2.1 Admissible groups

Let k be a number field, and consider an absolutely simple algebraic k-group $\mathbf G$ such that

(2.1) $$ \begin{align} \mathbf G(k \otimes_{\mathbb Q} \mathbb R) \cong \operatorname{\mathrm{Sp}}(n,1) \times K \end{align} $$

for some compact group K. Then $\mathbf G$ is simply connected of type $\mathrm {C}_{n+1}$ , and k is totally real. Moreover, we can fix an embedding $k \subset \mathbb R$ so that $\mathbf G(\mathbb R)$ identifies with $\operatorname {\mathrm {Sp}}(n,1)$ . It follows from the classification of simple algebraic group (see [Reference Tits36] and Remark 2.2 below) that $\mathbf G$ is isomorphic over k to some unitary group $\operatorname {\mathrm {\mathbf {U}}}(V, h)$ , where

• ○ D is a quaternion algebra over k, with the standard involution $x \mapsto \overline {x}$ .

• ○ V is a right D-vector space.

• ○ h is a nondegenerate hermitian form on V (sesquilinear with respect to the standard involution).

Such a k-group $\mathbf G = \operatorname {\mathrm {\mathbf {U}}}(V, h)$ satisfying equation (2.1) will be called admissible, and in this case, we shall use the same terminology for the Hermitian space $(V, h)$ . We call D the defining algebra of $\mathbf G$ . Facts concerning quaternion algebras will be recalled along the lines; we refer to [Reference Vignéras38] or [Reference Maclachlan and Reid19, Ch. 2 and 6].

2.2 Admissible defining algebras

We denote by $\mathcal {V}_k = \mathcal {V}_k^{\infty } \cup \mathcal {V}_k^{\mathrm {f}}$ the set of (infinite or finite) places of k, and, for any $v \in \mathcal {V}_k$ , by $D_v$ the quaternion algebra $D_{k_v} = D \otimes _k k_v$ . The algebra D is completely determined by the set of places $v \in \mathcal {V}_k$ where it ramifies (i.e., for which $D_v$ is a division algebra), and the set of such places is of even (finite) cardinality. There is no other obstruction to the existence of a quaternion algebra D with prescribed localizations $\left \{ D_v \, |\, v \in V \right \}$ ; see [Reference Maclachlan and Reid19, Sect. 7.3].

Let $\mathbf G = \operatorname {\mathrm {\mathbf {U}}}(V, h)$ as above, with V over D. The following (well-known) result appears as a special case of Lemma 5.2 below. Recall that $D_v$ is said split if it is isomorphic to $M_2(k_v)$ , and this happens exactly when $D_v$ is not ramified.

A pair $(k, D)$ with $k \subset \mathbb R$ a totally real number field and D a quaternion algebra over k will be called admissible if D satisfies the necessary condition of Corollary 2.5. More simply, we say that ‘D is admissible’.

2.3 The classification of lattices

Let $\mathbf G$ be an admissible k-group, and let $\mathcal O_k$ denote the ring of integers in k. By the Theorem of Borel and Harish-Chandra, any subgroup $\Gamma \subset \mathbf G(\mathbb R)$ that is commensurable with $\mathbf G(\mathcal O_k)$ (for some embedding $\mathbf G \subset \operatorname {\mathrm {GL}}_N$ ) is a lattice in $\operatorname {\mathrm {Sp}}(n,1)$ . Such a subgroup is called arithmetic. The work of Margulis [Reference Margulis20] has shown that superrigidity for $\operatorname {\mathrm {Sp}}(n,1)$ (later proved by Gromov and Schoen [Reference Gromov and Schoen13] in the non-Archimedean case and Corlette [Reference Corlette6] in the Archimedean) implies the arithmeticity of any lattice in $\operatorname {\mathrm {Sp}}(n,1)$ , i.e., any lattice in $\operatorname {\mathrm {Sp}}(n,1)$ can be constructed as an arithmetic subgroup, as above. A pair $(k, \mathbf G)$ with $\mathbf G$ admissible determines exactly one commensurability class of lattices in $\operatorname {\mathrm {Sp}}(n,1)$ (up to conjugacy); see [Reference Prasad and Rapinchuk29, Prop. 2.5]. We will say that the lattices $\Gamma $ in such a commensurability class are defined over k. Moreover, with Corollary 2.4, we see that the defining k-algebra D of $\mathbf G$ is an invariant of the commensurability class. We take over the terminology and say that that D is the defining algebra of $\Gamma $ . Conversely, by Proposition 2.6, the pair $(k,D)$ uniquely determines the commensurability class.

We will describe in Section 5 a concrete way to construct arithmetic subgroups in $\operatorname {\mathrm {Sp}}(n,1)$ .

3.1 Parahoric subgroups

For any $v \in \mathcal {V}_k^{\mathrm {f}}$ , the group $\mathbf G(k_v)$ acts on its associated Bruhat–Tits building $X_v$ . A parahoric subgroup $P_v \subset \mathbf G(k_v)$ is by definition a stabilizer of a facet of $X_v$ (note that we are working with $\mathbf G$ simply connected). Maximal parahoric subgroups correspond to stabilizers of vertices on $X_v$ . If $\Delta _v$ denotes the affine root system of $\mathbf G(k_v)$ , then the conjugacy classes of parahoric subgroups $P_v \subset \mathbf G(k_v)$ are in correspondence with the subsets $\Theta _v \subsetneq \Delta _v$ ; then $\Theta _v$ is called the type of $P_v$ . The correspondence preserves the inclusion, and thus, maximal subgroups have types that omit exactly one element in $\Delta _v$ .

Assume first that $\mathbf G_{k_v}$ is split. Then its affine root system is given by the following local Dynkin diagram:

(3.1)

The parahoric subgroups of maximal volume in $\mathbf G(k_v)$ are those that are hyperspecial, i.e., of type $\Delta _v \smallsetminus \left \{\alpha \right \}$ , where $\alpha $ is any of the two hyperspecial vertices (labelled ‘hs’ in equation (3.1)).

There is exactly one nonsplit form of $\mathbf G_{k_v}$ of type $\mathrm {C}_{n+1}$ , and it splits over the maximal unramified extension $\widehat {k}_v/k_v$ . If $\mathbf G_{k_v}$ is not split, it corresponds to the local type named $^2\mathrm {C}_{n+1}$ in [Reference Tits37, Sect. 4.3]; we have reproduced in Table 3 the corresponding local indices (their description depends on the parity of n). It is a general fact, proved in [Reference Borel and Prasad4, App. A], that parahoric subgroups of maximal volume in $\mathbf G(k_v)$ are those of type $\Delta _v \smallsetminus \left \{ \alpha \right \}$ , where $\alpha $ is very special (see loc. cit. for the definition). In our case, the very special vertex in $\Delta _v$ is $\alpha _0$ for n even, respectively, $\alpha _1$ for n odd (as shown in Table 3; note that $\alpha _1$ is defined only for n odd). Thus, for $s = n \mod 2$ , a parahoric subgroup of $\mathbf G(k_v)$ is of maximal volume exactly when it is of type $\Theta _v = \Delta _v \smallsetminus \left \{ \alpha _s \right \}$ .

Table 3 Local indices for the type $^2\mathrm {C}_{n+1}$ .

3.2 The group scheme structure

Let $\mathbf G(k_v)$ as above, split or not. Each parahoric subgroup $P_v \subset \mathbf G(k_v)$ can be written as $P_v = \mathscr G(\mathfrak o_v)$ for some canonical smooth group scheme $\mathscr G$ over $\mathfrak o_v$ . Let $\mathfrak f_v$ be the residue field of $k_v$ . For a fixed $P_v$ , following the notation of [Reference Prasad28], we denote by $\overline {\mathrm M}_{v}$ the maximal reductive quotient of the $\mathfrak f_v$ -group $\mathscr G_{f_v}$ obtained from $\mathscr G$ by base change; $\overline {\mathrm M}_{v}$ is connected. The structure of $\overline {\mathrm M}_{v}$ can be obtained from the type of $P_v$ by using the procedure explained in [Reference Tits37, Sect. 3.5.2]. If $\mathbf G_{k_v}$ is split, then $P_v$ is hyperspecial if and only if $\overline {\mathrm M}_{v}$ is simple of type $\mathrm {C}_{n+1}$ (i.e., of the same type as $\mathbf G_{k_v}$ ). In this case, we will write $\overline {\mathrm M}_{v} = \overline {\mathscr M}_{\!\! v}$ .

3.3 The Galois cohomology action

For $\mathbf G$ admissible over k, we let $\mathbf C$ denote its center and $\overline {\mathbf G} = \mathbf G/\mathbf C$ its adjoint quotient. For any field extension $K/k$ , the group $\overline {\mathbf G}(K)$ identifies with the group of inner K-automorphisms of $\mathbf G$ . We denote by $\delta $ the connecting map in the Galois cohomology exact sequence:

(3.2) $$ \begin{align} 1 \to \mathbf C(K) \to \mathbf G(K) \stackrel{\pi}{\to} \overline{\mathbf G}(K) \stackrel{\delta}{\to} H^1(K, \mathbf C) \to H^1(K, \mathbf G). \end{align} $$

For $K = k_v$ non-Archimedean, this provides an action of $H^1(k_v, \mathbf C)$ on the local Dynkin diagram $\Delta _v$ (see [Reference Borel and Prasad4, Sect. 2.8]); the action respects the symmetries of $\Delta _v$ so that $H^1(k_v, \mathbf C) \to \operatorname {\mathrm {Aut}}(\Delta _v)$ . Note in particular that, for $\mathbf G_{k_v}$ nonsplit of type $\mathrm {C}_{n+1}$ , we have $\operatorname {\mathrm {Aut}}(\Delta _v) = 1$ . In the split case, there is exactly one nontrivial symmetry. We denote by $\xi $ the induced ‘global’ map $H^1(k, \mathbf C) \to \prod _{v\in \mathcal {V}_k^{\mathrm {f}}} \operatorname {\mathrm {Aut}}(\Delta _v)$ (the image actually lies in the direct product).

Of particular interest to us is the subgroup $\delta (\overline {\mathbf G}(k))' = \delta (\overline {\mathbf G}(k) \cap \pi (\mathbf G(\mathbb R)))$ (the notation follows [Reference Borel and Prasad4, Sect. 2.8]; recall that we have fixed an inclusion $k \subset \mathbb R$ ). We will make use of the following alternative description.

Lemma 3.1. The group $\delta (\overline {\mathbf G}(k))'$ coincides with the kernel of the diagonal map

$$ \begin{align*} H^1(k, \mathbf C) \to \prod_{v \in \mathcal{V}_k^{\infty}} H^1(k_v, \mathbf C). \end{align*} $$

4.1 Principal arithmetic subgroups

For $\mathbf G$ an admissible k-group, we will denote by $P = (P_v)_{v \in \mathcal {V}_k^{\mathrm {f}}}$ a collection of parahoric subgroups $P_v \subset \mathbf G(k_v)$ ( $v \in \mathcal {V}_k^{\mathrm {f}}$ ). Such a collection is called coherent if the product $\prod _{v\in \mathcal {V}_k^{\mathrm {f}}} P_v$ is open in the group $\mathbf G(\mathbb A_{\mathrm {f}})$ , where $\mathbb A_{\mathrm {f}}$ denotes the finite adèles of k. This condition implies that $P_v$ is hyperspecial for all but finitely many $v \in \mathcal {V}_k^{\mathrm {f}}$ . Moreover, one has that the subgroup $\Lambda _P = \mathbf G(k) \cap \prod _{v \in \mathcal {V}_k^{\mathrm {f}}} P_v$ is an arithmetic subgroup of $\mathbf G(k)$ , called principal.

4.2 The normalized Haar measure

The covolume of the principal arithmetic subgroup $\Lambda _P \subset \mathbf G(k)$ can be computed with Prasad’s volume formula [Reference Prasad28] in terms of the volumes of the parahoric subgroups $P_v$ ( $v\in \mathcal {V}_k^{\mathrm {f}}$ ). In the notation of loc. cit., our situation corresponds to the case $\mathbf G_S = \mathbf G(\mathbb R)$ (i.e., S contains a single infinite place corresponding to the inclusion $k \subset \mathbb R$ ). We write $\mu = \mu _S$ for the normalization of the Haar measure on $\mathbf G(\mathbb R)$ used in [Reference Prasad28, Sect. 3.6]. Then for the Euler–Poincaré characteristic (in the sense of C.T.C. Wall) of $\Gamma \subset \mathbf G(\mathbb R)$ , one has $|\chi (\Gamma )| = |\chi (X_u)|\, \mu (\Gamma \backslash \mathbf G(\mathbb R))$ , where $X_u$ is the compact dual symmetric space associated with $\mathbf H^n_{\mathbb H}$ (see [Reference Borel and Prasad4, §4]). Explicitly, $X_u = \operatorname {\mathrm {Sp}}(n+1)/(\operatorname {\mathrm {Sp}}(n) \times \operatorname {\mathrm {Sp}}(1))$ is the quaternionic projective space $\mathbb H P^n$ , for which $\chi (X_u) = n+1$ . Moreover, since the symmetric space of $\operatorname {\mathrm {Sp}}(n,1)$ has dimension $4n$ , it follows from [Reference Serre33, Prop. 23] that $\chi (\Gamma )$ is positive. Thus,

(4.1) $$ \begin{align} \chi(\Gamma) &= (n+1) \cdot \mu(\Gamma\backslash \mathbf G(\mathbb R)). \end{align} $$

4.3 Prasad’s volume formula

To state the volume formula for $\Lambda _P \subset \mathbf G(k)$ in an explicit way, we need to introduce some notation; we mostly follow [Reference Prasad28]. The symbol $\mathscr D_k$ denotes the absolute value of the discriminant of k, and we write $d = [k:\mathbb Q]$ for the degree. For each $v \in \mathcal {V}_k^{\mathrm {f}}$ , let $\mathfrak f_v$ be the residue field of $k_v$ , and let $q_v$ be the cardinality of $\mathfrak f_v$ . For each parahoric subgroup $P_v$ , the reductive $\mathfrak f_v$ -group $\overline {\mathrm M}_{v}$ is defined in Section 3.2. The reductive $\mathfrak f_v$ -group corresponding to a hyperspecial parahoric subgroup in the split form of $\mathbf G$ is denoted by $\overline {\mathscr M}_{\!\! v}$ . For all but finitely many $v \in \mathcal {V}_k^{\mathrm {f}}$ , we have that $P_v$ is hyperspecial and thus $\overline {\mathrm M}_{v} \cong \overline {\mathscr M}_{\!\! v}$ . In our situation, Prasad’s volume formula [Reference Prasad28, Theorem 3.7] takes the following form (note that in our case $\ell = k$ since $\mathbf G$ is of type $\mathrm {C}$ and thus has no outer symmetries):

(4.2) $$ \begin{align} \mu(\Lambda_P\backslash \mathbf G(\mathbb R)) &= \mathscr D_k^{\dim \mathbf G /2} \left( \prod_{j = 1}^{n+1} \frac{(2j-1)!}{(2 \pi)^{2j}} \right)^d \mathscr{E}(P), \end{align} $$

where the ‘Euler product’ $\mathscr {E}(P)$ is given by

(4.3) $$ \begin{align} \mathscr{E}(P) &= \prod_{v\in\mathcal{V}_k^{\mathrm{f}}} \frac{q_v^{(\dim \overline{\mathrm M}_{v} + \dim \overline{\mathscr M}_{\!\! v})/2}}{|\overline{\mathrm M}_{v}(\mathfrak f_v)|}. \end{align} $$

4.4 The Euler product and zeta functions

Let T be the finite set of places $v \in \mathcal {V}_k^{\mathrm {f}}$ such that $P_v$ is not hyperspecial. For $v \not \in T$ , we have that $\overline {\mathrm M}_{v} \cong \overline {\mathscr M}_{\!\! v}$ , which is an $\mathfrak f_v$ -split simple group of type $\mathrm {C}_{n+1}$ , for which $|\overline {\mathscr M}_{\!\! v}(\mathfrak f_v)| = q_v^{(n+1)^2} \prod _{j=1}^{n+1}(q_v^{2j} -1)$ (see [Reference Ono24, Tab. 1]), and $\dim \overline {\mathscr M}_{\!\! v} = \dim \mathbf G = (n+1)(2n+3)$ . By a direct computation, we may rewrite $\mathscr {E}(P)$ as the following:

(4.4) $$ \begin{align} \mathscr{E}(P) &= \prod_{v \in T} e'(P_v) \prod_{v \in \mathcal{V}_k^{\mathrm{f}}} \nonumber \frac{q_v^{\dim \overline{\mathscr M}_{\!\! v}}}{|\overline{\mathscr M}_{\!\! v}(\mathfrak f_v)|}\\ &= \prod_{v \in T} e'(P_v) \prod_{v \in \mathcal{V}_k^{\mathrm{f}}} \prod_{j=1}^{n+1} \frac{1}{1 - q_v^{-2j}} \nonumber \\ &= \prod_{v \in T} e'(P_v) \prod_{j=1}^{n+1} \zeta_k(2j), \end{align} $$

where $\zeta _k$ is the Dedekind zeta function of k, and the correcting factors $e'(P_v)$ (so called ‘lambda factors’ in [Reference Belolipetsky and Emery3]) are given by

(4.5) $$ \begin{align} e'(P_v) &= q_v^{(\dim \overline{\mathrm M}_{v} - \dim \overline{\mathscr M}_{\!\! v})/2} \frac{|\overline{\mathscr M}_{\!\! v}(\mathfrak f_v)|}{|\overline{\mathrm M}_{v}(\mathfrak f_v)|}. \end{align} $$

Putting together equations (4.1), (4.2) and (4.4), we can finally write (where the second line is obtained from the functional equation of $\zeta _k$ ; see [Reference Neukirch22, Ch.VII (5.11)]):

(4.6) $$ \begin{align} \chi(\Lambda_P) &= (n+1) \mathscr D_k^{\dim \mathbf G /2} \prod_{v \in T} e'(P_v) \prod_{j=1}^{n+1} \left(\frac{(2j-1)!}{(2 \pi)^{2j}}\right)^d \zeta_k(2j) \end{align} $$

(4.7) $$ \begin{align} &= (n+1) \prod_{v \in T} e'(P_v) \prod_{j=1}^{n+1} 2^{-d} |\zeta_k(1-2j)|.\qquad\end{align} $$

4.5 The nonsplit local factors

We compute in the following lemma the local factors $e'(P_v)$ of interest to us.

Lemma 4.1. Suppose that $\mathbf G_{k_v}$ is nonsplit, and let $P^t_v \subset \mathbf G(k_v)$ be a special parahoric subgroup of type $\Delta _v \smallsetminus \left \{ \alpha _t \right \}$ (assuming n odd if $t=1$ ). Then

(4.8) $$ \begin{align} e'(P^0_v) &= \prod_{j=1}^{n+1} (q_v^j + (-1)^j); \end{align} $$

(4.9) $$ \begin{align} e'(P^1_v) &= \frac{\prod_{j=1}^{2m} (q_v^{2j}-1)}{\prod_{j=1}^m(q_v^{4j} -1)}, \end{align} $$

where $n+1 = 2m$ in the latter.

5.1 Hermitian lattices over orders

Let us fix an order $\mathcal O_D$ in an admissible quaternion k-algebra D, and consider the right $\mathcal O_D$ -module $L = \mathcal O_D^{n+1}$ . We set $\mathcal O_{D, R} = \mathcal O_D \otimes _{\mathcal O_k} R$ . Then $L_R = L \otimes _{\mathcal O_k} R$ is a right $\mathcal O_{D,R}$ -module. Consider a Hermitian form h on L, described as follows:

(5.1) $$ \begin{align} h(x, y) &= \sum_{i=0}^n a_i \overline{x}_i y_i, \end{align} $$

where the coefficients $a_i$ are taken in $\mathcal O_k$ . Note that the standard involution on D restricts to $\mathcal O_D$ (use the trace) so that $(L, h)$ is indeed a Hermitian module in the sense of [Reference Scharlau32, Ch. 7] (and so is $(L_R, h)$ for any ring extension R of $\mathcal O_k$ ). We write $V = L_k = D^{n+1}$ , and we will assume that $(V, h)$ is admissible. We say in this case that the module $(L, h)$ itself is admissible.

Recall that by definition the Hermitian module $(L_R, h)$ is regular (or nonsingular) if the map $\phi _h: x \mapsto h(x, \cdot )$ induces an isomorphism of $\mathcal O_{D,R}$ -modules from $L_R$ onto its dual module $(L_R)^*$ , seen as a right module via $f \alpha = \overline {\alpha } f$ (see [Reference Scharlau32, Sect. 7.1]). When R is a field this is equivalent to $(L_R, h)$ being nondegenerate, i.e., $(L_R)^{\perp } = 0$ . For more general R, we will need the following result.

5.2 A key lemma

For $(L, h)$ and $(V, h)$ as above, consider the stabilizer of L in $\mathbf G(k) = \operatorname {\mathrm {\mathbf {U}}}(V, h)$ , i.e., the subgroup

(5.2) $$ \begin{align} \operatorname{\mathrm{\mathbf{U}}}(L, h) &= \left\{ g \in \operatorname{\mathrm{\mathbf{U}}}(V, h) \,|\, g L = L \right\}. \end{align} $$

This is an arithmetic subgroup of $\mathbf G(k)$ . More generally, we will denote by $\operatorname {\mathrm {\mathbf {U}}}(L_R, h) \subset \mathbf G(K)$ the stabilizer of $L_R \subset V_K$ . The following lemma is the key result that will be used in Section 5.3.

Proof. We adapt the discussion from [Reference Scharlau32, pp. 361-362] (which considers skew-Hermitian spaces, only over fields) to our setting. First, we may fix an identification $\mathcal O_{D, R} = M_2(R)$ ; the standard involution is then given by

(5.3) $$ \begin{align} \overline{\left(\begin{array}{cc}a & b \\ c & d \end{array}\right)} &= \left(\begin{array}{cc}d & -b \\ -c & a \end{array}\right). \end{align} $$

Let $e_1 = \left (\begin {array}{cc}1 & 0 \\ 0 & 0 \end {array}\right )$ and $e_2 = \left (\begin {array}{cc}0 & 0 \\ 0 & 1 \end {array}\right )$ so that $V_K$ has the following splitting: $V_K = V_K e_1 \oplus V_K e_2$ . We set $V_1 = V_K e_1$ . As for loc. cit., we obtain from h a bilinear form $b_h$ on $V_1$ determined by

(5.4) $$ \begin{align} h(x e_1, y e_1) = \left(\begin{array}{cc}0 & 0 \\ b_h(x e_1, y e_1) & 0 \end{array}\right), \end{align} $$

and in our case, $b_h$ is easily seen to be antisymmetric. Since $(L_R, h)$ is a Hermitian module, the form $b_h$ actually restricts to a bilinear (antisymmetric) form on the R-lattice $L_1 = L_R e_1$ of $V_1$ . Note that $L_1$ is free over R of rank $2n+2$ . If $f \in L_1^*$ , then we can extend f to $L_R$ by setting for any $x \in L_R$ :

(5.5) $$ \begin{align} \tilde{f}(x) = f(x e_1) + f(x e e_1) e, \end{align} $$

where $e = \left (\begin {array}{cc}0 & 1 \\ 1 & 0 \end {array}\right )$ . One computes that this is indeed an extension, which is actually $\mathcal O_{D,R}$ -linear, i.e., $\tilde {f} \in (L_R)^*$ . In particular, we have that the symplectic module $(L_1, b_h)$ is regular, as $(L_R, h)$ itself is assumed to be regular. Since by assumption R is a principal ideal domain (PID), we can now deduce that $(L_1, h)$ is a orthogonal sum of hyperbolic modules (see, for instance, [Reference Kirkwood and McDonald16, Prop. 2.1]), and thus, its isometry group is isomorphic to $\operatorname {\mathrm {Sp}}_{2n+2}(R)$ .

An analogous formula to equation (5.5) can be used to extend any isometry $\sigma $ of $(L_1, h)$ to an isometry $\tilde {\sigma } \in \operatorname {\mathrm {\mathbf {U}}}(L_R, h)$ (see [Reference Scharlau32, p.362]). This shows that $g \mapsto g|_{L_1}$ yields an isomorphism from $\operatorname {\mathrm {\mathbf {U}}}(L_R, h)$ to $\operatorname {\mathrm {Sp}}_{2n+2}(R)$ . The same construction with $R = K$ thus provides the isomorphism $\phi $ in the statement.

5.3 The local structure of lattice stabilizers

Let again $(L, h)$ denote an admissible lattice over $\mathcal O_D$ , with D defined over the number field k and $\mathcal O_D \subset D$ an order. The following (nonstandard) terminology will be convenient for us.

For each finite place $v \in \mathcal {V}_k^{\mathrm {f}}$ , we shall abbreviate the notation from Section 5.1 (with $R = \mathfrak o_v$ ) as follows: $L_v = L_{\mathfrak o_v}$ . As above, $\mathcal {R}$ denotes the set of finite places $v \in \mathcal {V}_k^{\mathrm {f}}$ , where $D_v$ ramifies, and $\mathbf G = \operatorname {\mathrm {\mathbf {U}}}(V, h)$ .

We now turn our attention to the case of places where D ramifies.

Proof. Let $\widehat {k}_v$ be the maximal unramified extension of $k_v$ , with ring of integers $\widehat {\mathfrak o}_v$ . Let $\mathcal O_{D,v} = \mathcal O_{D, \mathfrak o_v}$ and $\widehat {\mathcal O} = \mathcal O_{D,v} \otimes _{\mathfrak o_v} \widehat {\mathfrak o}_v$ . The latter in an order in $\widehat {D}_v = D_v \otimes \widehat {k}_v$ , and we consider a maximal order $\mathcal O' \subset \widehat {D}_v$ containing $\widehat {\mathcal O}$ . That is,

(5.6) $$ \begin{align} \mathcal O_{D,v} \subset \widehat{\mathcal O} \subset \mathcal O'\,. \end{align} $$

Thus $\mathcal O' \cap D_v$ is an order in $D_v$ , which equals $\mathcal O_{D,v}$ since the latter is maximal. Note that $\widehat {D}_v$ is split (see [Reference Maclachlan and Reid19, Theorem 2.6.5]).

From the inclusions (5.6), we may interpret the subgroup $P_v = \operatorname {\mathrm {\mathbf {U}}}(L_v, h) \cong \operatorname {\mathrm {\mathbf {U}}}(\mathcal O_{D,v}^{n+1}, h)$ of $\mathbf G(k_v)$ as a subgroup of the matrix group $P_v' = \operatorname {\mathrm {\mathbf {U}}}((\mathcal O')^{n+1}, h)$ . The latter is a hyperspecial parahoric of $\mathbf G(\widehat {k}_v)$ by Lemma 5.5. From the equality $\mathcal O' \cap D_v = \mathcal O_{D,v}$ , we deduce $P_v' \cap \mathbf G(k_v) = P_v$ . But in view of the local indices in Table 3, this means that $P_v$ is a special parahoric subgroup of type $\Delta _v \smallsetminus \left \{ \alpha _0 \right \}$ (since $\alpha _0$ is the unique affine root in $\Delta _v$ that appears as the restriction of hyperspecial roots of $\mathbf G(\widehat {k}_v)$ ).

5.4 The volume formula for the maximal type

Lattices of maximal type are particularly interesting because of the following result. Recall that $q_v$ denotes the cardinality of the residue field of $k_v$ (for $v \in \mathcal {V}_k^{\mathrm {f}}$ ). See Definition 5.3 for ‘maximal type’.

Theorem 5.7. Let $(L, h)$ be an admissible Hermitian $\mathcal O_D$ -lattice of maximal type. Then $\operatorname {\mathrm {\mathbf {U}}}(L, h)$ is a principal arithmetic subgroup of $\mathbf G(k) = \operatorname {\mathrm {\mathbf {U}}}(V, h)$ , and

(5.7) $$ \begin{align} \chi(\operatorname{\mathrm{\mathbf{U}}}(L, h)) &= (n+1) \prod_{j=1}^{n+1} \left( \frac{\zeta_k(1-2j)}{2^{\scriptscriptstyle [k:\mathbb Q]}} \prod_{v \in \mathcal{R}} q_v^j + (-1)^j \right), \end{align} $$

where $\mathcal {R}$ is the set of finite places where D ramifies.

Proof. We can write $L = V_k \cap \prod _{v \in \mathcal {V}_k^{\mathrm {f}}} L_v$ , from which we obtain (for $\mathbf G(k)$ diagonally embedded in $\prod _v \mathbf G(k_v)$ ):

$$ \begin{align*} \operatorname{\mathrm{\mathbf{U}}}(L, h) &= \mathbf G(k) \cap \prod_{v \in \mathcal{V}_k^{\mathrm{f}}} \operatorname{\mathrm{\mathbf{U}}}(L_v, h). \end{align*} $$

With Lemmas 5.5 and 5.6, this shows that $\operatorname {\mathrm {\mathbf {U}}}(L, h)$ is principal, and the formula for $\chi (\operatorname {\mathrm {\mathbf {U}}}(L, h))$ is deduced from equation (4.7) and Lemma 4.1.

We emphasize the special case of the standard Hermitian form over the Hurwitz integers in the next corollary. It implies the formula in equation (1.2) since we have $\chi (\Gamma ^0_n) = \chi (\operatorname {\mathrm {Sp}}(n,1, \mathscr H)) / 2$ by construction.

Corollary 5.8. The arithmetic subgroup $\operatorname {\mathrm {Sp}}(n,1, \mathscr H)$ is principal, and

(5.8) $$ \begin{align} \chi(\operatorname{\mathrm{Sp}}(n,1, \mathscr H)) &= (n+1) \prod_{j=1}^{n+1} \frac{2^j + (-1)^j}{4j} |B_{2j}|, \end{align} $$

where $B_m$ is the m-th Bernoulli number.

Proof. We have that $\mathscr H$ is a maximal order in $D = \mathscr H \otimes \mathbb Q = \left ( \frac {-1, -1}{\mathbb Q} \right )$ , and the latter is the quaternion $\mathbb Q$ -algebra that ramifies exactly at $p = 2$ and $p = \infty $ (see [Reference Vignéras38, p.79]). By Lemma 5.1, it is clear that $(\mathscr H^{n+1}, h)$ , with h given in equation (1.1), is of maximal type. Thus, we can apply the theorem, and the formula in equation (5.8) follows immediately from the known expression:

$$ \begin{align*} \zeta(-m) &= (-1)^m \frac{B_{m+1}}{m+1}.\\[-38pt] \end{align*} $$

5.5 The covolume of $\Delta _n$

Let $L = \mathscr I^{n+1}$ , where $\mathscr I$ is the icosian ring. The Hermitian form equation (1.4) has been chosen so that $(L, h)$ is of maximal type. By definition, $\Delta _n = \operatorname {\mathrm {\mathbf {U}}}(L, h)$ . The formula in equation (1.5) for $\chi (\Delta _n)$ is thus an immediate consequence of Theorem 5.7 since in this case $\mathcal {R} = \emptyset $ (see [Reference Vignéras38, p.150]).

6.1 Normalizers of minimal covolume

Let $\Gamma \subset \operatorname {\mathrm {Sp}}(n,1)$ be a maximal lattice, i.e., maximal with respect to inclusion as in Section 2.3. We have $\Gamma \subset \mathbf G(\overline {k} \cap \mathbb R)$ for some admissible k-group $\mathbf G$ , but the stricter inclusion $\Gamma \subset \mathbf G(k)$ does not hold in general. By [Reference Borel and Prasad4, Prop. 1.4], we have that $\Gamma $ is a normalizer $N_{\mathbf G(\mathbb R)}(\Lambda _P)$ , where $\Lambda _P \subset \mathbf G(k)$ is a principal arithmetic subgroup.

Let $P = (P_v)$ be a coherent collection such that any parahoric subgroup $P_v$ is of maximal volume. Then $\Lambda _P$ is of minimal covolume among arithmetic lattices contained in $\mathbf G(k)$ . It is a priori not clear—but turns out to be true—that the normalizer $N_{\mathbf G(\mathbb R)}(\Lambda _P)$ for such a choice of P is of minimal covolume in its commensurability class in $\mathbf G(\mathbb R)$ . This can be proved in the same way as in [Reference Belolipetsky and Emery3, Sect. 4.3] (see also [Reference Emery7, Sect. 12.3] for a more detailed exposition). We state the result in the following lemma. In the rest of this section, $\mathcal {R}$ denotes the set of finite places where $\mathbf G$ does not split (equivalently, where its defining algebra ramifies).

Recall that for $v \in \mathcal {R}$ a parahoric subgroup $P_v \subset \mathbf G(k_v)$ is special of maximal volume exactly when it has type $\Delta _v \smallsetminus \left \{ \alpha _s \right \}$ , where $s = n \mod 2$ . Using Lemmas 5.5 and 5.6, we thus have:

6.2 The index computation

We will need to estimate the index $[\Gamma : \Lambda _P]$ for $\Gamma = N_{\mathbf G(\mathbb R)}(\Lambda _P)$ of minimal covolume in its commensurability class. For this, we state the following lemma, which considers a slightly more general situation. The symbol $h_k$ denotes the class number of k, and $U_k$ (respectively, $U_k^+$ ) are the units (respectively, totally positive units) in $\mathcal O_k$ .

Lemma 6.3. Let $P = (P_v)$ such that $P_v$ is hyperspecial for any $v \in \mathcal {V}_k^{\mathrm {f}} \smallsetminus \mathcal {R}$ . Then

$$ \begin{align*} [\Gamma : \Lambda_P] &\le 2^{\#\mathcal{R}} \cdot h_k \cdot |U^+_{k}/U_k^2|. \end{align*} $$

Proof. We assume the notation of Section 3.3; in particular, $\mathbf C$ is the center of $\mathbf G$ . We set $A = \delta (\overline {\mathbf G}(k))'$ . Let $\Theta = (\Theta _v)_{v\in \mathcal {V}_k^{\mathrm {f}}}$ be the type of the coherent collection P. From the assumption, it follows that none of the types $\Theta _v$ has symmetries, and thus, the stabilizer of $\Theta $ in A equals the kernel $A_{\xi }$ of $\xi $ . By [Reference Borel and Prasad4, Prop. 2.9], we thus have the exact sequence

(6.1) $$ \begin{align} 1 \to \mathbf C(\mathbb R)/(\mathbf C(k) \cap \Lambda_P) \to \Gamma/\Lambda_P \to A_{\xi} \to 1. \end{align} $$

In our case, $\mathbf C = \mu _2$ , and it follows that the left part vanishes. Hence, $[\Gamma :\Lambda _P] = |A_{\xi }|$ . We now use the identification $H^1(k, \mu _2) = k^{\times }/(k^{\times })^2$ . For $S \subset \mathcal {V}_k^{\mathrm {f}}$ any finite set of places, we define

$$ \begin{align*} k_{2, S} = \left\{\left. x \in k^{\times} \; \right| \; v(x) \in 2{\mathbb Z} \;\; \forall v \in \mathcal{V}_k^{\mathrm{f}} \smallsetminus S \right\}, \end{align*} $$

and $k_2 = k_{2, \emptyset }$ . Since $\operatorname {\mathrm {Aut}}(\Delta _v)$ is trivial for any $v \in \mathcal {R}$ , it follows from [Reference Borel and Prasad4, Prop. 2.7] that $H^1(k, \mathbf C)_{\xi } = k_{2, \mathcal {R}}/(k^{\times })^2$ . See [Reference Borel and Prasad4] for the definitions. For $k_{2, S}^+ \subset k_{2, S}$ denoting the subgroup consisting of totally positive elements, we conclude from Lemma 3.1 that $A_{\xi } = k_{2, \mathcal {R}}^+/(k^{\times })^2$ . This group contains $k_2^+/(k^{\times })^2$ with index at most $2^{\#\mathcal {R}}$ , and the order of the latter can be bounded by $h_k \cdot |U^+_{k}/U_k^2|$ by using the same argument as in the proof of [Reference Borel and Prasad4, Prop. 0.12].

The following is obtained as a corollary of the proof.

6.3 The nonuniform lattices $\Gamma _n^s$

By definition, $\Gamma ^0_n = \left < g, \operatorname {\mathrm {Sp}}(n,1, \mathscr H) \right>$ with g of order $2$ that normalizes $\operatorname {\mathrm {Sp}}(n,1, \mathscr H)$ . Let D be the defining algebra of $\Gamma _n^0$ (i.e., $D = \mathscr H \otimes \mathbb Q$ ). It ramifies precisely at $p=2$ and $p=\infty $ . Thus, we can apply Corollary 6.4, and it follows that $\Gamma _n^0$ coincides with the normalizer of $\operatorname {\mathrm {Sp}}(n,1, \mathscr H)$ in $\operatorname {\mathrm {Sp}}(n,1)$ . For n even, Corollary 6.2 then shows that $\Gamma ^0_n$ is of minimal covolume in its commensurability class. On the other hand, Rohlfs’ criterion [Reference Rohlfs30, Satz 3.5] shows that $\Gamma _n^0$ is maximal (w.r.t. inclusion) for any $n>1$ .

For $n>1$ odd, we construct $\Gamma _n^1$ as follows. Let $\mathbf G$ be the $\mathbb Q$ -group that contains $\operatorname {\mathrm {Sp}}(n,1, \mathscr H)$ . We choose a coherent collection $P = (P_v)$ with $P_v$ hyperspecial for $v \neq 2$ , and $P_v = P^1_v$ of type $\Delta _v \smallsetminus \left \{ \alpha _1 \right \}$ for $v = 2$ . Let $\Gamma _n^1 = N_{\mathbf G(\mathbb R)}(\Lambda _P)$ . By Lemma 6.1, it is of minimal covolume in its commensurability class. Moreover, by Corollary 6.4, we have $[\Gamma _n^1:\Lambda _P] = 2$ . In particular,

(6.2) $$ \begin{align} \chi(\Gamma_n^1) &= \frac{e'(P^1_2)}{e'(P^0_2)} \chi(\Gamma_n^0), \end{align} $$

from which we obtain the formula in equation (1.3) with Lemma 4.1.

The following proposition now implies—up to the uniqueness—Theorems 1 and 2.

Proof. We have seen that $\Gamma ^s_n$ is of minimal covolume in its commensurability class, so it suffices to prove the commensurability. By Section 2, $\Gamma $ nonuniform is constructed as an arithmetic subgroup of $\mathbf G = \operatorname {\mathrm {\mathbf {U}}}(V, h)$ for $(V, h)$ admissible, and V a vector space over a quaternion $\mathbb Q$ -algebra D. Let $\mathcal {R}$ be the set of places v such that $D_v$ ramifies. Being of minimal covolume, we may write $\Gamma = N_{\mathbf G(\mathbb R)}(\Lambda _P)$ , with $P_v$ hyperspecial unless $v \in \mathcal {R}$ (by Lemma 6.1). Moreover, for $v \in \mathcal {R}$ the subgroup $P_v$ is of maximal volume and thus of type $\Delta \smallsetminus \left \{ \alpha _s \right \}$ by Section 3.1. By Lemma 6.3 (with $k=\mathbb Q$ ), we have $[\Gamma :\Lambda _P] \le 2^{\#\mathcal {R}}$ . Together with equation (4.7), this gives

(6.3) $$ \begin{align} \chi(\Gamma) &\ge \frac{\chi(\Lambda_P)}{[\Gamma : \Lambda_P]} \nonumber\\ &\ge (n+1) \prod_{v \in \mathcal{R}} \frac{e'(P_v)}{2} \prod_{j=1}^{n+1} \frac{\zeta(1-2j)}{2}. \end{align} $$

Only the middle factor in equation (6.3) depends on the choice of $\Lambda _P$ , and it takes the smallest possible value for $\mathcal {R} = \left \{ 2 \right \}$ (note that $\mathcal {R} = \emptyset $ cannot appear here; see Section 2.2). But in that case, this lower bound is precisely $\chi (\Gamma ^s_n)$ , whence $\chi (\Gamma ) = \chi (\Gamma ^s_n)$ (by minimality of $\chi (\Gamma ))$ . Since D is now ramified exactly at $p = 2$ and $p=\infty $ , it has same defining algebra as $\Gamma ^0_n$ , and the commensurability follows from Section 2.3.

6.4 The minimality of $\chi (\Delta _n)$

We now discuss the uniform case. Recall that the covolume of $\Delta _n$ has been discussed in Section 5.5. Lemma 6.3 (with $k = \mathbb Q(\sqrt {5})$ and $\mathcal {R} = \emptyset $ ) implies that $\Delta _n$ coincides with its own normalizer in $\operatorname {\mathrm {Sp}}(n,1)$ . Corollary 6.2 thus implies that $\Delta _n$ is of minimal covolume in its commensurability class. The following proposition proves the first statement in Theorem 3. In the proof, we omit details that should be clear from the proof of Proposition 6.5.

Proof. Let $\Gamma $ be a uniform lattice of minimal covolume, and let k be its field of definition. Then k is totally real, of degree $d \ge 2$ . Assume first that $k= \mathbb Q(\sqrt {5})$ , and let D be the defining algebra of $\Gamma $ . It is clear that any nontrivial local factor $e'(P_v)$ would contribute to increase the volume formula, and this shows that D does not ramify at any finite place; this implies that $\Gamma $ is commensurable to $\Delta _n$ . Thus, it suffices to prove that $k = \mathbb Q(\sqrt {5})$ , i.e., that $d = 2$ and $\mathscr D_k = 5$ (recall that $\mathscr D_k$ denotes the discriminant in absolute value).

Let $\mathbf G$ be the algebraic k-group used to construct the arithmetic subgroup $\Gamma $ , and let us write $\Gamma = N_{\mathbf G(\mathbb R)}(\Lambda _P)$ with $P = (P_v)$ a coherent collection of parahoric subgroups $P_v \subset \mathbf G(k_v)$ (of maximal volume). Combining equation (4.6) and Lemma 6.3, we find

$$ \begin{align*} \chi(\Gamma) &\ge \frac{(n+1) \mathscr D_k^{\dim\mathbf G/2}}{2^{\#\mathcal{R}}\; h_k \; |U_k^+/U_k^2|} \; C(n)^d \; \prod_{v \in \mathcal{R}} e'(P_v) \prod_{j=1}^{n+1} \zeta_k(2j), \end{align*} $$

with

(6.4) $$ \begin{align} C(n) &= \prod_{j=1}^{n+1} \frac{(2j-1)!}{(2\pi)^{2j}}. \end{align} $$

We clearly have $\zeta _k(2j)> 1$ , and $|U_k^+/U_k^2| \le 2^{d-1}$ (by Dirichlet’s unit theorem). Moreover, $e'(P_v)> 2$ for any $v \in \mathcal {R}$ , so that the factor $2^{-\#\mathcal {R}}$ is compensated by the product of those local factors. We can use the bound $h_k \le 16 \left ( \pi /12 \right )^d \mathscr D_k$ (see [Reference Belolipetsky and Emery3, Sect. 7.2]: the argument given there for a non-totally real field $\ell $ provides the same bound for k; see [Reference Emery7, Sect. 15.2] for details). This gives

$$ \begin{align*} \chi(\Gamma) &\ge \frac{(n+1)}{16\; \mathscr D_k\; 2^{d-1}}\; \left( \frac{12}{\pi} \right)^d \mathscr D_k^{\dim\mathbf G/2} \; C(n)^d. \end{align*} $$

On the other hand, we have

$$ \begin{align*} \chi(\Delta_n) &= (n+1)\, 5^{\dim\mathbf G/2}\, C(n)^2 \prod_{j=1}^{n+1} \zeta_{\mathbb Q(\sqrt{5})}(2j) \\ &< 1.2 \cdot (n+1)\, 5^{\dim\mathbf G/2}\, C(n)^2. \end{align*} $$

Here we bound the product of zeta functions by the value $1.2$ by adapting the proof in [Reference Belolipetsky2, p.760: proof of ( $\ast$ )] as follows (see loc. cit. for details):

$$ \begin{align*} \prod_{j=1}^{n+1} \zeta_{\mathbb Q(\sqrt{5})}(2j) &< \zeta_{\mathbb Q(\sqrt{5})}(2)\, \zeta_{\mathbb Q(\sqrt{5})}(4)\, \zeta_{\mathbb Q(\sqrt{5})}(6)\, \prod_{j=4}^{\infty} \left(1 + \frac{2}{2^{2j}}\right)^2 \\ &< \zeta_{\mathbb Q(\sqrt{5})}(2) \,\zeta_{\mathbb Q(\sqrt{5})}(4) \,\zeta_{\mathbb Q(\sqrt{5})}(6) \,e^{1/48}, \end{align*} $$

and we evaluate this last bound with Pari/GP [25].

For the quotient, this gives

(6.5) $$ \begin{align} \frac{\chi(\Gamma)}{\chi(\Delta_n)} &> \frac{1}{39\, \mathscr D_k} \left( \frac{12}{\pi} \right)^d \left( \frac{\mathscr D_k}{5} \right)^{\frac{(n+1)(2n+3)}{2}} \left(\frac{C(n)}{2} \right)^{d-2}. \end{align} $$

Let us write $f(n, d, \mathscr D_k)$ for the bound on the right-hand side. We have to show that $f(n, d, \mathscr D_k) \ge 1$ unless $d=2$ and $\mathscr D_k=5$ .

The constant $C(n)/2$ is larger than $1$ for $n\ge 13$ , and grows monotone from that point. Thus, $f(n, d, \mathscr D_k) \ge f(13, d, \mathscr D_k)$ , and it suffices to consider the range $n \in \left \{ 2, \dots , 13 \right \}$ . For $d=2$ , the smallest discriminant after $5$ is $\mathscr D_k = 8$ , and numerical evaluation shows that $f(n, 2, 8)> 1$ for all $n \le 13$ . This shows $d> 2$ . For k totally real of degree $d = 3$ , the lowest discriminant is $\mathscr D_k = 49$ . Again, we check that $f(n, 3, 49)> 1$ . And similarly with $d = 4$ and $\mathscr D_k \ge 725$ .

It remains to exclude $d \ge 5$ . In that case, we use the following bound due to Odlyzko (see [Reference Odlyzko23, Tab. 4]): $\mathscr D_k> (6.5)^d$ . Then equation (6.5) transforms into

(6.6) $$ \begin{align} \frac{\chi(\Gamma)}{\chi(\Delta_n)} &> \frac{1}{39 \cdot 5} \left( \frac{12}{\pi} \right)^2 \left( \frac{(6.5)^2}{5} \right)^{\delta(n)} a(n)^{d-2}, \end{align} $$

where $a(n) = (6/\pi ) C(n) (6.5)^{\delta (n)}$ and $\delta (n) = \dim \mathbf G/2 -1$ . We check that $a(n)> 1$ for all $n \in \left \{ 2, \dots , 13 \right \}$ . The product preceding $a(n)$ in equation (6.6) is also easily seen to be (much) larger than $1$ . This finishes the proof.

6.5 The proof of Corollary 1

We have that each of $\chi (\Delta _n)$ , $\chi (\Gamma ^s_n)$ and their quotients contains a factor $C(n)$ (which is given in (6.4)); see equation (4.6). For n large enough, it is easily seen that this factor grows faster than (say) $(2n+1)!$ , i.e., it grows superexponentially. This implies immediately that $\chi (\Delta _n)$ and $\chi (\Gamma ^s_n)$ grow superexponentially, as their remaining factors also increase with n. Moreover, the other factors appearing in $\chi (\Gamma ^s_n)$ grow at most exponentially so that the $\chi (\Delta _n)/\chi (\Gamma ^s_n)$ has a superexponentially growth as well.

7.1 The surjectivity of the adjoint map

We start by proving the following auxiliary result.

7.2 Counting the conjugacy classes

Let $\Gamma \subset \operatorname {\mathrm {Sp}}(n,1)$ be a nonuniform (respectively, uniform) lattice that realizes the smallest covolume. Then by Section 6, we have $\Gamma = N_{\operatorname {\mathrm {Sp}}(n,1)}(\Lambda _P)$ , where $P = (P_v)$ is a coherent collection of parahoric subgroups $P_v \subset \mathbf G(k_v)$ of maximal volume for each $v \in \mathcal {V}_k^{\mathrm {f}}$ , and $\mathbf G$ is precisely the admissible k-group that determines $\Gamma ^s_n$ (resp. $\Delta _n$ ). For $v \in \mathcal {R}$ , this determines the type $\Theta _v$ of $P_v$ uniquely (see Lemma 6.1), and for $v \notin \mathcal {R}$ we have that $\Theta _v$ is one of the two conjugate hyperspecial types. These two hyperspecial types are conjugate by $\overline {\mathbf G}(k_v)$ . Up to $\overline {\mathbf G}(k)$ -conjugacy, the number of principal arithmetic subgroups $\Lambda _P$ with such a type $\Theta = (\Theta _v)$ is given by the order of following class group (see, for instance, [Reference Belolipetsky and Emery3, Sect. 6.2]):

(7.1) $$ \begin{align} \mathfrak{C}_P = \frac{\prod'_{v\in\mathcal{V}_k^{\mathrm{f}}} H^1(k_v, \mathbf C)}{\delta(\overline{\mathbf G}(k)) \prod_{v \in \mathcal{V}_k^{\mathrm{f}}} \delta(\overline{P}_v)}, \end{align} $$

where $\overline {P}_v \subset \overline {\mathbf G}(k_v)$ is the stabilizer of $P_v$ , and in the numerator, $\prod '$ denotes the restricted product with respect to the collection of subgroups $\delta (\overline {P}_v)$ .

By Lemma 7.1, this order also gives an upper bound on the number of $\mathbf G(\mathbb R)$ -conjugacy classes of $\Lambda _P$ (note that $k \subset \mathbb R$ ). Thus, the uniqueness in Theorems 1–3 follows immediately from the following.

Proof. We have $H^1(k_v, \mathbf C) = k_v^{\times }/(k_v^{\times })^2$ . The subgroup $\delta (\overline {P}_v) \subset H^1(k_v, \mathbf C)$ equals the type stabilizer $H^1(k_v, \mathbf C)_{\Theta _v}$ . If $v \in \mathcal {R}$ , then $\operatorname {\mathrm {Aut}}(\Delta _v) = 1$ (see Table 3) so that this stabilizer is trivially the whole $H^1(k_v, \mathbf C)$ . For $v \notin \mathcal {R}$ , we have $\delta (\overline {P}_v) = \mathfrak o^{\times }_v (k_v^{\times })^2/(k_v^{\times })^2$ by [Reference Borel and Prasad4, Prop. 2.7]. Thus, $\mathfrak {C}_P$ is a quotient of

(7.2) $$ \begin{align} \mathfrak{C}_P' &= \frac{\prod'_{v\in\mathcal{V}_k^{\mathrm{f}}}k_v^{\times}/(k_v^{\times})^2}{\delta(\overline{\mathbf G}(k)) \prod_{v \in \mathcal{V}_k^{\mathrm{f}}} \mathfrak o^{\times}_v (k_v^{\times})^2/(k_v^{\times})^2}. \end{align} $$

Now by Lemma 7.1 and the proof of Lemma 3.1, we have that $\delta (\overline {\mathbf G}(k)) = \delta (\overline {\mathbf G}(k))' = k^{(+)}/(k^{\times })^2$ , where

(7.3) $$ \begin{align} k^{(+)}=\left\{x \in k^{\times} \;|\; x_v> 0 \quad \forall\, v \in \mathcal{V}_k^{\infty} \right\}. \end{align} $$

From equation (7.2), we obtain an isomorphism

$$ \begin{align*} \mathfrak{C}_P' &\cong \mathcal{J}_k/\left(k^{(+)} \mathcal{J}_k^{\infty} \mathcal{J}_k^2 \right), \end{align*} $$

where $\mathcal {J}_k$ is the group of finite idèles of k, and $\mathcal {J}_k^{\infty } \subset \mathcal {J}_k$ its subgroup consisting of integral idèles. For both $k = \mathbb Q$ and $k = \mathbb Q(\sqrt {5})$ , the unit group $U_k$ contains a representative of each class of $k^{\times }/k^{(+)}$ . Thus,

$$ \begin{align*} k^{(+)} \mathcal{J}_k^{\infty} &= k^{(+)} U_k \mathcal{J}_k^{\infty} \\ &= k^{\times} \mathcal{J}_k^{\infty} \end{align*} $$

so that

$$ \begin{align*} \mathfrak{C}_P' &\cong \mathcal{J}_k/\left(k^{\times} \mathcal{J}_k^{\infty} \mathcal{J}_k^2 \right). \end{align*} $$

But the latter is a quotient of the class group of k (see [Reference Platonov and Rapinchuck27, Sect. 1.2.1]), which is trivial here.