Volume formula

Abstract

In this chapter, we exhibit a formula for the covolume of a quaternionic group, a formula with many applications.

In this chapter, we exhibit a formula for the covolume of a quaternionic group, a formula with many applications.

1 $\triangleright$ Statement

We saw in (35.1.5) that the hyperbolic area of the quotient ${{\,\mathrm{SL}\,}}_2(\mathbb Z ) \backslash \mathbf{\textsf {H} }^2$ can be computed directly from the fundamental domain

$$\begin{aligned} =\{z \in \mathbf{\textsf {H} }^2: |{{\,\mathrm{Re}\,}}z \,| \le 1/2\text { and }|z \,| \ge 1\} \end{aligned}$$

as

(39.1.1)

In fact, given a Fuchsian or Kleinian group $\Gamma$, the hyperbolic area or volume of the quotient $\Gamma \backslash \mathcal H$ (where $\mathcal H =\mathbf{\textsf {H} }^2,\mathbf{\textsf {H} }^3$, respectively) can be computed without recourse to a fundamental domain: it is given in terms of the arithmetic invariants of the order and quaternion algebra that give rise to $\Gamma$.

To begin, we consider the already interesting case where the quaternion algebra is defined over $\mathbb Q$.

Theorem 39.1.2

Let B be a quaternion algebra over $\mathbb Q$ of discriminant D and let $\mathcal O \subseteq B$ be a maximal order. Let $\Gamma _0^1(D) \le {{\,\mathrm{PSL}\,}}_2(\mathbb R )$ be the Fuchsian group associated to the group ${{\,\mathrm{P\!}\,}}\mathcal O ^1=\mathcal O ^1/\{\pm 1\}$ of units of reduced norm 1.

Then

$$\begin{aligned} {{\,\mathrm{area}\,}}(\Gamma _0^1(D) \backslash \mathbf{\textsf {H} }^2) = \frac{\pi }{3} \varphi (D) \end{aligned}$$

(39.1.3)

where

$$\begin{aligned} \varphi (D) :=\prod _{p \mid D}(p-1)=D\prod _{p \mid D}\left( 1-\frac{1}{p}\right) \end{aligned}$$

(39.1.4)

Recall that the discriminant $D \in \mathbb Z$ is a squarefree positive integer, so the function $\varphi$ is just the Euler totient function. Theorem 39.1.2 recovers (39.1.1) with $D=1$.

Example 39.1.5

Recall the case $X_0^1(6)$ from section 38.1. We confirm that the hyperbolic area computed from the fundamental domain agrees with the formula in Theorem 39.1.2:

$$\begin{aligned} {{\,\mathrm{area}\,}}(X_0^1(6,1)) = \frac{\pi }{3}\varphi (6)\psi (1)=\frac{2\pi }{3}. \end{aligned}$$

The expression (39.1.3) is quite similar to the Eichler mass formula (Theorem 25.3.15). Indeed, the method of proof is the same, involving the zeta function of the order $\mathcal O$. For the case of a definite quaternion algebra, when the unit group was finite, we used Theorem 26.2.12 to relate the mass to a residue of the zeta function (see in particular Proposition 26.5.10); for an indefinite quaternion algebra, to carry this out in general would involve a multivariable integral whose evaluation is similar to the proof of Proposition 26.2.18 (the commutative case)—not an appealing prospect. That being said, this type of direct argument with the zeta function was carried out by Shimizu [Shz65, Appendix] over a totally real field F.

We prefer instead to use idelic methods; we already computed the normalized Tamagawa measure $\tau ^{1}(B^{1} \backslash \underline{B}^{1})=1$ (see Theorem 29.11.3), and this number has everything we need! It is a much simpler computation to relate this to the volume of the hyperbolic quotient: we carry this out in section 39.3.

The following notation will be used throughout.

39.1.6

Let F be number field of degree $n=[F:\mathbb Q ]$ with r real places and c complex places, so $r+2c=n$. Let B be a quaternion algebra over F of discriminant $\mathfrak D$ that is split at t real places. Suppose that B is indefinite (i.e., $t+c>0$).

Let $R=\mathbb Z _F$ be the ring of integers of F. Let $\mathcal O \subseteq B$ be an R-order of reduced discriminant $\mathfrak N$ that is locally norm-maximal. Let

$$\begin{aligned} \mathcal H :=(\mathbf{\textsf {H} }^2)^t \times (\mathbf{\textsf {H} }^3)^c \end{aligned}$$

and let

$$\begin{aligned} \Gamma ^1(\mathcal O ) \le {{\,\mathrm{PSL}\,}}_2(\mathbb R )^t \times {{\,\mathrm{PSL}\,}}_2(\mathbb C )^c \circlearrowright \mathcal H \end{aligned}$$

be the discrete group associated to the group ${{\,\mathrm{P\!}\,}}\mathcal O ^1 = \mathcal O ^1/\{\pm 1\}$ of units of $\mathcal O$ of reduced norm 1.

For a prime $\mathfrak p \mid \mathfrak N$ with , let be the Eichler symbol (Definition 24.3.2), and let

(39.1.7)

Main Theorem 39.1.8

With notation as in 39.1.6, we have

$$\begin{aligned} \begin{aligned} {{\,\mathrm{vol}\,}}(\Gamma ^1(\mathcal O ) \backslash \mathcal H )&= \frac{2(4\pi )^t}{(4\pi ^2)^r (8\pi ^2)^c} \zeta _F(2) d_F^{3/2}{{\,\mathrm{Nm}\,}}(\mathfrak N )\prod _\mathfrak{p \mid \mathfrak N } \lambda (\mathcal O ,\mathfrak p ). \end{aligned} \end{aligned}$$

(39.1.9)

Main Theorem 39.1.8 is proven as Main Theorem 39.3.1. Since $\zeta _F(2) \approx 1$, for algebras B with fixed signature r, s, t, we can roughly estimate

$$\begin{aligned} {{\,\mathrm{vol}\,}}(\Gamma ^1(\mathcal O )) \approx d_F^{3/2} {{\,\mathrm{Nm}\,}}({{\,\mathrm{discrd}\,}}\mathcal O ). \end{aligned}$$

(39.1.10)

39.1.11

The constant factor in (39.1.9) is written to help in remembering the formula; multiplying out, we have

$$\begin{aligned} \frac{2(4\pi )^t}{(4\pi ^2)^r (8\pi ^2)^c} = \frac{1}{2^{2r+3c-2t-1} \pi ^{2r+2c-t}}. \end{aligned}$$

Note that the right-hand side of (39.1.9) is independent of the choice of order $\mathcal O$ in the genus of $\mathcal O$, as it depends only on $\widehat{\mathcal{O }}$.

Remark 39.1.12. If $\mathcal O$ is not locally norm-maximal, then one corrects the formula by inserting factors $[R_\mathfrak p ^\times : {{\,\mathrm{nrd}\,}}(\mathcal O _\mathfrak p ^\times )]$ as in (39.2.10); since $R_\mathfrak p ^\times \subseteq \mathcal O _\mathfrak p ^\times$ we have $R_\mathfrak p ^{\times 2} \subseteq {{\,\mathrm{nrd}\,}}(\mathcal O _\mathfrak p ^\times )$, so these factors are at most 2 for each $\mathfrak p$.

An important special case of Main Theorem 39.1.8 is the case where $\mathcal O$ is an Eichler order, generalizing Theorem 39.1.2.

Theorem 39.1.13

Suppose that $\mathcal O =\mathcal O _0(\mathfrak M )$ is an Eichler order of level $\mathfrak M$ and ${{\,\mathrm{disc}\,}}B = \mathfrak D$, so $\mathfrak N =\mathfrak D \mathfrak M$. Write $\Gamma _0^1(\mathfrak M )=\Gamma ^1(\mathcal O )$ and $\Gamma ^1(1)$ for the group associated to a maximal order $\mathcal O (1) \supseteq \mathcal O _0(\mathfrak M )$. Then

$$\begin{aligned} {{\,\mathrm{vol}\,}}(\Gamma _0^1(\mathfrak M ) \backslash \mathcal H ) = \frac{2(4\pi )^t}{(4\pi ^2)^r (8\pi ^2)^c} \zeta _F(2)d_F^{3/2}\varphi (\mathfrak D )\psi (\mathfrak M ) \end{aligned}$$

where

$$\begin{aligned} \begin{aligned} \varphi (\mathfrak D )&= \#(\mathbb Z _F/\mathfrak D )^\times = {{\,\mathrm{Nm}\,}}\mathfrak D \prod _\mathfrak{p \mid \mathfrak D } \left( 1-\frac{1}{{{\,\mathrm{Nm}\,}}\mathfrak p }\right) \\ \psi (\mathfrak M )&= [\Gamma ^1(1) : \Gamma _0^1(\mathfrak M )] = {{\,\mathrm{Nm}\,}}\mathfrak M \prod _\mathfrak{p ^e \parallel \mathfrak M } \left( 1+\frac{1}{{{\,\mathrm{Nm}\,}}\mathfrak p }\right) . \end{aligned} \end{aligned}$$

(39.1.14)

We write $\mathfrak p ^e \parallel \mathfrak M$ to mean $\mathfrak p ^e$ exactly divides $\mathfrak M$, i.e., ${{\,\mathrm{ord}\,}}_\mathfrak p (\mathfrak M )=e$, and we take the product over all prime power divisors of $\mathfrak M$ in the definition of $\psi$.

In particular, when $\mathcal O$ is a maximal order, then $\mathfrak M =R$ and $\psi (\mathfrak M )=1$.

Remark 39.1.15. Theorem 39.1.13 is often attributed to Borel [Bor81], who derived it under the (highly nontrivial!) assumption that $\tau ^1(B^1 \backslash \underline{B}^1)=1$.

Example 39.1.16

Let $\Gamma ^1={{\,\mathrm{PSL}\,}}_2(\mathbb Z [i])$. Then

$$\begin{aligned} {{\,\mathrm{vol}\,}}(X^1) = \frac{2}{8\pi ^2} 4^{3/2} \zeta _F(2) = \frac{2}{\pi ^2} \zeta _F(2) = 0.3053\ldots . \end{aligned}$$

This agrees with the computation we did with a fundamental domain (see 36.6.8), since for $\chi$ the character 36.6.14 of conductor 4,

$$\begin{aligned} \frac{2}{\pi ^2} \zeta _F(2) = \frac{2}{\pi ^2} \zeta (2)L(2,\chi ) = \frac{1}{3} L(2,\chi ). \end{aligned}$$

2 Volume setup

In this section, we setup a few calculations in preparation for the volume formula.

Let F be number field with discriminant $d_F$, let B be a quaternion algebra over F of discriminant $\mathfrak D ={{\,\mathrm{disc}\,}}B$, and let $\mathcal O \subseteq B$ be an order. The key input to the proof is the following ingredient: from Theorem 29.11.3,

$$\begin{aligned} \underline{\tau }^1(B^1 \backslash \underline{B}^1)=1. \end{aligned}$$

(39.2.1)

We will convert the volume (39.2.1) into the desired form by separating out the contribution from the finite places and the infinite (real) ramified places: what remains is the volume of the orbifold we seek, which we then renormalize from the adelic to the standard hyperbolic volume.

We define

$$\begin{aligned} \underline{\mathcal{O }}:=\widehat{\mathcal{O }} \times B_\infty \subseteq \underline{B}. \end{aligned}$$

Then

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

and

$$\begin{aligned} \underline{\mathcal{O }}^1 = \widehat{\mathcal{O }}^1 \times B_\infty ^1. \end{aligned}$$

Now we apply the important assumption: we suppose that B is indefinite. The hypothesis that B is indefinite is necessary even to get a nontrivial space $\mathcal H =(\mathbf{\textsf {H} }^2)^t \times (\mathbf{\textsf {H} }^3)^c$ for the group to act upon! The case where B is definite was handled in the proof of the Eichler mass formula (Main Theorem 25.3.19).

39.2.2

By strong approximation (Corollary 28.5.12) we have $\underline{B}^1=B^1 \underline{\mathcal{O }}^1$. Therefore, the natural inclusion

$$\begin{aligned} \mathcal O ^1 \backslash \underline{\mathcal{O }}^1 \hookrightarrow B^1 \backslash \underline{B}^1 \end{aligned}$$

is also surjective, hence an isomorphism. Thus by (39.2.1)

$$\begin{aligned} \underline{\tau }^1(\mathcal O ^1 \backslash \underline{\mathcal{O }}^1) = \underline{\tau }^1(B^1 \backslash \underline{B}^1)=1. \end{aligned}$$

(39.2.3)

We have an embedding $\mathcal O ^1 \hookrightarrow B_\infty ^1$, so

$$\begin{aligned} 1 = \underline{\tau }^1(\mathcal O ^1 \backslash \underline{\mathcal{O }}^1) = \widehat{\tau }^1(\widehat{\mathcal{O }}^1) \tau _\infty ^1(\mathcal O ^1 \backslash B_\infty ^1). \end{aligned}$$

(39.2.4)

39.2.5

If $\mathcal O$ is maximal, then from (29.7.25) we have

$$\begin{aligned} \widehat{\tau }^1(\widehat{\mathcal{O }}^1) = \prod _\mathfrak{p } \tau _\mathfrak p ^1 ( \mathcal O _\mathfrak p ^1) = |d_F \,|^{-3/2}\zeta _F(2)^{-1} \prod _\mathfrak{p \in {{\,\mathrm{Ram}\,}}(B)} ({{\,\mathrm{Nm}\,}}\mathfrak p -1)^{-1} \end{aligned}$$

so that

$$\begin{aligned} \widehat{\tau }^1(\widehat{\mathcal{O }}^1)^{-1} = |d_F \,|^{3/2}\zeta _F(2) \varphi (\mathfrak D ). \end{aligned}$$

(39.2.6)

So by (39.2.6), we conclude that

$$\begin{aligned} \tau _\infty ^1(\mathcal O ^1 \backslash B_\infty ^1) = d_F^{3/2}\zeta _F(2) \varphi (\mathfrak D ). \end{aligned}$$

(39.2.7)

39.2.8

In general, if $\mathcal O \subseteq \mathcal O '$ with $\mathcal O '$ maximal, then

so similarly

(39.2.9)

If $\mathcal O$ is locally norm-maximal, then further

(39.2.10)

and by Lemma 26.6.7, with $\mathfrak N ={{\,\mathrm{discrd}\,}}\mathcal O$ we have

(39.2.11)

(as in (26.6.8)).

Next, we relate the measures on ${{\,\mathrm{SL}\,}}_2(\mathbb R )$ and ${{\,\mathrm{SL}\,}}_2(\mathbb C )$ to the corresponding measures on $\mathbf{\textsf {H} }^2$ and $\mathbf{\textsf {H} }^3$.

39.2.12

Recall the symmetric space identification

$$\begin{aligned} \mathbf{\textsf {H} }^2\rightarrow {{\,\mathrm{SL}\,}}_2(\mathbb R )/{{\,\mathrm{SO}\,}}(2) \end{aligned}$$

(39.2.13)

The hyperbolic plane $\mathbf{\textsf {H} }^2$ is equipped with the hyperbolic measure $\mu$, the unique measure invariant under the action of ${{\,\mathrm{PSL}\,}}_2(\mathbb R )$; the group ${{\,\mathrm{SL}\,}}_2(\mathbb R )$ has the Haar measure $\tau ^1$, also invariant under the left action of ${{\,\mathrm{SL}\,}}_2(\mathbb R )$. Therefore, the identification (39.2.13) relates these two measures up to a constant $v({{\,\mathrm{SO}\,}}(2))$ that gives a total measure to ${{\,\mathrm{SO}\,}}(2)$ (normalizing its Haar measure). Ditto for $\mathbf{\textsf {H} }^3$ and ${{\,\mathrm{SU}\,}}(2)$, with a constant $v({{\,\mathrm{SU}\,}}(2))$.

Lemma 39.2.14

We have

$$\begin{aligned} v({{\,\mathrm{SO}\,}}(2)) = \pi \qquad \text {and} \qquad v({{\,\mathrm{SU}\,}}(2))=8\pi ^2. \end{aligned}$$

Proof. One could compute the relevant constant by doing a (compatible) integral, but we prefer just to refer to an example where both sides are computed. We consider a fundamental domain for the action of ${{\,\mathrm{SL}\,}}_2(\mathbb Z )\, \circlearrowright \, {{\,\mathrm{SL}\,}}_2(\mathbb R )$ that is invariant under ${{\,\mathrm{SO}\,}}(2)$: for example, we can lift a fundamental domain for ${{\,\mathrm{PSL}\,}}_2(\mathbb Z )\, \circlearrowright \, \mathbf{\textsf {H} }^2$ under (39.2.13). The difference between ${{\,\mathrm{SL}\,}}_2(\mathbb Z )$ and ${{\,\mathrm{PSL}\,}}_2(\mathbb Z )={{\,\mathrm{SL}\,}}_2(\mathbb Z )/\{\pm 1\}$ is annoyingly relevant here! We have

$$\begin{aligned} {{\,\mathrm{PSL}\,}}_2(\mathbb Z ) \backslash \mathbf{\textsf {H} }^2= {{\,\mathrm{PSL}\,}}_2(\mathbb Z ) \backslash {{\,\mathrm{SL}\,}}_2(\mathbb R ) / {{\,\mathrm{SO}\,}}(2); \end{aligned}$$

if we let ${{\,\mathrm{SO}\,}}(2)_2 \simeq {{\,\mathrm{SO}\,}}(2)/\{\pm 1\}$ be the rotation group acting by $2\theta$ instead of $\theta$, then we can lift from ${{\,\mathrm{PSL}\,}}_2(\mathbb Z )$ to ${{\,\mathrm{SL}\,}}_2(\mathbb Z )$ and

$$\begin{aligned} {{\,\mathrm{PSL}\,}}_2(\mathbb Z ) \backslash {{\,\mathrm{SL}\,}}_2(\mathbb R ) / {{\,\mathrm{SO}\,}}(2) = {{\,\mathrm{SL}\,}}_2(\mathbb Z ) \backslash {{\,\mathrm{SL}\,}}_2(\mathbb R ) / {{\,\mathrm{SO}\,}}(2)_2 \end{aligned}$$

which under compatible metrics gives

$$\begin{aligned} v({{\,\mathrm{SO}\,}}(2))=2v({{\,\mathrm{SO}\,}}(2)_2) = 2\frac{\tau ^1({{\,\mathrm{SL}\,}}_2(\mathbb Z ) \backslash {{\,\mathrm{SL}\,}}_2(\mathbb R ))}{\mu ({{\,\mathrm{PSL}\,}}_2(\mathbb Z ) \backslash \mathbf{\textsf {H} }^2)}. \end{aligned}$$

(39.2.15)

On the bottom we have $\mu ({{\,\mathrm{SL}\,}}_2(\mathbb Z ) \backslash \mathbf{\textsf {H} }^2)=\pi /3$ by (35.1.5) and on the top we have $\tau ^1({{\,\mathrm{SL}\,}}_2(\mathbb Z ) \backslash {{\,\mathrm{SL}\,}}_2(\mathbb R ))=\zeta (2)=\pi ^2/6$ by (39.2.9). Plugging in, we compute

$$\begin{aligned} v({{\,\mathrm{SO}\,}}(2)) = 2\frac{\pi ^2/6}{\pi /3} = \pi . \end{aligned}$$

(39.2.16)

Similarly,

$$\begin{aligned} v({{\,\mathrm{SU}\,}}(2))=2\frac{\tau ^1({{\,\mathrm{SL}\,}}_2(\mathbb Z [i]) \backslash {{\,\mathrm{SL}\,}}_2(\mathbb C ))}{\mu ({{\,\mathrm{PSL}\,}}_2(\mathbb Z [i]) \backslash \mathbf{\textsf {H} }^3)} = 2\frac{4^{3/2}\zeta _\mathbb{Q (i)}(2)}{(2/\pi ^2)\zeta _\mathbb{Q (i)}(2)} = 8\pi ^2 \end{aligned}$$

(39.2.17)

by Example 39.1.16 and again (39.2.9).

3 Volume derivation

We now establish the volume formula (Main Theorem 39.1.8) using the computation of the Tamagawa measure (Theorem 29.11.3, (39.2.1)), following Borel [Bor81, 7.3].

We continue with the notation from 39.1.6, so in particular $n=[F:\mathbb Q ]$ and F has r real places, c complex places, and B is split at t real places; B is indefinite, so $t+c>0$; and $\mathcal O \subseteq B$ is a locally norm-maximal order. We restate the formula for convenience.

Main Theorem 39.3.1

We have

$$\begin{aligned} \begin{aligned} {{\,\mathrm{vol}\,}}(\Gamma ^1(\mathcal O ) \backslash \mathcal H )&= \frac{2(4\pi )^t}{(4\pi ^2)^r (8\pi ^2)^c} \zeta _F(2) d_F^{3/2}{{\,\mathrm{Nm}\,}}(\mathfrak N )\prod _\mathfrak{p \mid \mathfrak N } \lambda (\mathcal O ,\mathfrak p ). \end{aligned} \end{aligned}$$

(39.3.2)

Proof. To summarize the previous section, we started with $\tau ^1(B^1 \backslash \underline{B}^1)$ and concluded $\tau ^1(\mathcal O ^1 \backslash \underline{\mathcal{O }}^1)=1$ by strong approximation; we factored this into finite and infinite parts, with the finite part computed in terms of the order and the infinite part. Then

$$\begin{aligned} \mathcal O ^1 \backslash B_\infty ^1 = \mathcal O ^1 \backslash \prod _{v \mid \infty } B_v^1 \simeq \prod _{v \in \Omega } B_v^1 \times \biggl (\mathcal O ^1 \backslash \prod _{\begin{array}{c} v \mid \infty \\ v \not \in \Omega \end{array}} B_v^1\biggr ). \end{aligned}$$

(39.3.3)

Each term contributes to the volume. For the first product, for each of the $r-t$ places $v \in \Omega$ we have $B_v^1 \simeq \mathbb H ^1$ and we computed in Lemma 29.5.9 that $\tau ^1(\mathbb H ^1)=4\pi ^2$. For the remaining terms, we employ the comparison formula between measures (Lemma 39.2.14), and are plagued by the same factor 2 coming from the fact that $\Gamma ^1(\mathcal O )$ arises from ${{\,\mathrm{P\!}\,}}\mathcal O ^1/\{\pm 1\}$. Putting these together, the decomposition (39.3.3) yields a volume

$$\begin{aligned} \begin{aligned} \tau _\infty ^1(\mathcal O ^1 \backslash B_\infty ^1)&= (4\pi ^2)^{r-t} \pi ^t (8\pi ^2)^c \frac{1}{2} {{\,\mathrm{vol}\,}}(\Gamma ^1 \backslash \mathcal H ) \\&= \frac{(4\pi ^2)^r(8\pi ^2)^c}{2(4\pi )^t} {{\,\mathrm{vol}\,}}(\Gamma ^1 \backslash \mathcal H ). \end{aligned} \end{aligned}$$

(39.3.4)

From (39.2.9) and (39.3.4) we conclude

(39.3.5)

Finally, the computation (39.2.11) of the local index completes the proof.

Remark 39.3.6. A similar proof works for the case where F is a function field or where $\Gamma$ is an ${{{\texttt {\textit{S}}}}}$-arithmetic group, but in both cases still under the hypothesis that B is ${{{\texttt {\textit{S}}}}}$-indefinite for an eligible set ${{{\texttt {\textit{S}}}}}$ (playing the role of the archimedean places above).

Example 39.3.7

Suppose F is totally real, and we take $B={{\,\mathrm{M}\,}}_2(F)$ and $\mathcal O ={{\,\mathrm{M}\,}}_2(\mathbb Z _F)$. Then $\mathcal H = (\mathbf{\textsf {H} }^2)^n$ and

4 Genus formula

In this section, we take the volume formula (Main Theorem 39.3.1) in the special case of a Fuchsian group and extend it to a formula for the genus of a Shimura curve.

We maintain our notation but now specialize to the case where $c=0$ and $t=1$: in particular, F is a totally real field, and B is indefinite. Thus $\Gamma =\Gamma ^1(\mathcal O ) \le {{\,\mathrm{PSL}\,}}_2(\mathbb R )$ is a Fuchsian group.

We suppose that $\mathcal O$ is an Eichler order of level $\mathfrak M$.

39.4.1

Recalling 37.7, let $(g;e_1,\dots ,e_k;\delta )$ be the signature of $\Gamma$. Then $Y(\Gamma )$ has genus g; has k elliptic cycles of orders $e_1,\dots ,e_k \in \mathbb Z _{\ge 2}$, corresponding to cone points on $Y(\Gamma )$ with given order; and has $\delta$ parabolic cycles, corresponding to the punctures of $Y(\Gamma )$. We have $\delta =0$ unless $B=M_2(\mathbb Q )$, corresponding to the case of classical modular curves.

By Proposition 37.7.4,

Rewriting this slightly, for $q \in \mathbb Z _{\ge 2}$, let $m_q$ be the number of elliptic cycles of order q. Then

(39.4.2)

and the sum is finite.

The numbers $m_q$ are determined by embedding numbers of quadratic orders into the quaternion order $\mathcal O$, as studied in chapter 30.

39.4.3

Let $q \in \mathbb Z _{\ge 2}$. Suppose that $m_q>0$, so that $\mathcal O ^1$ has a maximal finite subgroup $\langle \gamma \rangle \le \mathcal O ^1$ of order 2q. Then the field $K_q=F(\zeta _{2q}) \supset F$ is a quadratic field extension and $K_q \hookrightarrow B$ embeds, where $\zeta _{2q}$ is a primitive 2qth root of unity, and we have two optimal embeddings $S=F(\gamma ) \cap \mathcal O \hookrightarrow \mathcal O$ given by $\gamma$ and $\overline{\gamma }$. Conversely, to every embedding $\phi :K_q \hookrightarrow B$, we associate the order $S=\phi (K_q) \cap \mathcal O$ and the finite subgroup $S_{\text {tors}}^\times \subset \mathcal O ^1$.

Thus there is a two-to-one map

In the notation of 30.3.10, we have shown that

(39.4.4)

Our next major ingredient is the theory of selectivity, treated in chapter 31.

39.4.5

We claim that $K_q$ does not satisfy the selectivity condition (OS), defined in 31.1.6. If $\mathcal O$ is an Eichler order, we may appeal to Proposition (31.2.1) and condition (a): since F is totally real, $K_q$ is totally complex, and B is split at a real place, condition (a) fails.

Therefore by Main Theorem 31.1.7(a), ${{\,\mathrm{Gen}\,}}\mathcal O$ is not optimally selective. By Corollary 31.1.10, for every R-order $S \subseteq K_q$, we have

$$\begin{aligned} m(S,\mathcal O ;\mathcal O ^\times ) = \frac{h(S)}{\#{{\,\mathrm{Cl}\,}}_\Omega R} m(\widehat{S},\widehat{\mathcal{O }};\widehat{\mathcal{O }}^\times ) \end{aligned}$$

(39.4.6)

where we have substituted $\#{{\,\mathrm{Cls}\,}}\mathcal O = \#{{\,\mathrm{Cl}\,}}_\Omega R$ (by Corollary 28.5.17).

The adelic embedding numbers $m(\widehat{S},\widehat{\mathcal{O }};\widehat{\mathcal{O }}^\times )$, a product of (finitely many) local embedding numbers by 30.7.1, are computed in section 30.6.

We will need one lemma relating units to class numbers.

Lemma 39.4.7

We have $[R_{>_{\Omega } 0}^{\times }:R^{\times 2}] = 2[{{\,\mathrm{Cl}\,}}_{\Omega } R : {{\,\mathrm{Cl}\,}}R]$.

Proof. The index $[R_{>_{\Omega } 0}^{\times }:R^{\times 2}]$, which does not depend on S, is related to class numbers as follows. For each real place v, define ${{\,\mathrm{sgn}\,}}_v :F^\times \rightarrow \{\pm 1\}$ by the real sign ${{\,\mathrm{sgn}\,}}_v(a)={{\,\mathrm{sgn}\,}}(v(a))$ at v. Let

$$\begin{aligned} {{\,\mathrm{sgn}\,}}_\Omega :F^\times&\rightarrow \{\pm 1\}^\Omega \\ a&\mapsto ({{\,\mathrm{sgn}\,}}_v(a))_v \end{aligned}$$

collect the signs at the places $v \in \Omega$. Then we have an exact sequence

$$\begin{aligned} 1 \rightarrow \{\pm 1\}^\Omega /{{\,\mathrm{sgn}\,}}_\Omega R^\times \rightarrow {{\,\mathrm{Cl}\,}}_\Omega R \rightarrow {{\,\mathrm{Cl}\,}}R \rightarrow 1 \end{aligned}$$

(39.4.8)

where the map on the left is induced by mapping a tuple of signs in $\{\pm 1\}^\Omega$ to the principal ideal generated by any $a \in F^\times$ with the given signs. We have a second (tautological) exact sequence

$$\begin{aligned} 1 \rightarrow R_{>_{\Omega } 0}^{\times }/R^{\times 2} \rightarrow R^\times /R^{\times 2} \xrightarrow {{{\,\mathrm{sgn}\,}}_\Omega } \{\pm 1\}^\Omega \rightarrow \{\pm 1\}^\Omega /{{\,\mathrm{sgn}\,}}_\Omega R^\times \rightarrow 1 \end{aligned}$$

(39.4.9)

of elementary abelian 2-groups (or $\mathbb F _2$-vector spaces). Combining (39.4.9) with (39.4.8), and noting that $[R^\times :R^{\times 2}]=2^r$ by Dirichlet’s unit theorem and $\#\Omega =r-1$ by hypothesis, we conclude that

Definition 39.4.10

For a quadratic R-order $S \subseteq K$, the Hasse unit index is defined by

We have $Q(S)<\infty$ because $S^\times$ and $R^\times$ have the same $\mathbb Z$-rank.

Remark 39.4.11. Hasse [Hass52, Sätze 14–29] proved numerous theorems about $Q(\mathbb Z _K)$, including that $Q(\mathbb Z _K) \le 2$: see also Washington [Was97, Theorem 4.12].

We are now ready to write in a simplified way the count $m_q$ of elliptic cycles.

Proposition 39.4.12

We have

where $h(R)=\#{{\,\mathrm{Pic}\,}}R$ and $h(S) = \#{{\,\mathrm{Pic}\,}}S$.

Proof. Beginning with (39.4.4), we have

$$\begin{aligned} m_q = \frac{1}{2}\sum _{\begin{array}{c} S \subset K_q \\ \#S^\times _{tors }=2q \end{array}} m(S,\mathcal O ;\mathcal O ^1). \end{aligned}$$

(39.4.13)

By Lemma 30.3.14, we have

$$\begin{aligned} m(S,\mathcal O ;\mathcal O ^1) = m(S,\mathcal O ;\mathcal O ^\times ) [{{\,\mathrm{nrd}\,}}(\mathcal O ^\times ) : {{\,\mathrm{nrd}\,}}(S^\times )]. \end{aligned}$$

(39.4.14)

Since B is indefinite, by Corollary 31.1.11 we have ${{\,\mathrm{nrd}\,}}(\mathcal O ^\times )=R_{>_{\Omega } 0}^{\times }$. By Lemma 39.4.7, we have $[R_{>_{\Omega } 0}^{\times }:R^{\times 2}] = 2[{{\,\mathrm{Cl}\,}}_{\Omega } R : {{\,\mathrm{Cl}\,}}R]$. Thus

$$\begin{aligned} \begin{aligned}{}[{{\,\mathrm{nrd}\,}}(\mathcal O ^\times ) : {{\,\mathrm{nrd}\,}}(S^\times )]&= [R_{>_{\Omega } 0}^{\times }:R^{\times 2}][R^{\times 2}:{{\,\mathrm{nrd}\,}}(S^\times )]\\&=\frac{2[{{\,\mathrm{Cl}\,}}_{\Omega } R:{{\,\mathrm{Cl}\,}}R]}{Q(S)}. \end{aligned} \end{aligned}$$

(39.4.15)

Substituting (39.4.6) and (39.4.15) into (39.4.14), we find

$$\begin{aligned} \begin{aligned} m(S,\mathcal O ;\mathcal O ^1)&= m(S,\mathcal O ;\mathcal O ^\times ) [{{\,\mathrm{nrd}\,}}(\mathcal O ^\times ): {{\,\mathrm{nrd}\,}}(S^\times )] \\&= \frac{h(S)}{Q(S)}m(\widehat{S},\widehat{\mathcal{O }};\widehat{\mathcal{O }}^\times )\frac{[R_{>_{\Omega } 0}^{\times }:R^{\times 2}]}{\#{{\,\mathrm{Cl}\,}}_\Omega R} \\&= \frac{2}{h(R)}\frac{h(S)}{Q(S)}m(\widehat{S},\widehat{\mathcal{O }};\widehat{\mathcal{O }}^\times ). \end{aligned} \end{aligned}$$

(39.4.16)

Finally, plugging (39.4.16) into (39.4.13) and cancelling a factor 2 gives the result.

Corollary 39.4.17

The signature of a Shimura curve depends only on the discriminant $\mathfrak D$ and level $\mathfrak M$.

Proof. The ambiguity corresponds to a choice of Eichler order of level $\mathfrak M$ and choice of split real place; when $F=\mathbb Q$, there is no ambiguity in either case. So we may suppose that $\Gamma$ has no parabolic cycles. Then we simply observe that the formula (Proposition 39.4.12) for the number of elliptic cycles depends only on $\widehat{\mathcal{O }}$.

Example 39.4.18

As a special case of Proposition 39.4.12, suppose that $\mathfrak N =\mathfrak D \mathfrak M$ is coprime to q. Then as in Example 30.7.5, we have

We now have the ingredients to give a formula for a Shimura curve.

Theorem 39.4.19

Let $Y^1(\mathcal O ) = \Gamma ^1(\mathcal O ) \backslash \mathbf{\textsf {H} }^2$. Then $Y^1(\mathcal O )$ is an orbifold with genus g where

where $m_q$ are given in Proposition (39.4.12).

Proof. Combine the volume formula (Main Theorem 39.3.1) with (39.4.2).

The special case where $F=\mathbb Q$ is itself important.

Theorem 39.4.20

Let $D={{\,\mathrm{disc}\,}}B>1$ and let $\mathcal O \subseteq B$ be an Eichler order of level M, so $N=DM={{\,\mathrm{discrd}\,}}\mathcal O$ with D squarefree and $\gcd (D,M)=1$.

Then $X^1(\mathcal O ) = \Gamma ^1(\mathcal O ) \backslash \mathbf{\textsf {H} }^2$ is an orbifold with genus g where

$$\begin{aligned} 2g-2 = \frac{\varphi (D)\psi (M)}{6} - \frac{m_2}{2} - \frac{2m_3}{3} \end{aligned}$$

where the embedding numbers were computed in Example 30.7.7:

$$\begin{aligned} m_2&= m(\mathbb Z [i],\mathcal O ;\mathcal O ^\times ) = {\left\{ \begin{array}{ll} \displaystyle {\prod _{p \mid D}}\left( 1-\biggl (\displaystyle {\frac{-4}{p}}\biggr )\right) \displaystyle {\prod _{p \mid M}}\left( 1+\biggl (\displaystyle {\frac{-4}{p}}\biggr )\right) , &{} \text { if }4 \not \mid M; \\ 0, &{} \text { if } 4 \mid M. \end{array}\right. } \\ m_3&= m(\mathbb Z [\omega ],\mathcal O ;\mathcal O ^\times ) = {\left\{ \begin{array}{ll} \displaystyle {\prod _{p \mid D}}\left( 1-\biggl (\displaystyle {\frac{-3}{p}}\biggr )\right) \displaystyle {\prod _{p \mid M}}\left( 1+\biggl (\displaystyle {\frac{-3}{p}}\biggr )\right) , &{} \text { if }9 \not \mid M; \\ 0, &{} \text { if }9 \mid M; \end{array}\right. } \end{aligned}$$

Example 39.4.21

Suppose $D=6$ and $M=1$, so we are in the setting of Then $m_2=m_3=2$ and

$$\begin{aligned} 2g-2 = \frac{\phi (6)}{6} - 1 - \frac{4}{3} = \frac{1}{3} = -2 \end{aligned}$$

so $g=0$; this confirms that $X^1(\mathcal O )$ has signature (0; 2, 2, 3, 3) as in 37.9.10.

5 Exercises

• 1. Let F be the function field of a curve X over $\mathbb F _q$ of genus g. Let B be a quaternion algebra over F.

  • (a) Let $v \in {{\,\mathrm{Pl}\,}}F$ be place that is split in B. Let ${{{\texttt {\textit{S}}}}}=\{v\}$, let $R=R_{({{{\texttt {\textit{S}}}}})}$, and let $\mathcal O \subseteq B$ be an R-order. Let $\mathcal T$ be the Bruhat–Tits tree associated to $B_v \simeq {{\,\mathrm{M}\,}}_2(F_v)$. Via the embedding $\iota :B \hookrightarrow B_v$, show that the group $\Gamma ^1(\mathcal O ) = {{\,\mathrm{P\!}\,}}\iota (\mathcal O ^1) \circlearrowright \mathcal T$ acts on $\mathcal T$ by left multiplication as a discrete group acting properly.

  • (b) Continuing as in (a), compute the measure of $\Gamma ^1(\mathcal O ) \backslash \mathcal T$ using the methods of section 39.3.

• 2. Generalizing the previous exercise, let F be a global field, let B be a quaternion algebra over F, let ${{{\texttt {\textit{S}}}}}$ be an eligible set and suppose that B is ${{{\texttt {\textit{S}}}}}$-indefinite. Let $R=R_{({{{\texttt {\textit{S}}}}})}$ and let $\mathcal O \subseteq B$ be an R-order. Define a symmetric space $\mathcal H$ on which ${{\,\mathrm{P\!}\,}}\mathcal O ^1$ acts as a discrete group acting properly, and compute the measure of $\Gamma ^1(\mathcal O ) \backslash \mathcal H$.