The Eichler mass formula

Abstract

In Part II of this text, we investigated arithmetic and algebraic properties of quaternion orders and ideals. Now in Part III, continuing this investigation we turn to the use of analytic methods. In this first introductory chapter, we restrict to a special case and consider zeta functions of quaternion orders over the rationals; as an application, we obtain a formula for the mass of the class set of a definite quaternion order.

In Part II of this text, we investigated arithmetic and algebraic properties of quaternion orders and ideals. Now in Part III, continuing this investigation we turn to the use of analytic methods. In this first introductory chapter, we restrict to a special case and consider zeta functions of quaternion orders over the rationals; as an application, we obtain a formula for the mass of the class set of a definite quaternion order.

1 $\triangleright$ Weighted class number formula

Gauss conjectured (in the language of binary quadratic forms) that there are finitely many imaginary quadratic orders of class number 1 [Gau86, Article 303]. Approaches to this problem involve beautiful and deep mathematics. Given that we want to prove some kind of lower bound for the class number (in terms of the discriminant of the order), it is natural to seek an analytic expression for it. The analytic class number formula of Dirichlet provides such an expression, turning the class number problem of Gauss into a (still very hard, but tractable) problem of estimation.

We recall section 14.1, which gives a classification of quaternion algebras over $\mathbb Q$, and sections 16.1 and 17.1, providing background on ideal classes in quaternion orders. With this motivation in hand, we are led to ask: what are the definite quaternion orders of class number 1? The method to prove Dirichlet’s formula generalizes to definite quaternion orders as well, as pursued by Eichler in his mass formula. This chapter gives an overview of the Eichler mass formula in the simplest case for a maximal order in a definite quaternion algebra over $\mathbb Q$. (The reader who is already motivated and ready for action may consider skipping to the next chapter.)

Theorem 25.1.1

(Eichler mass formula over $\mathbb Q$,maximal orders). Let B be a definite quaternion algebra over $\mathbb Q$ of discriminant D and let $\mathcal {O}\subset B$ be a maximal order. Then

$$\begin{aligned} \sum _{[J] \in {{\,\mathrm{Cls}\,}}\mathcal {O}} \frac{1}{w_J} = \frac{\varphi (D)}{12} \end{aligned}$$

where $w_J :=\#\mathcal {O}{}_{\textsf {\tiny {L}} }(J)^\times /\{\pm 1\}$ and $\varphi (D) :=\#(\mathbb Z /D\mathbb Z )^\times =\prod _{p \mid D}(p-1)$ is the Euler totient function.

The Eichler mass formula does not quite give us a formula for the class number—rather, it gives us a formula for a “weighted” class number. That being said, we remark that $w_J \le 24$ (see Theorem 11.5.14), and very often $w_J=1$ (i.e., $\mathcal {O}{}_{\textsf {\tiny {L}} }(J)=\{\pm 1\}$. In order to convert the Eichler mass formula into a formula for the class number itself, one needs to understand the unit groups of left orders: this can be understood either as a problem in representation numbers of ternary quadratic forms or of embedding numbers of quadratic orders into quaternion orders, and we will take this subject up in earnest in Chapter 30.

Over $\mathbb Q$, the Eichler mass formula was first proven by Hey [Hey29, II, (80)], a Ph.D. student of Artin, along the same lines as the proof sketched below. This formula was also stated by Brandt [Bra28, §67]. We gradually warm up to this theorem by considering a broader analytic context. We see the analytic class number for an imaginary quadratic field as coming from the residue of its zeta function, and we then pursue a quaternionic generalization.

2 $\triangleright$ Imaginary quadratic class number formula

To introduce the circle of ideas, let $K:=\mathbb Q (\sqrt{d})$ be a quadratic field of discriminant $d \in \mathbb Z$ and let $\mathbb Z _K$ be its ring of integers. We encode information about the field K by its zeta function.

25.2.1

Over $\mathbb Q$, we define the Riemann zeta function

$$\begin{aligned} \zeta (s) :=\sum _{n=1}^{\infty } \frac{1}{n^s} \end{aligned}$$

(25.2.2)

as the prototypical such function; this series converges for ${{\,\mathrm{Re}\,}}s>1$, by the comparison test. By unique factorization, there is an Euler product

$$\begin{aligned} \zeta (s) = \prod _p \left( 1-\frac{1}{p^s}\right) ^{-1} \end{aligned}$$

(25.2.3)

where the product is over all primes p. The function $\zeta (s)$ can be meromorphically continued to the right half-plane ${{\,\mathrm{Re}\,}}s>0$ using the fact that the sum

$$\begin{aligned} \zeta _2(s) = \sum _{n=1}^{\infty } \frac{(-1)^n}{n^s} \end{aligned}$$

converges for ${{\,\mathrm{Re}\,}}s>0$ and

$$\begin{aligned} \zeta (s) + \zeta _2(s) = 2^{1-s} \zeta (s) \end{aligned}$$

so that

$$\begin{aligned} \zeta (s)=\frac{1}{2^{1-s} -1} \zeta _2(s) \end{aligned}$$

and the right-hand side makes sense for ${{\,\mathrm{Re}\,}}s>0$ except for possible poles where $2^{1-s}=1$. For real values of $s>1$, we have

$$\begin{aligned} \frac{1}{s-1}=\int _1^{\infty } \frac{\mathrm{d {x}}}{x^s} \le \zeta (s) \le 1+ \int _1^{\infty } \frac{\mathrm{d {x}}}{x^s} = \frac{s}{s-1} \end{aligned}$$

so

$$\begin{aligned} 1 \le (s-1)\zeta (s) \le s; \end{aligned}$$

therefore, as s approaches 1 from above, we have $\lim _{s \searrow 1} (s-1) \zeta (s)=1$, so $\zeta (s)$ has a simple pole at $s=1$ with residue

$$\begin{aligned} {{\,\mathrm{res}\,}}_{s=1} \zeta (s) = 1. \end{aligned}$$

(25.2.4)

25.2.5

For the quadratic field K, modeled after (25.2.2) we define the Dedekind zeta function by

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

(25.2.6)

where $\mathsf N (\mathfrak a ) :=\#(\mathbb Z _K/\mathfrak a )$ is the absolute norm, the sum is over all nonzero ideals of $\mathbb Z _K$, and the series is defined for $s \in \mathbb C$ with ${{\,\mathrm{Re}\,}}s>1$. (We recall that $\mathsf N (\mathfrak a )$ is the positive generator of ${{\,\mathrm{Nm}\,}}_{K|\mathbb Q }(\mathfrak a )$, so we could equivalently work with the algebra norm, if desired.)

We can also write the Dedekind zeta function as a Dirichlet series

$$\begin{aligned} \zeta _K(s) = \sum _{n=1}^{\infty } \frac{a_n}{n^s} \end{aligned}$$

(25.2.7)

where $a_n :=\#\{\mathfrak{a } \subseteq \mathbb Z _K : \mathsf{N }(\mathfrak{a })=n\}$ is the number of ideals in $\mathbb Z _K$ of norm $n \ge 1$. By unique factorization of ideals, we again have an Euler product expansion

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

(25.2.8)

the product over all nonzero prime ideals $\mathfrak p \subset \mathbb Z _K$.

In order to introduce a formula that involves the class number, we group the ideals in (25.2.6) by their ideal class: for $[\mathfrak b ] \in {{\,\mathrm{Cl}\,}}(K)$, we define the partial zeta function

so that

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

(25.2.9)

In general, for $[\mathfrak b ] \in {{\,\mathrm{Cl}\,}}(K)$, we have $[\mathfrak a ]=[\mathfrak b ]$ if and only if there exists $a \in K^\times$ such that $\mathfrak a =a\mathfrak b$, but since $\mathfrak a \subseteq \mathbb Z _K$, in fact

$$\begin{aligned} a \in \mathfrak b ^{-1} = \{a \in \mathbb Z _K : a\mathfrak b \subseteq \mathbb Z _K\}; \end{aligned}$$

this gives a bijection

$$\begin{aligned} \{\mathfrak{a } \subseteq \mathbb Z _K : [\mathfrak{a }]=[\mathfrak{b }]\} \leftrightarrow \mathfrak{b }^{-1}/\mathbb Z _K^\times , \end{aligned}$$

(since the generator of an ideal is unique up to units). Thus

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

(25.2.10)

for each class $[\mathfrak b ] \in {{\,\mathrm{Cl}\,}}(K)$.

Everything we have done so far works equally as well for real as for imaginary quadratic fields. But to make sense of $\mathfrak b ^{-1}/\mathbb Z _K^\times$ in the simplest case, we want $\mathbb Z _K^\times$ to be a finite group, which by Dirichlet’s unit theorem means exactly that K is $\mathbb Q$ or an imaginary quadratic field. So from now on in this section, we suppose $K=\mathbb Q (\sqrt{d})$ with $d<0$. Then $w :=\#\mathbb Z _K^\times = 2$, except when $d=-3,-4$ where $w=6,4$, respectively.

Under this hypothesis, the sum (25.2.10) can be transformed into a sum over lattice points with the fixed factor w of overcounting. Before estimating the sum over reciprocal norms, we first estimate the count. Let $\Lambda \subset \mathbb C$ be a lattice. We can estimate the number of lattice points $\lambda \in \Lambda$ with $|\lambda \,|\le x$ by the ratio $\pi x^2/A$, where A is the area of a fundamental parallelogram P for $\Lambda$: roughly speaking, this says that we can tile a circle of radius x with approximately $\pi x^2/A$ parallelograms P.

More precisely, the following lemma holds.

Lemma 25.2.11

Let $\Lambda \subset \mathbb C$ be a lattice with ${{\,\mathrm{area}\,}}(\mathbb C /\Lambda )=A$. Then there is a constant C such that for all $x > 1$,

$$\begin{aligned} \left|\#\{\lambda \in \Lambda : |\lambda |\le x\} - \frac{\pi x^2}{A}\right|\le Cx. \end{aligned}$$

We leave this lemma as an exercise (Exercise 25.3) in tiling a circle with radius x with fundamental parallelograms for the lattice $\Lambda$. With a bit of manipulation (Exercise 25.4), this lemma can be used to prove the analytic class number formula.

Theorem 25.2.12

(Analytic class number formula, imaginary quadratic field). Let $K=\mathbb Q (\sqrt{d})$ be an imaginary quadratic field with discriminant $d<0$. Then

$$\begin{aligned} {{\,\mathrm{res}\,}}_{s=1}\zeta _K(s) = \frac{2\pi h}{w\sqrt{|d \,|}} \end{aligned}$$

where h is the class number of K and w is the number of roots of unity in K.

This formula simplifies slightly if we cancel the pole at $s=1$ with $\zeta (s)$, as follows. Like in the Dirichlet series, we can combine terms in (25.2.8) to get

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

where

$$\begin{aligned} L_p(T) :={\left\{ \begin{array}{ll} (1-T)^{2}, &{}\text { if }p\text { splits in }K; \\ 1-T, &{}\text { if }p\text { ramifies in }K;\text { and} \\ 1-T^2, &{}\text { if }p\text { is inert in }{} { K}. \end{array}\right. } \end{aligned}$$

(25.2.13)

The condition of being split, ramified, or inert in K is recorded in a function:

$$\begin{aligned} \chi (p) :=\chi _d(p)= {\left\{ \begin{array}{ll} 1, &{} \text { if }p\text { splits in }K; \\ 0, &{} \text { if }p\text { ramifies in }K;\text { and} \\ -1, &{} \text { if }p\text { is inert in }K \end{array}\right. } \end{aligned}$$

(25.2.14)

for prime p and extended to all positive integers by multiplicativity. If $p \not \mid d$ is an odd prime, then

$$\begin{aligned} \chi (p) = \biggl (\displaystyle {\frac{d}{p}}\biggr ) \end{aligned}$$

is the usual Legendre symbol, equal to 1 or $-1$ according as if d is a quadratic residue or not modulo p. Then in all cases, we have

$$\begin{aligned} L_p(T) = (1-T)(1-\chi (p)T). \end{aligned}$$

Expanding the Euler product term-by-term and taking a limit, we conclude

$$\begin{aligned} \zeta _K(s) = \zeta (s) L(s,\chi ) \end{aligned}$$

(25.2.15)

where

$$\begin{aligned} L(s,\chi ) :=\prod _p \left( 1-\frac{\chi (p)}{p^s}\right) ^{-1} = \sum _n \frac{\chi (n)}{n^s}. \end{aligned}$$

(25.2.16)

The function $L(s,\chi )$ is in fact holomorphic for all ${{\,\mathrm{Re}\,}}s>0$; this follows from the fact that the partial sums $\sum _{n \le x} \chi (n)$ are bounded and the mean value theorem. So in particular the series

$$\begin{aligned} L(1,\chi ) = 1 + \frac{\chi (2)}{2} + \frac{\chi (3)}{3} + \frac{\chi (4)}{4} + \dots \end{aligned}$$

converges (slowly). Combining (25.2.15) with the analytic class number formula yields:

$$\begin{aligned} L(1,\chi ) = \frac{2\pi h}{w\sqrt{|d \,|}} \ne 0. \end{aligned}$$

(25.2.17)

For example, taking $d=-4$, so $\chi (2)=0$ and $\chi (p)=(-1/p)=(-1)^{(p-1)/2}$,

$$\begin{aligned} L(1,\chi )&= 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} - \frac{1}{11} + \dots \\&= \prod _{p \ge 3} \left( 1-\frac{(-1)^{(p-1)/2}}{p}\right) ^{-1} = \frac{\pi }{4} = 0.7853\ldots . \end{aligned}$$

Remark 25.2.18. The fact that $L(1,\chi ) \ne 0$, and its generalization to complex characters $\chi$, is the key ingredient to prove Dirichlet’s theorem on primes in arithmetic progression (Theorem 14.2.9), used in the classification of quaternion algebras over $\mathbb Q$. The arguments to complete the proof are requested in Exercise 26.11.

Remark 25.2.19. To approach the class number problem of Gauss, we would then seek lower bounds on $L(1,\chi )$ in terms of the absolute discriminant $|d \,|$. Indeed, the history of class number problems is both long and beautiful. The problem of determining all positive definite binary quadratic forms with small class number was first posed by Gauss [Gau86, Article 303]. This problem was later seen to be equivalent to finding all imaginary quadratic fields of small class number (as in section 19.1). It would take almost 150 years of work, with important work of Heegner [Heeg52] and culminating in the results of Stark [Sta67] and Baker [Bak71], to determine those fields with class number 1: there are exactly nine, namely $d=-3,-4,-7,-8,-11,-19,-43,-67,-163$. See Goldfeld [Gol85] or Stark [Sta2007] for a history of this problem. For more specifically on the analytic class number formula for imaginary quadratic fields, see the survey by Weston [Wes] as well as the book by Serre [Ser73, Chapter VI].

3 $\triangleright$ Eichler mass formula: over the rationals

We are now prepared to consider the analogue of the analytic class number formula (Theorem 25.2.12) for quaternion orders: the Eichler mass formula, which is a weighted class number formula. We follow Eichler [Eic55-56, Eic56a], and in this section we give an overview with proofs omitted—a full development will be given in the next chapter, in more generality.

Let B be a quaternion algebra over $\mathbb Q$ of discriminant D and let $\mathcal {O}\subset B$ be an order. We define the zeta function  of $\mathcal {O}$ to be

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

(25.3.1)

where the sum over all invertible (nonzero, integral) right $\mathcal {O}$-ideals and

$$\begin{aligned} \mathsf{N }(I) :=[\mathcal {O}:I] = \#(\mathcal {O}/I) \in \mathbb Z _{>0}. \end{aligned}$$

(By Main Theorem 16.1.3, we have $\mathsf N (I)$ the totally positive generator of ${{\,\mathrm{nrd}\,}}(I)^2$, so we could equivalently work with the reduced norm.)

Let $a_n$ be the number of invertible right $\mathcal {O}$-ideals of reduced norm $n>0$ (with positive generator chosen, as usual). Then $\mathsf{N }(I)={{\,\mathrm{Nm}\,}}(I)={{\,\mathrm{nrd}\,}}(I)^2$ by 16.4.10, so

$$\begin{aligned} \zeta _{\mathcal {O}}(s) = \sum _{n=1}^{\infty } \frac{a_n}{n^{2s}}. \end{aligned}$$

(25.3.2)

To establish an Euler product for $\zeta _{\mathcal {O}}(s)$, in due course we will give a kind of factorization formula for right ideals of $\mathcal {O}$—but by necessity, writing an ideal as a compatible product will involve the entire set of orders connected to $\mathcal {O}$! A direct consequence of the local-global dictionary for lattices (Theorem 9.4.9) is that

$$\begin{aligned} a_{mn}=a_m a_n \end{aligned}$$

(25.3.3)

whenever m, n are coprime. Next, we will count the ideals of a given reduced norm $q=p^e$ a power of a prime: the answer will depend on the local structure of the order $\mathcal {O}_p$. Indeed, $\zeta _{\mathcal {O}}(s)$ has an Euler product

$$\begin{aligned} \zeta _{\mathcal {O}}(s) = \prod _p \zeta _{\mathcal {O},p}(p^{-s})^{-1} \end{aligned}$$

(25.3.4)

with $\zeta _{\mathcal {O},p}(T) \in 1+T\mathbb Z [T]$. In particular, $\zeta _{\mathcal {O}}(s)$ only depends on the genus (local isomorphism classes) of $\mathcal {O}$.

For simplicity, we first consider the case where $\mathcal {O}$ is a maximal order. Since there is a unique genus of maximal orders, the zeta function is independent of the choice of $\mathcal {O}$ and so we will write $\zeta _B(s) :=\zeta _\mathcal {O}(s)$ for $\mathcal {O}$ maximal. Then by a local count, we will show that

$$\begin{aligned} \zeta _{B,p}(T) = (1-T^2) \cdot {\left\{ \begin{array}{ll} 1, &{}\text { if }p \mid D; \\ 1-pT^2, &{} \text { if }p \not \mid D. \end{array}\right. } \end{aligned}$$

(25.3.5)

From (25.3.5),

$$\begin{aligned} \zeta _{B}(s) = \zeta (2s)\zeta (2s-1) \prod _{p \mid D} \left( 1-\frac{1}{p^{2s-1}}\right) . \end{aligned}$$

(25.3.6)

In particular, since $\zeta (s)$ has a simple pole at $s=1$ with residue 1 and $\zeta (2)=\pi ^2/6$ (Exercise 25.1),

$$\begin{aligned} {{\,\mathrm{res}\,}}_{s=1} \zeta _{B}(s) = \lim _{s \searrow 1} (s-1) \zeta _{B}(s) = \frac{\pi ^2}{12} \prod _{p \mid D} \left( 1-\frac{1}{p}\right) \end{aligned}$$

(25.3.7)

(We could also look to cancel the poles of $\zeta _{B}(s)$ in a similar way to define an L-function for B, holomorphic for ${{\,\mathrm{Re}\,}}s>0$.)

Now we break up the sum (25.3.1) according to right ideal class:

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

where

(25.3.8)

Since $[I]=[J]$ if and only if $I=\alpha J$ for some invertible $\alpha \in J^{-1}$, and $\mu J = J$ if and only if $\mu \in \mathcal {O}{}_{\textsf {\tiny {L}} }(J)^\times$, we conclude that

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

(25.3.9)

where the sum is taken over the nonzero elements $\alpha \in J^{-1}$ up to right multiplication by units $\mathcal {O}{}_{\textsf {\tiny {L}} }(J)^\times$ in the left order.

In order to proceed, we now suppose that B is definite (ramified at $\infty$) and hence that $\#\mathcal {O}{}_{\textsf {\tiny {L}} }(J)^\times <\infty$ (see Lemma 17.7.13); this is the analogue with the case of an imaginary quadratic field, and each J has the structure of a lattice in the Euclidean space $\mathbb R ^4$ via the embedding

$$\begin{aligned} J \hookrightarrow B \hookrightarrow B \otimes _\mathbb Q \mathbb R \simeq \mathbb H \simeq \mathbb R ^4. \end{aligned}$$

(25.3.10)

Let $w_J=\#\mathcal {O}{}_{\textsf {\tiny {L}} }(J)^\times /\{\pm 1\}$. We again argue by counting lattice points to prove the following proposition.

Proposition 25.3.11

The function $\zeta _{B,[J]}(s)$ has a simple pole at $s=1$ with residue

$$\begin{aligned} {{\,\mathrm{res}\,}}_{s=1} \zeta _{B,[J]}(s) = \frac{\pi ^2}{w_J D}. \end{aligned}$$

Proof sketch. From a more general result (Theorem 26.2.12, proven in the next section and used to prove the analytic class number formula itself), we will show that

$$\begin{aligned} {{\,\mathrm{res}\,}}_{s=1} \zeta _{B,[J]}(s) = \frac{1}{2w_J{\mathsf{N }}(J)}\frac{{{\,\mathrm{vol}\,}}((\mathbb R ^4)_{\le 1})}{{{\,\mathrm{covol}\,}}(J)} \end{aligned}$$

(25.3.12)

where under (25.3.10) we have

$$\begin{aligned} {{\,\mathrm{vol}\,}}((\mathbb R ^4)_{\le 1})={{\,\mathrm{vol}\,}}(\{x \in \mathbb R ^4 : |x \,| \le 1\}) = \frac{\pi ^2}{2} \end{aligned}$$

and

$$\begin{aligned} {{\,\mathrm{covol}\,}}(J)=\frac{{{\,\mathrm{covol}\,}}(\mathcal {O})}{\mathsf{N }(J)}=\frac{D/4}{\mathsf{N }(J)}. \end{aligned}$$

Putting all of these facts together,

$$\begin{aligned} {{\,\mathrm{res}\,}}_{s=1} \zeta _{B,[J]}(s) = \frac{\pi ^2}{4w_J\mathsf{N }(J)} \frac{4\mathsf{N }(J)}{D} = \frac{\pi ^2}{w_J D}. \qquad \square \end{aligned}$$

(25.3.13)

In particular the pole of each zeta function $\zeta _{B,[J]}(s)$ is almost independent of the class [J], with the only relevant term being $w_J$ the number of units.

Combining Proposition 25.3.13 with Proposition 25.3.11,

$$\begin{aligned} {{\,\mathrm{res}\,}}_{s=1} \zeta _{B}(s) = \frac{\pi ^2}{D} \sum _{[J] \in {{\,\mathrm{Cls}\,}}\mathcal {O}} \frac{1}{w_J} = \frac{\pi ^2}{12} \prod _{p \mid D} \left( 1-\frac{1}{p}\right) \end{aligned}$$

(25.3.14)

and we conclude the following theorem.

Theorem 25.3.15

(Eichler mass formula, maximal orders). Let B be a definite quaternion algebra over $\mathbb Q$ of discriminant D and let $\mathcal {O}\subset B$ be a maximal order. Then

$$\begin{aligned} \sum _{[J] \in {{\,\mathrm{Cls}\,}}\mathcal {O}} \frac{1}{w_J} = \frac{\varphi (D)}{12}. \end{aligned}$$

(25.3.16)

Remark 25.3.17. The Eichler mass formula is also very similar to the mass formula for the number of isomorphism classes of supersingular elliptic curves: this is no coincidence, and its origins will be explored in section 42.2.

To extend the Eichler mass formula to a more general class of orders, one only needs to replace the local calculation in 25.3.5 by a count of invertible ideals in the order. First we treat the important case of Eichler orders (see 23.1.3).

Theorem 25.3.18

(Eichler mass formula, Eichler orders over $\mathbb Q$). Let $\mathcal {O}\subset B$ be an Eichler order of level M in a definite quaternion algebra B of discriminant D. Then

$$\begin{aligned} \sum _{[J] \in {{\,\mathrm{Cls}\,}}\mathcal {O}} \frac{1}{w_J} = \frac{\varphi (D)\psi (M)}{12} \end{aligned}$$

where

$$\begin{aligned} \psi (M)=\prod _{p^e \parallel M}(p^e+p^{e-1}) = M \prod _{p \mid M}\left( 1+\frac{1}{p}\right) . \end{aligned}$$

The most general formula is written in terms of the Eichler symbol 24.3. Just to recall two important cases: if $p \mid N={{\,\mathrm{discrd}\,}}(\mathcal {O})$, then $\biggl (\displaystyle {\frac{\mathcal {O}}{p}}\biggr )=-1$ if $\mathcal {O}_p$ is the maximal order in the division algebra $B_p$ and $\biggl (\displaystyle {\frac{\mathcal {O}}{p}}\biggr )=1$ if $\mathcal {O}_p$ is an Eichler order.

Main Theorem 25.3.19

(Eichler mass formula, general case over $\mathbb Q$). Let B be a definite quaternion algebra over $\mathbb Q$ and $\mathcal {O}\subset B$ be an order with ${{\,\mathrm{discrd}\,}}(\mathcal {O})=N$. Then

$$\begin{aligned} \sum _{[J] \in {{\,\mathrm{Cls}\,}}\mathcal {O}} \frac{1}{w_J} = \frac{N}{12} \prod _{p \mid N} \lambda (\mathcal {O},p) \end{aligned}$$

where

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

(25.3.20)

Main Theorem 25.3.19 was proven by Brzezinski [Brz90, (4.6)] and more generally over number rings by Körner [Kör87, Theorem 1].

4 Class number one and type number one

The Eichler mass formula can be used to solve the class number 1 problem for definite quaternion orders over $\mathbb Z$, and in fact it is much easier than for imaginary quadratic fields! We begin with the case of maximal orders.

Theorem 25.4.1

Let $\mathcal {O}$ be a maximal order in a definite quaternion algebra over $\mathbb Q$ of discriminant D. Then $\#{{\,\mathrm{Cls}\,}}\mathcal {O}=1$ if and only if $D=2,3,5,7,13$.

Proof A calculation by hand; see Exercise 25.5. $\square$

Remark 25.4.2. The primes $p=2,3,5,7,13$ in Theorem 25.4.1 are also the primes p such that the modular curve $X_0(p)$ has genus 0. This is not a coincidence, and reflects a deep correspondence between classical and quaternionic modular forms (the Eichler–Shimizu–Jacquet–Langlands correspondence): see Remark 41.5.13.

The list of all definite quaternion orders (over $\mathbb Z$) of class number 1 was determined by Brzezinski [Brz95]. (Brzezinski mistakenly lists an order of class number 2, and so he counts 25, not 24; he corrects this later in a footnote [Brz98, Footnote 1].)

Theorem 25.4.3

(Brzezinski). There are exactly 24 isomorphism classes of definite quaternion orders over $\mathbb Z$ with $\#{{\,\mathrm{Cls}\,}}\mathcal {O}=1$.

The list of orders with class number 1 is given in Table 25.4.4. We provide $N :={{\,\mathrm{discrd}\,}}(\mathcal {O})$, $D :={{\,\mathrm{disc}\,}}(B)$, we list the Eichler symbols $({\mathcal {O}} \mid {p})$ for the relevant primes $p \le 13$, we say if the order is maximal, hereditary (but not maximal), Eichler/residually split or residually inert (but not hereditary), Bass (but not Eichler), or non-Gorenstein. This list includes the three norm Euclidean maximal orders (Exercise 25.6) of discriminant $D=2,3,5$.

Table 25.4.4: Definite quaternion orders over $\mathbb {Z}$ with class number 1

Proof of Theorem 25.4.3

Suppose $\#{{\,\mathrm{Cls}\,}}\mathcal {O}=1$. We apply the mass formula (Main Theorem 25.3.19). We note that $\lambda (\mathcal {O},p) \ge 1-1/p$, in all three cases, so

$$\begin{aligned} 1 \ge \frac{1}{w}=\frac{N}{12}\prod _{p \mid N}\lambda (\mathcal {O},p) \ge \frac{N}{12} \prod _{p \mid N}\left( 1-\frac{1}{p}\right) = \frac{\varphi (N)}{12} \end{aligned}$$

(25.4.5)

and therefore $\varphi (N) \le 12$. By elementary number theory, this implies that

$$\begin{aligned} 2 \le N \le 16 \text { or } N=18,20,21,22,24,26,28,30,36,42. \end{aligned}$$

This immediately gives a finite list of possibilities for the discriminant $D={{\,\mathrm{disc}\,}}B \in \{2,3,5,7,11,13,30,42\}$, as D must be a squarefree product of an odd number of primes.

By Exercise 17.3, if $\mathcal {O}\subseteq \mathcal {O}'$ then there is a natural surjection ${{\,\mathrm{Cls}\,}}\mathcal {O}\rightarrow {{\,\mathrm{Cls}\,}}\mathcal {O}'$, which is to say an order has at least as large a class number as any superorder. So we must have $\#{{\,\mathrm{Cls}\,}}\mathcal {O}'=1$ for a maximal order $\mathcal {O}' \subseteq B$, and by Theorem 25.4.1, this then reduces us to $D=2,3,5,7,13$. Because $\#{{\,\mathrm{Cls}\,}}\mathcal {O}'=1$, the maximal order $\mathcal {O}'$ is unique up to conjugation, and so fixing a choice of such a maximal order $\mathcal {O}'$ up to isomorphism we may suppose $\mathcal {O}\subseteq \mathcal {O}'$. But now the index $[\mathcal {O}':\mathcal {O}] = M=D/N$ is explicitly given, and there are only finitely many suborders of bounded index; for each, we may compute representatives of the class set in a manner similar to the example in section 17.6. (We are aided further by deciding what assignment of Eichler symbols and unit orders would be necessary in each case.) $\square$

There is similarly an interest in definite quaternion orders $\mathcal {O}$ of type number 1: these are the orders with the property that the “local-to-global principle applies for isomorphisms”, i.e., if $\mathcal {O}'_\mathfrak p \simeq \mathcal {O}_\mathfrak p$ for all primes $\mathfrak p$ then $\mathcal {O}' \simeq \mathcal {O}$. If an order has class number 1 then it has type number 1, by Lemma 17.4.13, but one may have $\#{{\,\mathrm{Typ}\,}}\mathcal {O}=1$ but $\#{{\,\mathrm{Cls}\,}}\mathcal {O}> 1$. Since an order has the same type number as its Gorenstein saturation, i.e. ${{\,\mathrm{Typ}\,}}\mathcal {O}= {{\,\mathrm{Typ}\,}}{{\,\mathrm{Gor}\,}}(\mathcal {O})$, it suffices to classify the Gorenstein orders with this property.

By the bijection between ternary forms and quaternion orders, this is equivalently the problem of enumerating one-class genera of primitive ternary quadratic forms. The list was drawn up by Jagy–Kaplansky–Schiemann [JKS97] (with early work due to Watson [Wats75]), and has been independently confirmed by Lorch–Kirschmer [LK2013].

Theorem 25.4.6

(Watson, Jagy–Kaplansky–Schiemann, Lorch–Kirschmer). There are exactly 794 primitive ternary quadratic forms of class number 1, corresponding to 794 Gorenstein quaternion orders of type number 1. The largest prime dividing a discriminant is 23, and the largest (reduced) discriminant is $2^83^37^2=338688$. There are exactly 9 corresponding to maximal quaternion orders: they have reduced discriminants

$$\begin{aligned} D = 2, 3, 5, 7, 13, 30, 42, 70, 78. \end{aligned}$$

Remark 25.4.7. The generalization of the class number 1 problem to quadratic forms of more variables was pursued by Watson, who showed that one-class genera do not exist in more than ten variables [Wats62]. Watson also tried to compile complete lists in low dimensions, followed by work of Hanke, and recently the complete list has been drawn up in at least 3 variables over $\mathbb Q$ by Lorch–Kirschmer [LK2013] and over totally real fields for maximal lattices by Kirschmer [Kir2014].

Exercises

• 1. A short and fun proof of the equality $\zeta (2)=\pi ^2/6$ is due to Calabi [BCK93].

  • (a) Expand $(1-x^2y^2)^{-1}$ in a geometric series and integrate termwise over $S=[0,1] \times [0,1]$ to obtain

    $$\begin{aligned} \int \!\!\!\int _S (1-x^2y^2)^{-1}\,\mathrm d {x}\,\mathrm d {y} = 1 + \frac{1}{3^2} + \frac{1}{5^2} + \ldots = \left( 1-\frac{1}{4}\right) \zeta (2). \end{aligned}$$

  • (b) Show that the substitution

    $$\begin{aligned} (x,y) :=\left( \frac{\sin u}{\cos v},\frac{\sin v}{\cos u}\right) \end{aligned}$$

    has Jacobian $1-x^2y^2$ and maps the open triangle

    $$\begin{aligned} T :=\{(u,v) : u,v > 0\text { and }u+v<\pi /2\} \end{aligned}$$

    bijectively to the interior of S.

  • (c) Conclude that

    $$\begin{aligned} \int \!\!\!\int _S (1-x^2y^2)^{-1}\,\mathrm d {x}\,\mathrm d {y} = \int \!\!\!\int _T \mathrm d {u}\,\mathrm d {v} = \frac{\pi ^2}{8} \end{aligned}$$

    and thus $\zeta (2)=\pi ^2/6$.

• 2. In this exercise, we give a very jazzy proof that $\zeta (k) \in \mathbb Q \pi ^{k}$ for all $k \in 2\mathbb Z _{\ge 1}$, due to Zagier [Zag94, p. 498].

  • (a) We start with $k=4$. Define

    $$\begin{aligned} f(m,n) :=\frac{1}{mn^3} + \frac{1}{2m^2n^2} + \frac{1}{m^3n} \end{aligned}$$

    for $m,n \in \mathbb Z _{>0}$, and prove that

    $$\begin{aligned} f(m,n) - f(m+n,n) - f(m,m+n) = \frac{1}{m^2n^2}. \end{aligned}$$

  • (b) Prove

    $$\begin{aligned} \zeta (2)^2 = \sum _{n=1}^{\infty } f(n,n) = \frac{5}{2}\zeta (4). \end{aligned}$$

    Conclude that $\zeta (4)=\pi ^4/90$ using Exercise 25.1.

  • (c) In general, for $k \in 2\mathbb Z _{\ge 2}$, let

    $$\begin{aligned} f(m,n)=\frac{1}{mn^{k-1}} + \frac{1}{2}\sum _{r=2}^{k-2} \frac{1}{m^r n^{k-r}} + \frac{1}{m^{k-1}n} \end{aligned}$$

    and check that

    $$\begin{aligned} f(m,n)-f(m+n,n)-f(m,m+n)=\sum _{\begin{array}{c} 0<j<k \\ j\text { even} \end{array}} \frac{1}{m^j n^{k-j}}. \end{aligned}$$

    Conclude in a similar way as in (b) that

    $$\begin{aligned} \sum _{\begin{array}{c} 0<j<k \\ j\text { even} \end{array}} \zeta (j)\zeta (k-j) = \frac{k+1}{2}\zeta (k) \end{aligned}$$

    so by induction $\zeta (k) \in \mathbb Q \pi ^k$.

$\triangleright$ 3.:

Prove Lemma 25.2.11 as follows.

(a):

Let P be a fundamental parallelogram for $\Lambda$, and for $\lambda \in \Lambda$ let $P_\lambda :=P+\lambda$. For $x > 1$, let

$$\begin{aligned} D(x) :=\{z \in \mathbb C : |z \,| \le x\} \end{aligned}$$

and

$$\begin{aligned} N(x)&:=\#\{\lambda \in \Lambda : \lambda \in D(x)\} \\ N_P(x)&:=\#\{\lambda \in \Lambda : P_\lambda \subseteq D(x)\} \\ N_P^+(x)&:=\#\{\lambda \in \Lambda : P_\lambda \cap D(x) \ne \emptyset \}. \end{aligned}$$

Show that

$$\begin{aligned} N_P(x) \le N(x) \le N_P^+(x). \end{aligned}$$

(b):

Show that $N_P(x) \le \pi x^2/A \le N_P^+(x)$.

(c):

Let l be the length of a long diagonal in P. Show that for all $\lambda \in \Lambda \cap D(x)$, we have $P_\lambda \subseteq D(x+l)$, so

$$\begin{aligned} N(x) \le N_P(x+l) \le \frac{\pi (x+l)^2}{A}. \end{aligned}$$

Similarly, show that if $P_\lambda \cap D(x-l) \ne \emptyset$ then $P_\lambda \subseteq D(x)$ and $\lambda \in D(x)$, so

$$\begin{aligned} \frac{\pi (x-l)^2}{A} \le N_P^+(x-l) \le N(x). \end{aligned}$$

(d):

Conclude that Lemma 25.2.11 holds with $C :=\pi (2l+l^2)/A$.

$\triangleright$ 4.:

Using the previous exercise, we now prove the analytic class number formula (Theorem 25.2.12).

(a):

Let $\mathfrak b \subset \mathbb C$ be a fractional ideal, and let

$$\begin{aligned} b_n :=\#\{a \in \mathfrak b ^{-1} : {{\,\mathrm{Nm}\,}}(a)=n\}. \end{aligned}$$

Show that

$$\begin{aligned} \left|\sum _{n \le x} b_n - \frac{\pi x}{A}\right|\le C\sqrt{x} \end{aligned}$$

for a constant C that does not depend on x and

$$\begin{aligned} A={{\,\mathrm{Nm}\,}}(\mathfrak b ^{-1})\frac{\sqrt{|d \,|}}{2}. \end{aligned}$$

[Hint: Apply Lemma 25.2.11 to $\mathfrak b ^{-1}$.]

(b):

Consider the Dirichlet series

$$\begin{aligned} f(s) :=\frac{1}{wN(\mathfrak b )^s} \sum _{n=1}^{\infty } \left( b_n - \frac{\pi }{A}\right) \frac{1}{n^s}. \end{aligned}$$

Show by the comparison test that f(s) converges for all ${{\,\mathrm{Re}\,}}s > 1/2$.

(c):

Show that

$$\begin{aligned} {{\,\mathrm{res}\,}}_{s=1}\zeta _K(s) = \lim _{s \searrow 1} (s-1)\zeta _{K,[\mathfrak b ]}(s) = \frac{2\pi }{w\sqrt{|d \,|}}. \end{aligned}$$

(d):

Sum the residues over $[\mathfrak b ] \in {{\,\mathrm{Cl}\,}}(K)$ to derive the theorem.

$\triangleright$ 5.:

In this exercise, we prove Theorem 25.4.1: if $\mathcal {O}$ is a maximal order in a definite quaternion algebra over $\mathbb Q$ of discriminant D, then $\#{{\,\mathrm{Cls}\,}}\mathcal {O}=1$ if and only if $D=2,3,5,7,13$. By the Eichler mass formula (Theorem 25.3.15), we have $\#{{\,\mathrm{Cls}\,}}\mathcal {O}=1$ if and only if

$$\begin{aligned} \frac{1}{w} = \frac{\varphi (D)}{12} \end{aligned}$$

where $w=\#\mathcal {O}/\{\pm 1\}$.

(a):

Show (cf. 11.5.13) that if $D>3$ then $w \le 3$.

(b):

Show that $\#{{\,\mathrm{Cls}\,}}\mathcal {O}=1$ for $D=2,3$. (The case $D=2$ is the Hurwitz order and $D=3$ is considered in Exercise 11.11. In fact, these orders are Euclidean with respect to the reduced norm: see the next exercise.)

(c):

Show that if D is a squarefree positive integer with an odd number of prime factors and $\varphi (D)/12 \in \{1,1/2,1/3\}$, then $D \in \{5,7,13,42\}$.

(d):

Prove that $\#{{\,\mathrm{Cls}\,}}\mathcal {O}=1$ for $D=5,7,13$ (cf. Exercise 17.10).

(e):

Show that $\#{{\,\mathrm{Cls}\,}}\mathcal {O}=2$ for $D=42$.

• 6. Let $\mathcal {O}$ be a definite quaternion order over $\mathbb Z$. If $\mathcal {O}$ is Euclidean, then $\#{{\,\mathrm{Cls}\,}}\mathcal {O}=1$, and we saw in 11.3.1 that the Hurwitz order $\mathcal {O}$ of discriminant $D=2$ is Euclidean with respect to the reduced norm.

  • (a) Show that if $\mathcal {O}$ is norm Euclidean, then $\mathcal {O}$ is maximal.

  • (b) Show that $\mathcal {O}$ is Euclidean with respect to the reduced norm if and only if for all $\gamma \in B$, there exists $\mu \in \mathcal {O}$ such that ${{\,\mathrm{nrd}\,}}(\gamma -\mu )<1$.

  • (c) Show that if $\mathcal {O}$ is maximal, then $\mathcal {O}$ is norm Euclidean if and only if $D=2,3,5$.

• 7. Generalizing the previous exercise, we may ask for the Euclidean ideal classes in maximal orders. We will show that there are no nonprincipal Euclidean two-sided ideal classes in maximal definite quaternion orders over $\mathbb Z$. [This exercise was suggested by Pete L. Clark.]

  Let $\mathcal {O}$ be a maximal definite quaternion order over $\mathbb Z$ of discriminant D, and let I be a two-sided $\mathcal {O}$-ideal. We say that I is (norm) Euclidean  if for all $\gamma \in B$, there exists $\mu \in I$ such that ${{\,\mathrm{nrd}\,}}(\gamma -\mu )<{{\,\mathrm{nrd}\,}}(I)$.

  • (a) Show that if I is principal, then I is Euclidean if and only if $\mathcal {O}$ is Euclidean. In general, show that I is Euclidean if and only if $\alpha I$ is Euclidean for all $\alpha \in B^\times$, so we may ask if [I] is Euclidean.

  • (b) Show that if [I] is nontrivial and Euclidean, then $\#{{\,\mathrm{Cls}\,}}\mathcal {O}=2$ and ${{\,\mathrm{Pic}\,}}_\mathbb{Z }(\mathcal {O})$ is cyclic, generated by [I]. [Hint: argue by induction on ${{\,\mathrm{nrd}\,}}(J)$. Then use the fact that ${{\,\mathrm{Pic}\,}}(\mathcal {O})$ is a group of exponent 2.]

  • (c) Show that $\#{{\,\mathrm{Cls}\,}}\mathcal {O}=2$ if and only if $D=11,17,19,30,42,70,78$.

  • (d) Show that ${{\,\mathrm{Pic}\,}}_\mathbb{Z }(\mathcal {O})=1$ for $D=11,17,19$, and for the remaining discriminants $D=30,42,70,78$ that the nontrivial class [I] is not norm Euclidean.