(\mathfrak p )} :={\left\{ \begin{array}{ll} I_{(\mathfrak p )}=\mathcal {O}_{(\mathfrak p )}, &{} \text { if }\mathfrak p \not \mid \mathfrak m \mathfrak n ; \\ I_{(\mathfrak p )}, &{} \text { if }\mathfrak p \mid \mathfrak m ; \\ \mathcal {O}_{(\mathfrak p )}, &{} \text { if }\mathfrak p \mid \mathfrak n . \end{array}\right. } \end{aligned}$$

(26.3.7)

We have $\mathcal {O}{}_{\textsf {\tiny {R}} }(J)=\mathcal {O}$ and ${{\,\mathrm{nrd}\,}}(J)=\mathfrak m$, since these statements hold locally. Integrality and invertibility are local; since these are true for I they are true for J. Finally, we compute that $(IJ^{-1})_{(\mathfrak p )}=\mathcal {O}_{(\mathfrak p )}$ for all $\mathfrak p \not \mid \mathfrak n$ and $(IJ^{-1})_{(\mathfrak p )}=I_{(\mathfrak p )}$ for $\mathfrak p \mid \mathfrak n$, so $IJ^{-1}$ is locally integral and hence integral. The uniqueness of J can be verified directly (Exercise 26.4). $\square$

26.3.8

Consider the situation of Lemma 26.3.6. Let $I'=IJ^{-1}$. Then $I=I'J$, and $I'$ is integral, invertible (Paragraph 16.5.3) and compatible with J. Since ${{\,\mathrm{nrd}\,}}(I')=\mathfrak n$, we have “factored” I.

We have $\mathcal {O}{}_{\textsf {\tiny {R}} }(I')=\mathcal {O}{}_{\textsf {\tiny {L}} }(J)$ by compatibility, but this common order is only locally isomorphic to $\mathcal {O}$, since $I',J$ are locally principal but not necessarily principal. So in a sense, this factorization occurs not over $\mathcal {O}$ but over the genus of $\mathcal {O}$—but this is a harmless extension.

Proposition 26.3.9

If $\mathfrak m ,\mathfrak n$ are coprime, then $a_\mathfrak{m \mathfrak n }(\mathcal {O})=a_\mathfrak{m }(\mathcal {O})a_\mathfrak{n }(\mathcal {O})$.

Proof. Write $A_\mathfrak n (\mathcal {O})$ for the set of integral, invertible right $\mathcal {O}$-ideals I with ${{\,\mathrm{nrd}\,}}(I)=\mathfrak n$. Then $\#A_\mathfrak n (\mathcal {O})=a_\mathfrak n (\mathcal {O})$. According to Lemma 26.3.6, there is a map

$$\begin{aligned} \begin{aligned} A_\mathfrak{m \mathfrak n }(\mathcal {O})&\rightarrow A_\mathfrak n (\mathcal {O}) \\ I&\mapsto J \end{aligned} \end{aligned}$$

(26.3.10)

We claim that this map is surjective and that each fiber has cardinality $a_\mathfrak m (\mathcal {O})$. Indeed, these statements follow at the same time from the following observation: if $J \in A_\mathfrak n (\mathcal {O})$ with $\mathcal {O}'=\mathcal {O}{}_{\textsf {\tiny {L}} }(J)$, then for each $I' \in A_\mathfrak m (\mathcal {O}')$, we have $\overline{I'}$ compatible with J and $I=I'J \in A_\mathfrak m$, and conversely; so the fiber of (26.3.10) is identified with $A_\mathfrak m (\mathcal {O}')$, of cardinality $a_\mathfrak m (\mathcal {O}')=a_\mathfrak m (\mathcal {O})$ by Lemma 26.3.4. $\square$

26.3.11

From Proposition 26.3.9 and unique factorization of ideals in R, we find that $\zeta _\mathcal {O}$ has an Euler product

$$\begin{aligned} \zeta _\mathcal {O}(s) = \prod _\mathfrak{p } \zeta _{\mathcal {O}_\mathfrak p }(s) \end{aligned}$$

(26.3.12)

where

$$\begin{aligned} \zeta _{\mathcal {O}_\mathfrak p }(s) :=\sum _{I_\mathfrak p \subseteq \mathcal {O}_\mathfrak p } \frac{1}{\mathsf{N }(I_\mathfrak p )^s} = \sum _{e=0}^{\infty } \frac{a_\mathfrak{p ^e}(\mathcal {O})}{\mathsf{N }(\mathfrak p )^{2s}}. \end{aligned}$$

(26.3.13)

Remark 26.3.14. Zeta functions of semisimple algebras over a number field can be defined in the same way as in (26.3.2), following Solomon [Sol77]: see the survey on analytic methods in noncommutative number theory by Bushnell–Reiner [BR85].

Remark 26.3.15. The world of L-functions is rich and very deep: for a beautiful survey of the analytic theory of automorphic L-functions in historical perspective, see Gelbart–Miller [GM2004]. In particular, we have not given a general definition of zeta functions (or L-functions) in this section, but it is generally agreed that the Selberg class incorporates the minimal essential features: definition as a Dirichlet series, meromorphic continuation to the complex plane, Euler product, and functional equation. See e.g. Conrey–Ghosh [CG93] and the references therein.

4 Counting ideals in a maximal order

We now count ideals of prime power norm. By the local-global dictionary, there is a bijection

$$\begin{aligned} \{I \subseteq \mathcal {O}: {{\,\mathrm{nrd}\,}}(I)=\mathfrak p ^e\} \xrightarrow {\smash {{\sim }}}\{I_\mathfrak p \subseteq \mathcal {O}_\mathfrak p : {{\,\mathrm{nrd}\,}}(I_\mathfrak p )=\mathfrak p ^e\}. \end{aligned}$$

so it suffices to count the number of ideals in the local case. In this section, we carry out this count for maximal orders.

So let $\mathfrak p \subset R$ be a (nonzero) prime and let $q :={{\,\mathrm{Nm}\,}}(\mathfrak p )$. Let $\mathcal {O}_\mathfrak p \subset B_\mathfrak p$ be a maximal order. Let

$$\begin{aligned} a_\mathfrak{p ^e}(\mathcal {O}_\mathfrak p ) = \#\{I_\mathfrak p =\alpha _\mathfrak p \mathcal {O}_\mathfrak p \subseteq \mathcal {O}_\mathfrak p : {{\,\mathrm{nrd}\,}}(I_\mathfrak p )=\mathfrak p ^e\} \end{aligned}$$

count the number of right integral $\mathcal {O}_\mathfrak p$-ideals of norm $\mathfrak p ^e$. Since $\mathcal {O}_\mathfrak p$ is maximal, every nonzero ideal is invertible; and because $R_\mathfrak p$ is a DVR, all such invertible ideals are principal.

Lemma 26.4.1

Let $\mathcal {O}_\mathfrak p \subset B_\mathfrak p$ be a maximal order and let $e \in \mathbb Z _{\ge 0}$.

• (a) If $B_\mathfrak p$ is a division ring, then every right integral $\mathcal {O}_\mathfrak p$-ideal is a power of the maximal ideal and $a_\mathfrak{p ^e}(\mathcal {O}_\mathfrak p )=1$.

• (b) If $B_\mathfrak p \simeq {{\,\mathrm{M}\,}}_2(F_\mathfrak p )$, so that $\mathcal {O}_\mathfrak p \simeq {{\,\mathrm{M}\,}}_2(R_\mathfrak p )$, then the set of right integral $\mathcal {O}_\mathfrak p$-ideals of reduced norm $\mathfrak p ^e$ is in bijection with the set

  $$\begin{aligned} \left\{ \begin{pmatrix} \pi ^u &{} 0 \\ c &{} \pi ^v \end{pmatrix} : u,v \in \mathbb Z _{\ge 0}, u+v=e\text { and }c \in R/\mathfrak p ^v\right\} \end{aligned}$$

  and

  $$\begin{aligned} a_\mathfrak{p ^e}(\mathcal {O}_\mathfrak p )=1+q+\cdots +q^e. \end{aligned}$$

  (26.4.2)

Proof. For (a), if $\mathfrak p$ is ramified then by the work of section 13.3, there is a unique maximal order $\mathcal {O}_\mathfrak p$ with a unique (two-sided) maximal ideal $J_\mathfrak p$ having ${{\,\mathrm{nrd}\,}}(J_\mathfrak p )=\mathfrak p$, and all ideals of $\mathcal {O}_\mathfrak p$ are powers of $J_\mathfrak p$.

To prove (b), we appeal to the theory of elementary divisors (applying column operations, acting on the right). Suppose $\mathcal {O}_\mathfrak p ={{\,\mathrm{M}\,}}_2(R_\mathfrak p )$. Let $I_\mathfrak p = \alpha _\mathfrak p \mathcal {O}_\mathfrak p$ be a right integral $\mathcal {O}_\mathfrak p$-ideal of norm $\mathfrak p ^e$ and let $\pi$ be a uniformizer for $\mathfrak p$. Then by the theory of elementary divisors, we can write

$$\begin{aligned} \alpha _\mathfrak p = \begin{pmatrix} \pi ^{u} &{} 0 \\ c &{} \pi ^{v} \end{pmatrix} \end{aligned}$$

for unique $u,v\in \mathbb Z _{\ge 0}$ with $u+v=e$ and $c \in R$ is uniquely defined as element of $R/\mathfrak p ^v$ (Exercise 26.6). It follows that the number of such ideals is equal to $\sum _{v=0}^e q^v = 1+q + \dots + q^e$. $\square$

26.4.3

There is an alternate bijection that is quite useful. We say an integral right $\mathcal {O}$-ideal I is $\mathfrak p$-primitive if it does not contain $\mathfrak p \mathcal {O}$ (so we cannot write $I=\mathfrak p I'$ with $I'$ integral).

For a commutative ring A, we define the projective line over A to be the set

$$\begin{aligned} \mathbb P ^1(A) :=\{(x,y) \in A^2 : xA+yA=A\}/A^\times \end{aligned}$$

and write equivalence classes $(x:y) \in \mathbb P ^1(A)$.

Then for $\mathcal {O}_\mathfrak p ={{\,\mathrm{M}\,}}_2(R_\mathfrak p )$, there is a bijection

$$\begin{aligned} \begin{aligned} \mathbb P ^1(R/\mathfrak p ^e)&\rightarrow \{I_\mathfrak p \subseteq \mathcal {O}_\mathfrak p : I_\mathfrak p \text { primitive and }{{\,\mathrm{nrd}\,}}(I_\mathfrak p )=\mathfrak p ^e\} \\ (a:c)&\mapsto \begin{pmatrix} a &{} 0 \\ c &{} 0 \end{pmatrix}\mathcal {O}_\mathfrak p + \mathfrak p ^e \mathcal {O}_\mathfrak p \end{aligned} \end{aligned}$$

(26.4.4)

Any ideal of the form in the right-hand side of (26.4.4) is a primitive right integral $\mathcal {O}_\mathfrak p$-ideal with reduced norm $\mathfrak p ^e$. Conversely, suppose that $I_\mathfrak p =\alpha _\mathfrak p \mathcal {O}_\mathfrak p$ is primitive. We have ${{\,\mathrm{nrd}\,}}(\alpha _\mathfrak p ) \equiv 0 \pmod \mathfrak{p ^e}$. We find a “standard form” for $I_\mathfrak p$ by looking at the left kernel of $\alpha _\mathfrak p$. Let

$$\begin{aligned} L :=\{x \in (R/\mathfrak p ^e)^2 : x \alpha _\mathfrak p \equiv 0 ~(\text{ mod } ~\mathfrak{p ^e})\}. \end{aligned}$$

We claim that L is a free $R/\mathfrak p ^e$-module of rank 1. Indeed, L is one-dimensional over $R/\mathfrak p$ since $I_\mathfrak p$ is primitive and so $\alpha _\mathfrak p \not \equiv 0 \pmod \mathfrak{p }$; by Hensel’s lemma, it follows that L is also one-dimensional. Therefore, there is a unique generator $(a:c) \in \mathbb P ^1(R/\mathfrak p ^e)$ for L. We therefore define an map $I_\mathfrak p \mapsto (-c:a)$ and verify that this furnishes an inverse to (26.4.4).

Since $\#\mathbb P ^1(R/\mathfrak p ^e)=q^e+q^{e-1}$ for $e \ge 1$, we recover the count (26.4.2) as

$$\begin{aligned} a_\mathfrak{p ^e}(\mathcal {O}_\mathfrak p ) = \sum _{i=0}^{\lfloor e/2\rfloor } \#\mathbb P ^1(R/\mathfrak p ^{e-2i}) = q^e+q^{e-1}+\dots +q+1. \end{aligned}$$

26.4.5

Lemma 26.4.1 implies a factorization of $\zeta _{B_p}(s)=\zeta _{\mathcal {O}_\mathfrak p }(s)$. Write

$$\begin{aligned} \zeta _{F_p}(s) = \sum _{e=0}^{\infty } \frac{1}{q^{es}} = \left( 1-\frac{1}{q^s}\right) ^{-1} \end{aligned}$$

(26.4.6)

so that $\zeta _F(s)=\prod _\mathfrak p \zeta _{F_p}(s)$.

Corollary 26.4.7

We have

$$ \zeta _{B_\mathfrak p }(s)=\left( 1-\frac{1}{q^{2s}}\right) ^{-1} \cdot {\left\{ \begin{array}{ll} 1, &{} \text { if }\mathfrak p \text { is ramified;} \\ \left( 1-1/q^{2s-1}\right) ^{-1}, &{} \text { if }\mathfrak p \text { is split.} \end{array}\right. } $$

Equivalently,

$$ \zeta _{B_\mathfrak p }(s)= {\left\{ \begin{array}{ll} \zeta _{F_\mathfrak p }(2s), &{} \text { if }\mathfrak p \text { is ramified;} \\ \zeta _{F_\mathfrak p }(2s)\zeta _{F_\mathfrak p }(2s-1), &{} \text { if }\mathfrak p \text { is split.} \end{array}\right. } $$

Proof. We use Lemma 26.4.1. If $B_\mathfrak p$ is a division ring, then Lemma 26.4.1(a) applies, and the result is immediate. For the second case, we compute

$$\begin{aligned} \begin{aligned} \zeta _{B_\mathfrak p }(s)&= \sum _{e=0}^{\infty } \frac{1+q+\cdots +q^e}{q^{2es}} = \sum _{e=0}^{\infty } \frac{1-q^{e+1}}{(1-q)q^{2es}} \\&= \frac{1}{1-q}\left( \sum _{e=0}^{\infty } \frac{1}{q^{2es}} - q\sum _{e=0}^{\infty } \frac{1}{q^{(2s-1)e}}\right) \\&= \frac{1}{1-q}\left( \frac{1}{1-1/q^{2s}} - \frac{q}{1-1/q^{2s-1}}\right) \\&= \left( 1-\frac{1}{q^{2s}}\right) ^{-1}\left( 1-\frac{1}{q^{2s-1}}\right) ^{-1} \end{aligned} \end{aligned}$$

(26.4.8)

as claimed. $\square$

We have proven the following result.

Theorem 26.4.9

(Factorization of $\zeta _B(s)$, maximal order). Let B be a quaternion algebra of discriminant $\mathfrak D = {{\,\mathrm{disc}\,}}B$. Then

$$\begin{aligned} \zeta _B(s)=\prod _\mathfrak p \zeta _{B_\mathfrak p }(s) = \zeta _F(2s)\zeta _F(2s-1) \prod _\mathfrak{p \mid \mathfrak D } (1-\mathsf{N }(\mathfrak p )^{1-2s}). \end{aligned}$$

(26.4.10)

Proof. Combine the Euler product 26.3.11 with Corollary 26.4.7. $\square$

Corollary 26.4.11

$\zeta _B(s)$ has a simple pole at $s=1$ with residue

$$\begin{aligned} \zeta _B^*(1) = \lim _{s \rightarrow 1} (s-1) \zeta _B(s) = \zeta _F(2) \frac{\zeta _F^*(1)}{2} \prod _\mathfrak{p \mid \mathfrak D } (1-\mathsf{N }(\mathfrak p )^{-1}). \end{aligned}$$

(26.4.12)

Proof. Since $\zeta _F(s)$ has only a simple pole at $s=1$, with residue computed in Theorem 26.2.3, there is a single simple pole of $\zeta _B(s)$ at $s=1$. $\square$

5 Eichler mass formula: maximal orders

We now finish the proof of the Eichler mass formula (Main Theorem 26.1.5) for maximal orders (26.1.13). In the next section, we will deduce the general formula from it: for a nonmaximal order, there are extra factors at each prime dividing the discriminant, and it is simpler to account for those in a separate step.

In this section, we now suppose that B is definite, so F is a totally real field (and $d_F>0$). In particular, B is a division algebra. We saw in 26.1.3 that it was natural to weight ideal classes inversely by the size of their automorphism group (modulo scalars). To this end, and noting $R^\times \trianglelefteq \mathcal {O}^\times$ is central so normal, we prove the following lemma.

Lemma 26.5.1

The group $\mathcal {O}^\times /R^\times$ is finite.

Proof. In Lemma 17.7.13, we proved that

$$\begin{aligned} \mathcal {O}^1 :=\{\gamma \in \mathcal {O}^\times : {{\,\mathrm{nrd}\,}}(\gamma )=1\} \end{aligned}$$

is a finite group by embedding $\mathcal {O}\hookrightarrow B_\mathbb R \simeq \mathbb R ^{4n}$ as a Euclidean lattice with respect to the absolute reduced norm (see 17.7.10). Since $\mathcal {O}^1 \cap R^\times = \{\pm 1\}$, the reduced norm gives an exact sequence

$$\begin{aligned} 1 \rightarrow \frac{\mathcal {O}^1}{\{\pm 1\}} \rightarrow \frac{\mathcal {O}^\times }{R^\times } \xrightarrow {{{\,\mathrm{nrd}\,}}} \frac{R^\times }{R^{\times 2}}. \end{aligned}$$

(26.5.2)

By Dirichlet’s unit theorem, the group $R^\times$ is finitely generated (of rank $r+c-1$), so the group $R^\times /R^{\times 2}$ is a finite abelian 2-group. The result follows. $\square$

We will examine unit groups in detail in Chapter 32. With this finiteness statement in hand, we make the following definition.

Definition 26.5.3

Define the mass of $\mathcal {O}$ to be

$$\begin{aligned} {{\,\mathrm{mass}\,}}({{\,\mathrm{Cls}\,}}\mathcal {O}) :=\sum _{[J] \in {{\,\mathrm{Cls}\,}}\mathcal {O}} \frac{1}{w_J} \end{aligned}$$

where $w_J=[\mathcal {O}{}_{\textsf {\tiny {L}} }(J)^\times :R^\times ] \in \mathbb Z _{\ge 1}$.

Theorem 26.5.4

(Eichler’s mass formula). Let $\mathcal {O}$ be a maximal order in a totally definite quaternion algebra B of discriminant $\mathfrak D$. Then

$$\begin{aligned} {{\,\mathrm{mass}\,}}({{\,\mathrm{Cls}\,}}\mathcal {O}) = \frac{2}{(2\pi )^{2n}} h_F d_F^{3/2} \varphi (\mathfrak D ) \end{aligned}$$

where $\varphi (\mathfrak D )=\prod _\mathfrak{p \mid \mathfrak D }(\mathsf{N }(\mathfrak p )-1)$.

Following the strategy in the classical case (to prove the analytic class number formula), to prove Theorem 26.5.4 we will write $\zeta _{\mathcal {O}}(s)$ as a sum over right ideal classes and analyze its residue at $s=1$ by a volume computation.

26.5.5

For an integral invertible right $\mathcal {O}$-ideal J, let

(26.5.6)

Then

$$\begin{aligned} \zeta _{\mathcal {O}}(s) = \sum _{[J] \in {{\,\mathrm{Cls}\,}}\mathcal {O}} \zeta _{\mathcal {O},[J]}(s). \end{aligned}$$

We have $[I]=[J]$ if and only if $I \simeq J$ if and only if $I=\alpha J$ for nonzero $\alpha \in J^{-1}$. Since $\mu J=J$ if and only if $\mu \in \mathcal {O}{}_{\textsf {\tiny {L}} }(J)^\times$ (Exercise 16.3), it follows that

$$\begin{aligned} \zeta _{\mathcal {O},[J]}(s) = \frac{1}{\mathsf{N }(J)^s} \sum _{0 \ne \alpha \in J^{-1}/\mathcal {O}{}_{\textsf {\tiny {L}} }(J)^{\times }} \frac{1}{\mathsf{N }(\alpha )^s}. \end{aligned}$$

(26.5.7)

By Lemma 26.5.1, we have

$$\begin{aligned} w_J :=[\mathcal {O}{}_{\textsf {\tiny {L}} }(J)^\times : R^\times ] \in \mathbb Z _{>0}. \end{aligned}$$

(26.5.8)

Then (26.5.7) becomes

$$\begin{aligned} \zeta _{\mathcal {O},[J]}(s) = \frac{1}{w_J\mathsf{N }(J)^s} \sum _{0 \ne \alpha \in J^{-1}/R^{\times }} \frac{1}{\mathsf{N }(\alpha )^s}. \end{aligned}$$

(26.5.9)

Proposition 26.5.10

Let $\mathfrak N :={{\,\mathrm{discrd}\,}}(\mathcal {O})$. Then $\zeta _{\mathcal {O},[J]}(s)$ has a simple pole at $s=1$ with residue

$$\begin{aligned} \zeta _{\mathcal {O},[J]}^*(1) = \frac{2^n (2\pi )^{2n} {Reg }_F}{8w_J d_F^2\mathsf{N }(\mathfrak N )}. \end{aligned}$$

Proof. We relate residue to volumes using Theorem 26.2.12. We recall (again) 17.7.10: this gives

$$\begin{aligned} J^{-1} \hookrightarrow B \hookrightarrow B_\mathbb R :=B \otimes _\mathbb Q \mathbb R \simeq \mathbb H ^n \simeq \mathbb R ^{4n} \end{aligned}$$

the structure of a Euclidean lattice $\Lambda \subseteq \mathbb R ^{4n}$ with respect to the absolute reduced norm. We take the function N in Theorem 26.2.12 to be the absolute norm $\mathsf{N }$ (recalling 16.4.8).

We claim that

$$\begin{aligned} {{\,\mathrm{covol}\,}}(\mathcal {O})=\frac{d_F^2\mathsf{N }(\mathfrak N )}{2^n}. \end{aligned}$$

(26.5.11)

By compatible real scaling, it is enough to prove that this relation holds for a single order $\mathcal {O}$, and we choose the R-order

$$\begin{aligned} \mathcal {O}= R \oplus Ri \oplus Rj \oplus Rk. \end{aligned}$$

(26.5.12)

The lattice $R \subseteq F_\mathbb R$ has covolume $\sqrt{d_F}$, so $R^4$ has covolume $\sqrt{d_F}^4 = d_F^2$; the $\mathbb Z$-order $\mathbb Z \oplus \mathbb Z i \oplus \mathbb Z j \oplus \mathbb Z k$ has reduced discriminant 4 and covolume 1; and putting these together, the formula (26.5.11) is verified.

Then (26.5.11) and $\mathsf{N }(J)=[\mathcal {O}:J]_\mathbb{Z }=[J^{-1}:\mathcal {O}]$ imply that

$$\begin{aligned} {{\,\mathrm{covol}\,}}(\Lambda )=\frac{{{\,\mathrm{covol}\,}}(\mathcal {O})}{\mathsf{N }(J)}= \frac{d_F^2 \mathsf{N }(\mathfrak N )}{2^n\mathsf{N }(J)}. \end{aligned}$$

(26.5.13)

Next, the group $\mathcal {O}{}_{\textsf {\tiny {L}} }(J)^{\times }$ acts on $J^{-1}$ (and on $B_\mathbb R$); and this group contains $R^\times$ with finite index $w_J=[\mathcal {O}{}_{\textsf {\tiny {L}} }(J) : R^\times ]$, so

$$\begin{aligned} {{\,\mathrm{vol}\,}}(\mathcal {O}{}_{\textsf {\tiny {L}} }(J)^{\times } \backslash B_\mathbb R ) = \frac{1}{w_J} {{\,\mathrm{vol}\,}}(R^\times \backslash B_\mathbb R ). \end{aligned}$$

(26.5.14)

Multiplication provides an identification

$$\begin{aligned} B_\mathbb{R ,\le 1} \simeq F_\mathbb{R ,\le 1} \times (\mathbb H ^1)^n, \end{aligned}$$

so

$$\begin{aligned} X_{\le 1} = R^\times \backslash B_\mathbb{R ,\le 1} \simeq (E \backslash F_\mathbb{R ,\le 1}) \times (\{\pm 1\} \backslash (\mathbb H ^1)^n) \end{aligned}$$

(26.5.15)

where $E \le R^\times$ is acting by squares. Thus

$$\begin{aligned} {{\,\mathrm{vol}\,}}(R^\times \backslash F_\mathbb{R ,\le 1}) = \frac{2^{n-1}}{2(2^n)}{Reg }_F = \frac{1}{4}{Reg }_F. \end{aligned}$$

(26.5.16)

Therefore

$$\begin{aligned} {{\,\mathrm{vol}\,}}(X_{\le 1}) = \frac{(2\pi ^2)^n {Reg }_F}{8w_J}. \end{aligned}$$

(26.5.17)

From Theorem 26.2.12 together with (26.5.13) and (26.5.17),

$$\begin{aligned} \zeta _{\mathcal {O},[J]}^*(1) = \frac{4^n(2\pi ^2)^{n}{Reg }_F}{8w_J d_F^2 \mathsf{N }(\mathfrak N )} = \frac{2^n(2\pi )^{2n}{Reg }_F}{8w_J d_F^2 \mathsf{N }(\mathfrak N )}. \square \end{aligned}$$

(26.5.18)

We now conclude the proof.

Proof of Theorem 26.5.4

We now suppose that $\mathcal {O}\subset B$ is a maximal order, and write $\zeta _B(s)$ and $\zeta _{B,[J]}(s)$. We compare the evaluation of residues given by Corollary 26.4.11 and Proposition 26.5.10. Since $\zeta _B(s)$ and each $\zeta _{B,[J]}(s)$ have simple poles at $s=1$, we get

$$\begin{aligned} \zeta _B^*(1) = \sum _{[J] \in {{\,\mathrm{Cls}\,}}\mathcal {O}} \zeta _{B,[J]}^*(1). \end{aligned}$$

From (26.4.12),

$$\begin{aligned} \zeta _{B}^*(1) = \zeta _F(2)\frac{\zeta _F^*(1)}{2} \prod _\mathfrak{p \mid \mathfrak D } \left( 1-\frac{1}{\mathsf{N }(\mathfrak p )}\right) = \zeta _F(2)\frac{\zeta _F^*(1)}{2} \frac{\varphi (\mathfrak D )}{\mathsf{N }(\mathfrak D )}. \end{aligned}$$

(26.5.19)

From the analytic class number formula (Theorem 26.2.3),

$$\begin{aligned} \zeta _F^*(1)= \frac{2^n}{2\sqrt{d_F}} h_F {Reg }_F \end{aligned}$$

since $w_F=2$ (as F is totally real).

Adding the residues from Lemma 26.5.10, we find that

$$\begin{aligned} \frac{2^n\zeta _F(2)}{4\sqrt{d_F}} h_F {Reg }_F \frac{\varphi (\mathfrak D )}{\mathsf{N }(\mathfrak D )} = \frac{2^n (2\pi )^{2n} {Reg }_F}{8 d_F^2\mathsf{N }(\mathfrak D )} \sum _{[J] \in {{\,\mathrm{Cls}\,}}\mathcal {O}} \frac{1}{w_J}. \end{aligned}$$

(26.5.20)

Cancelling, we find

$$\begin{aligned} {{\,\mathrm{mass}\,}}({{\,\mathrm{Cls}\,}}\mathcal {O}) = \sum _{[J] \in {{\,\mathrm{Cls}\,}}\mathcal {O}} \frac{1}{w_J} = \frac{2}{(2\pi )^{2n}} \zeta _F(2) d_F^{3/2} h_F \varphi (\mathfrak D ) \end{aligned}$$

(26.5.21)

and this concludes the proof. $\square$

Remark 26.5.22. For an alternative direct approach in this setting using Epstein zeta functions, see Sands [San2017].

6 Eichler mass formula: general case

We now consider the general case of the Eichler mass formula, involving two steps. First, we relate the class set of a suborder to the class set of a (maximal) superorder; second, we compute the fibers of this map via a group action of the units.

For these steps, we refresh our notation and allow B to be a definite or indefinite quaternion algebra over F.

26.6.1

Let $\mathcal {O}' \supseteq \mathcal {O}$ be an R-superorder, and suppose that there is a prime $\mathfrak p$ such that $\mathcal {O}_\mathfrak q '=\mathcal {O}_\mathfrak q$ for all primes $\mathfrak q \ne \mathfrak p$. We refine the map from Exercise 17.3(b) as follows. For $I \subseteq \mathcal {O}$ a right $\mathcal {O}$-ideal, we define the right $\mathcal {O}'$-ideal $\rho (I)=I\mathcal {O}' \subseteq \mathcal {O}'$ obtained by extension. Then $\rho$ induces a map

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

(26.6.2)

that is well-defined and surjective (Exercise 26.5(a)). Let $[I'] \in {{\,\mathrm{Cls}\,}}\mathcal {O}'$ and consider the set

$$\begin{aligned} \rho ^{-1}(I')=\{I \subseteq \mathcal {O}: I\mathcal {O}=I'\}, \end{aligned}$$

the fiber of the extension map over $I'$.

We define an action of the group $\mathcal {O}_\mathfrak p '^\times$ on $\rho ^{-1}(I')$ as follows. Write $I'_\mathfrak p =\beta _\mathfrak p \mathcal {O}_\mathfrak p '$. Then to $\mu _\mathfrak p \in \mathcal {O}_\mathfrak p '^\times$, we associate the unique lattice $I\langle \mu _\mathfrak p \rangle$ (the notation to suggest “the lattice generated by $\mu _\mathfrak p$”) such that

$$\begin{aligned} I\langle \mu _\mathfrak p \rangle _\mathfrak p = \beta _\mathfrak p \mu _\mathfrak p \mathcal {O}_\mathfrak p \end{aligned}$$

and $I\langle \mu _\mathfrak p \rangle _\mathfrak q = I_\mathfrak q = I'_\mathfrak q$ for all $\mathfrak q \ne \mathfrak p$, using the local-global dictionary (Theorem 9.4.9). This defines a right action of $\mathcal {O}_\mathfrak p '^\times$; it acts simply transitively on $\rho ^{-1}(I')$, and the kernel of this action is visibly the subgroup $\mathcal {O}_\mathfrak p ^\times$. Therefore

$$\begin{aligned} \#\rho ^{-1}(I') = [\mathcal {O}_\mathfrak p '^\times :\mathcal {O}_\mathfrak p ^\times ]. \end{aligned}$$

We now look at the classes in the fiber. If $\mu _\mathfrak p ,\nu _\mathfrak p \in \mathcal {O}_\mathfrak p '^\times$ have $[I\langle \mu _\mathfrak p \rangle ]=[I\langle \nu _\mathfrak p \rangle ] \in {{\,\mathrm{Cls}\,}}\mathcal {O}$, then there exists $\alpha \in B^\times$ such that

$$\begin{aligned} \alpha I\langle \mu _\mathfrak p \rangle = I\langle \nu _\mathfrak p \rangle \end{aligned}$$

and by extension $\alpha I' = I'$, so $\alpha \in \mathcal {O}{}_{\textsf {\tiny {L}} }(I')$, and conversely. Therefore, we have a bijection

$$\begin{aligned} {{\,\mathrm{Cls}\,}}\mathcal {O}\leftrightarrow \bigsqcup _{[I'] \in {{\,\mathrm{Cls}\,}}\mathcal {O}'} \mathcal {O}{}_{\textsf {\tiny {L}} }(I')^\times \backslash \rho ^{-1}(I'). \end{aligned}$$

(26.6.3)

(See also Pacetti–Sirolli [PS2014, §3].)

Proposition 26.6.4

Let $\mathcal {O}' \supseteq \mathcal {O}$ be an R-superorder, and suppose that there is a prime $\mathfrak p$ such that $\mathcal {O}_\mathfrak q '=\mathcal {O}_\mathfrak q$ for all primes $\mathfrak q \ne \mathfrak p$. Then

$$\begin{aligned} {{\,\mathrm{mass}\,}}({{\,\mathrm{Cls}\,}}\mathcal {O}) = [\mathcal {O}_\mathfrak p '^\times : \mathcal {O}_\mathfrak p ^\times ] {{\,\mathrm{mass}\,}}({{\,\mathrm{Cls}\,}}\mathcal {O}'). \end{aligned}$$

Proof. By (26.6.3), we conclude that

$$\begin{aligned} \begin{aligned} {{\,\mathrm{mass}\,}}({{\,\mathrm{Cls}\,}}\mathcal {O})&= \sum _{[I] \in {{\,\mathrm{Cls}\,}}\mathcal {O}} \frac{1}{w_I} = \sum _{[I'] \in {{\,\mathrm{Cls}\,}}\mathcal {O}'} \sum _{I\mathcal {O}' = I'} \frac{1}{w_I} \left( \frac{w_{I'}}{w_I}\right) ^{-1} \\&= \sum _{[I'] \in {{\,\mathrm{Cls}\,}}\mathcal {O}'} [\mathcal {O}_\mathfrak p '^\times :\mathcal {O}_\mathfrak p ^\times ] \frac{1}{w_{I'}} \\&= [\mathcal {O}_\mathfrak p '^\times :\mathcal {O}_\mathfrak p ^\times ] {{\,\mathrm{mass}\,}}({{\,\mathrm{Cls}\,}}\mathcal {O}') \end{aligned} \end{aligned}$$

(26.6.5)

as claimed. $\square$

In order to apply Proposition 26.6.4, we need to compute the index of unit groups, a quantity that depends on the (locally defined) Eichler symbol. For a prime $\mathfrak p$, we define

$$\begin{aligned} \lambda (\mathcal {O},\mathfrak p ) :=\frac{1-{{\,\mathrm{Nm}\,}}(\mathfrak p )^{-2}}{1-\biggl (\displaystyle {\frac{\mathcal {O}}{\mathfrak{p }}}\biggr ){{\,\mathrm{Nm}\,}}(\mathfrak p )^{-1}}= {\left\{ \begin{array}{ll} 1+1/q, &{} \text {if }({\mathcal {O}} \mid {p})=1; \\ 1-1/q, &{} \text {if }({\mathcal {O}} \mid {p})=-1; \\ 1-1/q^2, &{} \text {if }({\mathcal {O}} \mid {p})=0. \end{array}\right. } \end{aligned}$$

(26.6.6)

Lemma 26.6.7

Let $\mathcal {O}' \supseteq \mathcal {O}$ be a containment of R-orders with $\mathcal {O}'$ maximal. Then

$$ [\mathcal {O}_\mathfrak p '^\times :\mathcal {O}_\mathfrak p ^\times ] = [\mathcal {O}_\mathfrak p ':\mathcal {O}_\mathfrak p ]\lambda (\mathcal {O},\mathfrak p )\cdot {\left\{ \begin{array}{ll} 1, &{} \text { if }\mathfrak p \text { is split in }B; \\ (1-1/q)^{-1}, &{} \text { if }\mathfrak p \text { is ramified in }B. \end{array}\right. } $$

Proof. We follow Körner [Kör85, §3]. To prove the lemma, we may localize at $\mathfrak p$ and so we drop the subscripts. Let $n \in \mathbb Z _{\ge 1}$ be such that $\mathfrak p ^n \mathcal {O}' \subseteq \mathfrak p \mathcal {O}$. Then

$$\begin{aligned}{}[\mathcal {O}'^\times :\mathcal {O}^\times ] = \frac{[\mathcal {O}'^\times :1+\mathfrak p \mathcal {O}'][1+\mathfrak p \mathcal {O}':1+\mathfrak p ^n\mathcal {O}']}{[\mathcal {O}^\times :1+\mathfrak p \mathcal {O}][1+\mathfrak p \mathcal {O}:1+\mathfrak p ^n\mathcal {O}']}. \end{aligned}$$

For $\gamma ,\delta \in 1+\mathfrak p \mathcal {O}$, we have $\gamma \delta ^{-1} \in 1+\mathfrak p ^n \mathcal {O}'$ if and only if $\gamma -\delta \in \mathfrak p ^n \mathcal {O}'$. Therefore

$$\begin{aligned}{}[1+\mathfrak p \mathcal {O}:1+\mathfrak p ^n \mathcal {O}'] = [\mathfrak p \mathcal {O}:\mathfrak p ^n \mathcal {O}'] = [\mathcal {O}:\mathfrak p ^{n-1} \mathcal {O}'] \end{aligned}$$

and similarly with $\mathcal {O}'$, all indices taken as abelian groups. Therefore

$$\begin{aligned} \frac{[1+\mathfrak p \mathcal {O}':1+\mathfrak p ^n\mathcal {O}']}{[1+\mathfrak p \mathcal {O}:1+\mathfrak p ^n\mathcal {O}']} = [\mathcal {O}':\mathcal {O}]. \end{aligned}$$

For the other terms, we recall Lemma 24.3.12. We divide up into the cases, noting that if$({\mathcal {O}'} \mid {\mathfrak{p }})=-1$ then we must have $\varepsilon =-1,0$ by classification (Exercise 24.3); this leaves 6 cases to compute. For example, if $({\mathcal {O}'} \mid {\mathfrak{p }})=*$ and $({\mathcal {O}'} \mid {\mathfrak{p }})=1$, then

$$ \frac{[\mathcal {O}'^\times :1+\mathfrak p \mathcal {O}']}{[\mathcal {O}^\times :1+\mathfrak p \mathcal {O}]} =\frac{q(q-1)^2(q+1)}{q^2(q-1)^2}=1+\frac{1}{q}. $$

The other cases follow similarly (Exercise 26.8). $\square$

We can now finish the job.

Proof of Main Theorem 26.1.5

We first invoke Theorem 26.5.4 for a maximal order $\mathcal {O}' \supseteq \mathcal {O}$ to get

$$\begin{aligned} {{\,\mathrm{mass}\,}}({{\,\mathrm{Cls}\,}}\mathcal {O}') = \frac{2}{(2\pi )^{2n}} h_F d_F^{3/2} {{\,\mathrm{Nm}\,}}(\mathfrak D ) \prod _\mathfrak{p \mid \mathfrak D }\left( 1-\frac{1}{{{\,\mathrm{Nm}\,}}(\mathfrak p )}\right) . \end{aligned}$$

By Proposition 26.6.4 and Lemma 26.6.7, we have

$$\begin{aligned} \begin{aligned} {{\,\mathrm{mass}\,}}({{\,\mathrm{Cls}\,}}\mathcal {O})&= {{\,\mathrm{mass}\,}}({{\,\mathrm{Cls}\,}}\mathcal {O}')\prod _\mathfrak{p \mid \mathfrak N } [\mathcal {O}_\mathfrak p '^\times :\mathcal {O}_\mathfrak p ^\times ] \\&= {{\,\mathrm{mass}\,}}({{\,\mathrm{Cls}\,}}\mathcal {O}') [\mathcal {O}':\mathcal {O}]_\mathbb{Z } \prod _\mathfrak{p \mid \mathfrak D } \left( 1-\frac{1}{{{\,\mathrm{Nm}\,}}(\mathfrak p )}\right) ^{-1} \prod _\mathfrak{p \mid \mathfrak N } \lambda (\mathcal {O},\mathfrak p ) \\&= \frac{2}{(2\pi )^{2n}} h_F d_F^{3/2} {{\,\mathrm{Nm}\,}}(\mathfrak N ) \prod _\mathfrak{p \mid \mathfrak N } \lambda (\mathcal {O},\mathfrak p ) \end{aligned} \end{aligned}$$

(26.6.8)

using ${{\,\mathrm{Nm}\,}}(\mathfrak N )={{\,\mathrm{Nm}\,}}(\mathfrak D )[\mathcal {O}':\mathcal {O}]_\mathbb{Z }$. $\square$

7 Class number one

It is helpful to get a sense of the overall size of the mass, as follows.

26.7.1

Let $m(\mathfrak D ,\mathfrak M )$ be the mass of an(y) Eichler order of level $\mathfrak M$. Then in analogy with the Brauer–Siegel theorem,

$$\begin{aligned} \log m(\mathfrak D ,\mathfrak M ) \sim \frac{3}{2}{\log d_F} + \log h_F + \log {{\,\mathrm{Nm}\,}}(\mathfrak D \mathfrak M ) \end{aligned}$$

(26.7.2)

as $d_F{{\,\mathrm{Nm}\,}}(\mathfrak D \mathfrak M ) \rightarrow \infty$ with the degree n fixed: see Exercise 26.9. In particular, for $F=\mathbb Q$,

$$\begin{aligned} \log m(D,M) \sim \log (DM). \end{aligned}$$

Since F is totally real, one typically expects $h_F$ to be small in comparison to $d_F$—but there is a family of real quadratic fields with small regulator first studied by Chowla with $\log h_F \sim \frac{1}{2}\log d_F$, a result due to Montgomery–Weinberger [MW77].

To conclude this section, as in section 25.4 (over $\mathbb Q$), the Eichler mass formula can now be used to solve class number one problems for quaternion orders for definite quaternion orders (over totally real fields). This effort was undertaken recently by Kirschmer–Lorch [KL2016]: a complete list of definite orders of type number one is given, and again because $\#{{\,\mathrm{Typ}\,}}\mathcal {O}\le \#{{\,\mathrm{Cls}\,}}\mathcal {O}$, the following theorem can be proven.

Theorem 26.7.3

(Kirschmer–Lorch). There are 4194 one-class genera of primitive, positive definite ternary quadratic forms (equivalently, definite quaternion orders $\mathcal {O}$ with $\#{{\,\mathrm{Typ}\,}}\mathcal {O}=1$, up to isomorphism): they occur over 30 possible base fields of degrees up to 5.

There are exactly 154 isomorphism classes of definite quaternion orders $\mathcal {O}$ with $\#{{\,\mathrm{Cls}\,}}\mathcal {O}=1$; of these, 144 are Gorenstein and 10 are non-Gorenstein.

Remark 26.7.4. Kirschmer–Lorch [KL2016] also enumerate two-class genera; a complete list is available online [KLwww]. Cerri–Chaubert–Lezowski [CCL2013] also consider totally definite Euclidean orders over totally real fields, giving the complete list over $\mathbb Q$ and over quadratic fields: all of them are Euclidean under the reduced norm.

8 Functional equation and classification

To conclude this chapter, we discuss the functional equation and important applications to the classification of quaternion algebras over number fields. This section serves as preview of the material in chapter 29 (motivation for it!) but we frame results in the same vein as the results of this chapter.

Following Riemann, we complete $\zeta (s) :=\sum _{n=1}^{\infty } n^{-s}$ to the function

$$\begin{aligned} \xi (s) :=\pi ^{-s/2}\zeta (s)\Gamma (s/2), \end{aligned}$$

where $\Gamma (s)$ is the complex $\Gamma$-function (Exercise 26.2). Riemann proved that $\xi (s)$ extends to a meromorphic function on $\mathbb C$ and satisfies the functional equation

$$\begin{aligned} \xi (1-s)=\xi (s). \end{aligned}$$

(26.8.1)

(It is also common to multiply $\xi (s)$ by $s(1-s)$ to cancel the poles at $s=0,1$.) We will prove this in section 29.1—the impatient reader is encouraged to flip ahead!

26.8.2

This result extends to Dedekind zeta functions (retaining the notation from section 26.2). Define

$$\begin{aligned} \Gamma _\mathbb R (s) :=\pi ^{-s/2} \Gamma (s/2), \qquad \Gamma _\mathbb C (s) :=2(2\pi )^{-s}\Gamma (s). \end{aligned}$$

(26.8.3)

We then define the completed  Dedekind zeta function to be

$$\begin{aligned} \xi _F(s) :=|d_F \,|^{s/2} \Gamma _\mathbb R (s)^r \Gamma _\mathbb C (s)^c \zeta _F(s). \end{aligned}$$

(26.8.4)

Then $\xi _F(s)$ satisfies the functional equation

$$\begin{aligned} \xi _F(1-s)=\xi _F(s) \end{aligned}$$

(26.8.5)

for all $s \in \mathbb C$. We prove (26.8.5) as Corollary 29.10.3(a) using idelic methods; this proof will also motivate the completion defined above. For now, we borrow from the future. The functional equation gives $\zeta _F(s)$ meromorphic continuation to $\mathbb C$ via

$$\begin{aligned} \zeta _F(1-s)=\zeta _F(s) \left( \frac{|d_F \,|}{4^c \pi ^n}\right) ^{s-1/2} \frac{\Gamma (s/2)^r \Gamma (s)^c}{\Gamma ((1-s)/2)^r \Gamma (1-s)^c}. \end{aligned}$$

(26.8.6)

26.8.7

Using the functional equation (26.8.6), we can rewrite (26.2.4) to obtain the tidier expression

$$\begin{aligned} \zeta _F^*(0) = \lim _{s \rightarrow 0} s^{-(r+c-1)} \zeta _F(0) = \frac{h_F {Reg }_F}{w_F}; \end{aligned}$$

(26.8.8)

in particular, $\zeta _F$ has a zero at $s=0$ of order $r+c-1$, the rank of the unit group of R by Dirichlet’s unit theorem.

In terms of the completed Dedekind zeta function, we find $\xi _F(s)$ has analytic continuation to $\mathbb C \smallsetminus \{0,1\}$ with simple poles at $s=0,1$ and residues

$$\begin{aligned} \xi _F^*(0)=\xi _F^*(1)=\frac{2^{r+c} h_F {Reg }_F}{w_F}. \end{aligned}$$

(26.8.9)

Example 26.8.10

When F is an imaginary quadratic field ($r=0$ and $c=1$) we have ${Reg }_F=1$ and $\zeta _F^*(0)=\zeta _F(0)$, so $h_F/w_F = \zeta _F(0)$, and in particular if $|d_F \,| > 4$ then $h_F = 2\zeta _F(0)$.

We now turn to a quaternionic generalization.

26.8.11

The factorization of $\zeta _B(s)$ in Theorem 26.4.9 implies a functional equation for $\zeta _B(s)$ via the functional equation for $\zeta _F(s)$. This functional equation is simplest to state for a completed zeta function. We recall that

$$\begin{aligned} \zeta _B(s) = \zeta _F(2s)\zeta _F(2s-1) \prod _\mathfrak{p \mid \mathfrak D } \left( 1-{{\,\mathrm{Nm}\,}}(\mathfrak p )^{1-2s}\right) ; \end{aligned}$$

just as the function $\zeta _F$ completes to $\xi _F$ with a simple functional equation, we analogously complete $\zeta _B$ to

$$\begin{aligned} \xi _B(s) :=\xi _F(2s)\xi _F(2s-1) \prod _\mathfrak{p \mid \mathfrak D } {{\,\mathrm{Nm}\,}}(\mathfrak p )^{s}\bigl (1-{{\,\mathrm{Nm}\,}}(\mathfrak p )^{1-2s}\bigr ) \prod _{v \in \Omega } (2s-1) \end{aligned}$$

(26.8.12)

where $\Omega \subseteq {{\,\mathrm{Ram}\,}}B$ be the set of real, ramified places in B. The definition (26.8.12) is motivated by the simplicity of the functional equation; see also Remark 26.8.15 below.

Written in a different way,

$$\begin{aligned} \xi _B(s) = (2\pi )^t (|d_F \,|^4 {{\,\mathrm{Nm}\,}}(\mathfrak D )^2)^{s/2} \Gamma _B(s) \zeta _B(s) \end{aligned}$$

(26.8.13)

where t is the number of split real places, so that $\#{{\,\mathrm{Pl}\,}}(F) = \#\Omega + t$,

$$\begin{aligned} \begin{aligned} \Gamma _B(s)&:=\Gamma _\mathbb R (2s)^{r} \Gamma _\mathbb R (2s+1)^{r-t} \Gamma _\mathbb R (2s-1)^t \Gamma _\mathbb C (2s)^c \Gamma _\mathbb C (2s-1)^c, \end{aligned} \end{aligned}$$

(26.8.14)

and we have used the formula

$$\begin{aligned} (2s-1)\Gamma _\mathbb R (2s-1)&= 2(s-1/2)\pi ^{-(2s-1)/2}\Gamma (s-1/2)\\&=(2\pi )\pi ^{-(2s+1)/2}\Gamma (s+1/2) \\&= 2\pi \Gamma _\mathbb R (2s+1). \end{aligned}$$

Remark 26.8.15. The completion factors (26.8.12) are not arbitrarily chosen; they have a natural interpretation from an idelic perspective. Perhaps this serves as a motivation for working idelically: namely, that it helps to nail down these kinds of quantities! For more, see section 29.8.

Some properties can be read off easily from (26.8.12).

Proposition 26.8.16

(Analytic continuation, functional equation). Let $m=\#{{\,\mathrm{Ram}\,}}B$. Then the following statements hold.

• (a) $\xi _B(s)$ has meromorphic continuation to $\mathbb C$ and is holomorphic in $\mathbb C \smallsetminus \{0,1/2,1\}$ with simple poles at $s=0,1$.

• (b) $\xi _B(s)$ satisfies the functional equation

  $$\begin{aligned} \xi _B(1-s) = (-1)^m \xi _B(s). \end{aligned}$$

  (26.8.17)

• (c) $\xi _B(s)$ has a pole of order $2-m$ at $s=1/2$; in particular, if $m \ge 2$ then $\xi _B(s)$ is holomorphic at $s=1/2$.

Proof. Statement (a) follows from (26.8.12), recalling that $\xi _F(s)$ is holomorphic in $\mathbb C \smallsetminus \{0,1\}$ by 26.8.7 with simple poles at $s=0,1$. Part (c) follows similarly from (b), since the other factors in (26.8.12) have a simple zero at $s=1/2$.

To prove (b), we consider each term in the definition of (26.8.12). The functional equation (26.8.5) for $\xi _F(s)$ with $s \leftarrow 1-s$ implies

$$\begin{aligned} \begin{aligned} \xi _F(2(1-s))\xi _F(2(1-s)-1)&= \xi _F(2-2s)\xi _F(1-2s) \\&=\xi _F(1-(2-2s))\xi _F(1-(1-2s)) \\&=\xi _F(2s-1)\xi _F(2s). \end{aligned} \end{aligned}$$

(26.8.18)

For

$$\begin{aligned} \ell (s)=q^{s}(1-q^{1-2s})=q^{s}-q^{1-s} \end{aligned}$$

and $q>0$ we have $\ell (1-s)=-\ell (s)$, so with $q={{\,\mathrm{Nm}\,}}(\mathfrak p )$ the factors $\mathfrak p \mid \mathfrak D$ are taken into account. Finally, $2(1-s)-1=-(2s-1)$ takes care of $v \in \Omega$, and (b) follows. $\square$

Proposition 26.8.16 shows how algebraic properties of B correspond to analytic properties of $\xi _B$. A deeper investigation ultimately will reveal the following fundamental result.

Theorem 26.8.19

(Sign of functional equation, holomorphicity). $\xi _B(s)$ satisfies the functional equation

$$\begin{aligned} \xi _B(1-s) = \xi _B(s). \end{aligned}$$

(26.8.20)

Moreover, if B is a division algebra, then $\xi _B(s)$ is holomorphic at $s=1/2$.

Proof. This theorem was proven by Hey [Hey29, §3] (more generally, for division algebras over $\mathbb Q$) following the same general script as in the proof of the functional equation for the Dedekind zeta function (26.8.5), as proven first by Hecke: the key ingredient is Poisson summation. The argument is also given by Eichler [Eic38a, Part V]. We instead prove this theorem in the language of ideles (Main Theorem 29.2.6), as it simplifies the calculations—and so for continuity of ideas in the exposition, we borrow from the future. $\square$

Assuming Theorem 26.8.19, we can now deduce the main classification theorem (Main Theorem 14.6.1) for quaternion algebras over number fields. First, we have Hilbert reciprocity as an immediate consequence.

Corollary 26.8.21

(Hilbert reciprocity, cf. Corollary 14.6.2). $\#{{\,\mathrm{Ram}\,}}B$ is even.

Proof. Immediate from (26.8.17) and (26.8.20). $\square$

Next we conclude the all-important local-global principle.

Corollary 26.8.22

We have $B \simeq {{\,\mathrm{M}\,}}_2(F)$ if and only if $B_v \simeq {{\,\mathrm{M}\,}}_2(F_v)$ for all (but one) places $v \in {{\,\mathrm{Pl}\,}}F$.

Proof. The implication $(\Rightarrow )$ is immediate. For the converse $(\Leftarrow )$, by Proposition 26.8.16(c), $\xi _B(s)$ has a pole of order $2-m$ at $s=1/2$, so if $m \le 1$ then $\xi _B(s)$ is not holomorphic at $s=1/2$; but then by Theorem 26.8.19, B is not a division algebra, so $B \simeq {{\,\mathrm{M}\,}}_2(F)$ (and the order of pole is necessarily 2, and $B_v \simeq {{\,\mathrm{M}\,}}_2(F_v)$ for all v). $\square$

From this corollary, we are able to deduce the Hasse norm theorem for quadratic extensions.

Theorem 26.8.23

(Hasse norm theorem). Let $K \supseteq F$ be a separable quadratic field extension and let $b \in F^\times$. Then $b \in {{\,\mathrm{Nm}\,}}_{K/F}(K^\times )$ if and only if $b \in {{\,\mathrm{Nm}\,}}_{K_v/F_v}(K_v^\times )$ for all (but one) places $v \in {{\,\mathrm{Pl}\,}}F$.

Proof. Consider the quaternion algebra $B=({K,b} \mid {F})$. Then by Main Theorem 5.4.4, we have $b \in {{\,\mathrm{Nm}\,}}_{K|F}(K^\times )$ if and only if $B \simeq {{\,\mathrm{M}\,}}_2(F)$. By Corollary 26.8.22, this holds if and only if $B_v \simeq {{\,\mathrm{M}\,}}_2(F_v)$ for all (but one) places v. Repeating the application of Main Theorem 5.4.4, this holds if and only if $B_v \simeq {{\,\mathrm{M}\,}}_2(F_v)$ for all (but one) v. $\square$

We may similarly conclude the all important local-global principle for quadratic forms.

Theorem 26.8.24

(Hasse–Minkowski theorem). Let Q be a quadratic form over F. Then Q is isotropic over F if and only if $Q_v$ is isotropic over $F_v$ for all places v of F.

Proof. The implication $(\Rightarrow )$ is immediate, so we prove $(\Leftarrow )$. We may suppose without loss of generality that Q is nondegenerate. If $n=\dim _F V=1$, the theorem is vacuous.

Suppose $n=2$. Then after scaling we may suppose $Q=\langle 1, -a \rangle$, and Q is isotropic if and only if a is a square. Suppose for purposes of contradiction that $K=F(\sqrt{a})$ is a field. Since $Q_v$ is isotropic for all v, we have $K_v \simeq F_v \times F_v$ for all v, and thus $\zeta _K(s) = \zeta _F(s)^2$. But as Dedekind zeta functions, both $\zeta _F(s)$ and $\zeta _K(s)$ have poles of order 1 at $s=1$ (we evaluated the residue in the analytic class number formula, Theorem 26.2.3), a contradiction.

Suppose $n=3$. Again after rescaling we may suppose $Q=\langle 1, -a, -b \rangle$, and Q is isotropic if and only if b is a norm from $F[\sqrt{a}]$: then the equivalence follows from Theorem 26.8.23.

Next, suppose $n=4$, and $Q=\langle 1, -a, -b, c \rangle$. Let $K=F(\sqrt{abc})$. By extension, Q is isotropic over K and all of its completions. But now $Q \simeq \langle 1,-a,-b,ab \rangle$ over K. Let $B=({a,b} \mid {K})$. Then by Main Theorem 5.4.4, we have $B_w \simeq {{\,\mathrm{M}\,}}_2(K_w)$ for all w; thus by Corollary 26.8.22 we have $B \simeq {{\,\mathrm{M}\,}}_2(K)$, so K splits B. By 5.4.7, we have $K \hookrightarrow B$, so there exist $x,y,z \in F$ such that

$$ {{\,\mathrm{nrd}\,}}(\alpha )={{\,\mathrm{nrd}\,}}(xi+yj+zij)=-ax^2-by^2+abz^2=-abc; $$

dividing by ab we have $z^2 - a(y/a)^2 - b(x/b)^2 + c = 0$ so $Q(z,y/a,x/b,1)=0$.

Finally, when $n \ge 5$, we make an argument like at the end of proof of Theorem 14.3.3: we follow Lam [Lam2005, Theorem VI.3.8] and Milne [Milne-CFT, Theorem VIII.3.5(b)], but we are brief. Write $Q = Q_1 \perp Q_2$ where $Q_1=\langle a,b\rangle$ and $\dim _F V_2 \ge 3$. Choosing a ternary subform and looking at its quaternion algebra, we find a finite set $T \subseteq {{\,\mathrm{Pl}\,}}F$ such that $Q_2$ is isotropic for all $v \not \in T$. For each $v \in T$, let $Q(z_v)=0$ and let $c_v=Q_1(z_v)=-Q_2(z_v)$. Choose $x,y \in F^\times$ close enough so that $z=Q_1(x)$ has $zz_v \in F_v^{\times 2}$. The form $Q'=\langle c \rangle \perp Q_2$ in $n-1$ variables is isotropic for all v: for $v \in T$ this was arranged, and for $v \not \in T$ already $Q_2$ was isotropic at v. By induction on n, we conclude that $Q'$ is isotropic; diagonalizing, we may write $Q = \langle d \rangle \perp Q'$, and it follows that Q is isotropic. $\square$

Corollary 26.8.25

Let $Q,Q'$ be quadratic forms over F in the same number of variables. Then $Q \simeq Q'$ if and only if $Q_v \simeq Q_v'$ for all places $v \in {{\,\mathrm{Pl}\,}}F$.

Proof. Apply the same method of proof as in Corollary 14.3.7: see Exercise 26.10. $\square$

We may now conclude the classification with one further input.

Theorem 26.8.26

(Infinitude of primes in arithmetic progression over number fields). Let $\mathfrak n \subseteq \mathbb Z _F$ be a nonzero ideal, let $a \in (\mathbb Z _F/\mathfrak n )^\times$, and for each $v \mid \infty$ real let $\epsilon _v \in \{\pm 1\}$. Then there are infinitely many prime elements $p \in \mathbb Z _F$ such that ${{\,\mathrm{sgn}\,}}(v(p))=\epsilon _v$ and $p \equiv a \pmod \mathfrak{n }$.

Proof. The theorem generalizes Dirichlet’s theorem on the infinitude of primes in arithmetic progression (Theorem 14.2.9): see Lang [Lang94, Theorem VIII.4.10] or Neukirch [Neu99, Theorem VII.13.4]. $\square$

Proof of Main Theorem 14.6.1, F a number field. First, the map $B \mapsto {{\,\mathrm{Ram}\,}}B$ has the correct codomain by Hilbert reciprocity (Corollary 26.8.21). Surjectivity follows by Exercise 14.17 (using Theorem 26.8.26. To conclude, we show injectivity. We refer to Corollary 5.2.6, giving a bijection between quaternion algebras over F up to isomorphism and ternary quadratic forms of discriminant 1 up to isometry; and we recall Theorem 12.3.4, that (rescaling) there is a unique anisotropic ternary quadratic form of discriminant 1 up to isometry. Therefore Corollary 26.8.25 implies that the map $B \mapsto {{\,\mathrm{Ram}\,}}B$ is injective, since the set ${{\,\mathrm{Ram}\,}}B$ records those places v where the ternary quadratic form attached to B is anisotropic. $\square$

We will give another proof of Main Theorem 14.6.1 over global fields using the characterization of idelic norms in Proposition 27.5.15 (avoiding fiddling with quadratic forms and the use of primes in arithmetic progression).

Remark 26.8.27. For the readers who accept the fundamental exact sequence of class field theory as in Remark 14.6.10, the arguments above can be run in reverse, and the analytic statement in Theorem 26.8.19 can be deduced as a consequence.

Exercises

1.:

Prove Proposition 26.2.18 that

$$\begin{aligned} {{\,\mathrm{vol}\,}}(X_{\le 1})= \frac{2^r (2\pi )^c {Reg }_F}{w_F} \end{aligned}$$

in the special case of a real quadratic field.

$\triangleright$ 2.:

Let

$$\begin{aligned} \Gamma (s) :=\int _0^\infty x^s e^{-x} \frac{\mathrm{d }{x}}{x} \end{aligned}$$

be the complex $\Gamma$-function, defined for ${{\,\mathrm{Re}\,}}s>0$. Verify the following basic properties of $\Gamma (s)$.

(a):

$\Gamma (1)=1$ and $\Gamma (1/2)=\sqrt{\pi }$.

(b):

$\Gamma (s+1)=s\Gamma (s)$ for all ${{\,\mathrm{Re}\,}}s>0$, and $\Gamma (n)=(n-1)!$ for $n \ge 1$.

(c):

$\Gamma (s)$ has meromorphic continuation to $\mathbb C$, holomorphic away from simple poles at $\mathbb Z _{\le 0}$.

(d):

$\Gamma (s)$ has no zeros in $\mathbb C$.

$\triangleright$ 3.:

In this exercise, we prove Proposition 26.2.18. Let F be a number field with ring of integers R, let $X \subseteq F_\mathbb R$ be a cone fundamental domain for $R^\times$. Let ${Reg }_F$ be the regulator of F and $w_F$ the number of roots of unity in F.

(a):

Let $V=F_\mathbb R$ be the ambient space. Let $\mu (R^\times ) \le R^\times$ be the group generated by a fundamental system of units, so $R^\times /\mu (R^\times ) \simeq R_{tors }^\times$. Show that

$$\begin{aligned} {{\,\mathrm{vol}\,}}((V/R^\times )_{\le 1}) = \frac{2^c}{w_F} \int _{V_{\le 1}/\mu (R^\times )} \mathrm d {x}\,\mathrm d {z} \end{aligned}$$

with $x_i,z_j$ standard coordinates on $\mathbb R ^r \times \mathbb C ^c$ in multi-index notation.

(b):

Let $\rho _j,\theta _j$ be polar coordinates on $\mathbb C ^c$, and restrict the domain V to the domain $V^+$ with $x_i>0$ for all i. Let $W^+$ be the projection of $V^+$ onto the $x,\rho$-coordinate plane and let $x_{r+j}=\rho _{j}^2$. Show that

$$\begin{aligned} \int _{V_{\le 1}/\mu (R^\times )} \mathrm d {x}\,\mathrm d {z} = 2^r \pi ^c \int _{W^{+,\le 1}/\mu (R^\times )} \mathrm d {x}. \end{aligned}$$

(c):

Apply the change of variables $u_i=\log x_i$ to obtain

$$\begin{aligned} \int _{W^{+}_{\le 1}/\mu (R^\times )} \mathrm d {x} = \int _{P} \mathrm d {u} \end{aligned}$$

where P is the fundamental parallelogram for the additive (logarithmic) action of $R^\times$. Conclude that

$$\begin{aligned} {{\,\mathrm{vol}\,}}(X_{\le 1})= \frac{2^r (2\pi )^c {Reg }_F}{w_F}. \end{aligned}$$

$\triangleright$ 4.:

Show that the ideal J in Lemma 26.3.6 is unique: more specifically, show that if I is an invertible, integral lattice and suppose that ${{\,\mathrm{nrd}\,}}(I)=\mathfrak m \mathfrak n$ with $\mathfrak m ,\mathfrak n \subseteq R$ coprime ideals, then an invertible, integral lattice J such that I is compatible with $J^{-1}$ with $IJ^{-1}$ integral and ${{\,\mathrm{nrd}\,}}(J)=\mathfrak m$ is unique.

$\triangleright$ 5.:

Let F be a number field with ring of integers R, let B be a quaternion algebra over F, and let $\mathcal {O}\subseteq \mathcal {O}' \subseteq B$ be R-orders. For $I \subseteq \mathcal {O}$ a right $\mathcal {O}$-ideal, we define the right $\mathcal {O}'$-ideal $\rho (I)=I\mathcal {O}' \subseteq \mathcal {O}'$ obtained by extension.

(a):

Show that $\rho$ induces a (well-defined) surjective map

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

of pointed sets with finite fibers.

(b):

For the case where $\mathcal {O}$ is the Lipschitz order and $\mathcal {O}'$ the Hurwitz order, show that the map in (a) is a bijection (cf. Lemma 11.2.9).

$\triangleright$ 6.:

Let R be a DVR with uniformizer $\pi$ and let I be a (invertible) integral right ${{\,\mathrm{M}\,}}_2(R)$-ideal. Show that I is generated by

$$\begin{aligned} x=\begin{pmatrix} \pi ^u &{} 0 \\ c &{} \pi ^v \end{pmatrix} \end{aligned}$$

where $u,v \in \mathbb Z _{\ge 0}$ and $c \in R/\pi ^v$ are unique.

7.:

Generalize Exercise 11.13 as follows. For $n \in \mathbb Z$, let

$$\begin{aligned} r_4(n) :=\#\{(t,x,y,z) \in \mathbb Z ^4 : t^2+x^2+y^2+z^2=n\} \end{aligned}$$

and let $r_4'(n) :=r_4(n)/8$.

(a):

Show that $r_4'(2^e)=1$ for all $e \ge 1$ and $r_4'(p^e)=1+p+\dots +p^e$ for all $e \ge 1$ and p odd. [Hint: relate the count to the number of right ideals and inspect the coefficients of the zeta function.]

(b):

Show that $r_4'$ is a multiplicative function: $r_4'(mn)=r_4'(m)r_4'(n)$ if $\gcd (m,n)=1$.

(c):

Conclude that

$$ r_4(n) = {\left\{ \begin{array}{ll} 8\sum _{d \mid n} d, &{} n\text { odd;} \\ 24\sum _{d \mid m} d, &{} n=2^e m\text { even with }m\text { odd.} \end{array}\right. } $$

$\triangleright$ 8.:

Finish the proof of Lemma 26.6.7 by checking the remaining cases.

9.:

Prove (26.7.2). Specifically, for a number field F and coprime ideals $\mathfrak D ,\mathfrak M$ with $\mathfrak D$ squarefree and coprime to $\mathfrak M$, define themass

$$\begin{aligned} m(F,\mathfrak D ,\mathfrak M ) :=\frac{2\zeta _F(2)}{(2\pi )^{2n}}d_F^{3/2} h_F \varphi (\mathfrak D )\psi (\mathfrak M ). \end{aligned}$$

Let $\mathfrak N :=\mathfrak D \mathfrak M$. Show for fixed n that

$$\begin{aligned} \log m(F,\mathfrak D ,\mathfrak M ) \sim \frac{3}{2}\log d_F + \log h_F + \log {{\,\mathrm{Nm}\,}}(\mathfrak N ) \end{aligned}$$

(26.8.28)

as $d_F{{\,\mathrm{Nm}\,}}(\mathfrak N ) \rightarrow \infty$, as follows.

(a):

Show that

$$\begin{aligned} \zeta _\mathbb{Q }(2)^n=\prod _p \left( 1-\frac{1}{p^2}\right) ^{-n} \le \zeta _F(2) \le \prod _p \left( 1-\frac{1}{p^{2n}}\right) ^{-1} = \zeta _\mathbb{Q }(2n) \end{aligned}$$

so $\zeta _F(2) \asymp 1$.

(b):

Show that

$$\begin{aligned} \frac{{{\,\mathrm{Nm}\,}}(\mathfrak N )}{\log \log {{\,\mathrm{Nm}\,}}(\mathfrak D )} \ll \varphi (\mathfrak D )\psi (\mathfrak M ) \ll {{\,\mathrm{Nm}\,}}(\mathfrak N )\left( \log \log {{\,\mathrm{Nm}\,}}(\mathfrak M )\right) . \end{aligned}$$

[Hint: you may need some elementary estimates from analytic number theory, adapted for this purpose; you may wish to start with the case $F=\mathbb Q$.]

(c):

Conclude (26.8.28).

10.:

Prove Corollary 26.8.25: if $Q,Q'$ are quadratic forms over F in the same number of variables, then $Q \simeq Q'$ if and only if $Q_v \simeq Q_v'$ for all places $v \in {{\,\mathrm{Pl}\,}}F$. [Hint: see Corollary 14.3.7.]

$\triangleright$ 11.:

Use Dirichlet’s analytic class number formula to prove the theorem on arithmetic progressions (Theorem 14.2.9) as follows.

(a):

Let $F=\mathbb Q (\zeta _m)$. Show that

$$\begin{aligned} \zeta _F(s)=\zeta (s)\prod _{\chi \ne 1} L(s,\chi ) \end{aligned}$$

where $\chi$ runs over all nontrivial Dirichlet characters $\chi :(\mathbb Z /m\mathbb Z )^\times \rightarrow \mathbb C ^\times$, and

$$\begin{aligned} L(s,\chi )\sum _{n=1}^{\infty } \frac{\chi (n)}{n^s}. \end{aligned}$$

[Hint: Factor according to the decomposition $m=ref$.]

(b):

Use partial summation and the fact that the partial sums are bounded to show that each $L(s,\chi )$ for $\chi \ne 1$ is holomorphic at $s=1$.

(c):

Conclude from the analytic class number formula that $L(1,\chi ) \ne 0$ for $\chi \ne 1$.

(d):

For $\gcd (a,m)=1$, using (c) show that as $s \searrow 1$ that

$$\begin{aligned} \sum _{p \equiv a ~(\text{ mod } ~{m})} p^{-s} = \frac{\log \zeta (s)}{\varphi (m)} + O(1) \end{aligned}$$

and conclude that the set of primes p with $p \equiv a \pmod {m}$ is infinite.

$\triangleright$ 12.:

Let F be a nonarchimedean local field with valuation ring R having maximal ideal $\mathfrak p$ and residue field k of size $q :=\#k$. For $n \ge 1$, let $B_n :={{\,\mathrm{M}\,}}_n(F)$ and $\mathcal {O}_n :={{\,\mathrm{M}\,}}_n(R)$. In this exercise we generalize Lemma 26.4.1(b).

(a):

Show that the set of right integral $\mathcal {O}_n$-ideals is in bijection with the set

$$ \left\{ \begin{pmatrix} \pi ^{u_1} &{} 0 &{} 0 &{} \cdots &{} 0 \\ c_{21} &{} \pi ^{u_2} &{} 0 &{} \cdots &{} 0 \\ c_{31} &{} c_{32} &{} \pi ^{u_3} &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \ddots &{} \ddots &{} 0 \\ c_{n1} &{} c_{n2} &{} c_{n3} &{} \cdots &{} \pi ^{u_n} \end{pmatrix} : u_1,\dots ,u_n \in \mathbb Z _{\ge 0}, c_{ij} \in R/\mathfrak p ^i \right\} . $$

[Hint: appeal to the theory of elementary divisors, applying column operations acting on the right.]

(b):

Let $a_\mathfrak{p ^e}(\mathcal {O}_n)$ be the number of right ideals of reduced norm $\mathfrak p ^e$ in $\mathcal {O}$. Show that

$$\begin{aligned} a_\mathfrak{p ^e}(\mathcal {O}_n) = \sum _{f=0}^e a_\mathfrak{p ^f}(\mathcal {O}_{n-1}) q^{(n-1)(e-f)}. \end{aligned}$$

(c):

Let

$$\begin{aligned} \zeta _{\mathcal {O}_n}(s) :=\sum _{I \subseteq \mathcal {O}_n} \frac{1}{\mathsf{N }(I)}, \end{aligned}$$

the sum over nonzero right ideals of $\mathcal {O}_n$. Show that $\zeta _{\mathcal {O}_1}(s) = \zeta _F(s) = (1-q^{-s})^{-1}$.

(d):

Show for $n \ge 2$ that

$$\begin{aligned} \zeta _{\mathcal {O}_n}(s) = \sum _{e=0}^{\infty } \frac{a_\mathfrak{p ^e}(\mathcal {O}_n)}{q^{nes}} = \zeta _{\mathcal {O}_{n-1}}(ns/(n-1)) \zeta _{\mathcal {O}_1}(ns-(n-1)). \end{aligned}$$

(e):

Conclude that

$$\begin{aligned} \zeta _{\mathcal {O}_n}(s) = \zeta _F(ns)\zeta _F(ns-1)\cdots \zeta _F(ns-(n-1)) = \prod _{i=0}^{n-1} \zeta _F(ns-i) \end{aligned}$$

(cf. Corollary 26.4.7).