$p$-adic equidistribution of modular geodesics and of Heegner points on Shimura curves

In [Duk88], W. Duke proves the equidistribution, inside the complex points of the modular curve endowed with the hyperbolic measure, of arithmetic objects attached to orders in quadratic extensions of $\mathbb{Q}$. The nature of these objects depends on the behavior at infinity of the extension. For quadratic imaginary fields these objects are Galois orbits of CM points and in the case of real quadratic fields his equidistribution theorem concerns packets of closed geodesics.

Let $p$ be a rational prime. Since CM points are defined over number fields, their Galois orbits can be naturally embedded inside the $p$-adic space of $\mathbb{C}_{p}$-points of the modular curve. In [HMRL20] and [HMRL20], the authors studied the distribution of such Galois orbits obtaining $p$-adic analogues to Duke’s theorem in the case of imaginary quadratic fields. In contrast with Duke’s result, different limit measures can appear, depending on the discriminants of the orders involved in the sequence of CM points.

The aim of this article is to state and prove a $p$-adic version of Duke’s theorem on the equidistribution of packets of closed geodesics. Also, we prove a $p$-adic version of the equidistribution of Galois orbits of Heegner points on Shimura curves, when the prime $p$ divides the discriminant of the underlying quaternion algebra. The complex version of the latter result is another extension of Duke’s theorem for which we refer to section 6 in [HM06] or to [Zha05].

We recall Duke’s result following section 2 of [ELMV12]. Let $\mathcal{H}$ be the complex upper half plane and denote by $Y_{0}$ the modular curve of level $1$. Then we have an identification between the quotient space $\mathrm{PSL}_{2}(\mathbb{Z})\backslash\mathcal{H}$ and $Y_{0}(\mathbb{C})$. Moreover, the quotient space $\mathrm{PSL}_{2}(\mathbb{Z})\backslash\mathrm{PGL}_{2}(\mathbb{R})^{+}$ is identified with $T^{1}(Y_{0}(\mathbb{C}))$, the unit tangent bundle of $Y_{0}(\mathbb{C})$. Here and elsewhere, the superscript ⁺ indicates that we consider matrices of positive determinant. Suppose that $\tau$ is a real quadratic irrational. We can view $\tau$ as an element in the boundary of $\mathcal{H}$ and we can consider the geodesic $\gamma_{\tau}$ in $\mathcal{H}$ connecting $\tau$ and its Galois conjugate. The projection of $\gamma_{\tau}$ to ${\mathrm{PSL}_{2}(\mathbb{Z})}\backslash{\mathcal{H}}$ is a closed geodesic. We lift this closed geodesic to a compact geodesic orbit $\widetilde{\gamma}_{\tau}$ in $T^{1}(Y_{0}(\mathbb{C}))$ that depends only on the $\mathrm{PSL}_{2}(\mathbb{Z})$-orbit of $\tau$. Also, we denote by $\mathcal{O}_{\tau}$ the ring of matrices in $M_{2}(\mathbb{Z})$ having the column $(\tau\,\,\,1)^{t}$ as an eigenvector. For $K$ a real quadratic field, we denote by $RQ(K)$ the set of $\tau$ as before such that $\mathcal{O}_{\tau}$ is isomorphic to the maximal order $\mathcal{O}_{K}$. The set $\mathrm{PSL}_{2}(\mathbb{Z})\backslash RQ(K)$ is finite and its cardinality equals the class number $h_{K}$ of $K$. We put $\mathcal{G}_{K}=\bigcup_{\tau}\widetilde{\gamma}_{\tau}$, with $\tau$ running through representatives of $\mathrm{PSL}_{2}(\mathbb{Z})\backslash RQ(K)$. Define by $\mu_{K}$ the unique probability measure in $T^{1}(Y_{0}(\mathbb{C}))$ supported on $\mathcal{G}_{K}$ that is invariant under the geodesic flow. Endow $T^{1}(Y_{0}(\mathbb{C}))$ with the unique probability measure $\mu_{L}$ coming from a Haar measure on $\mathrm{PGL}_{2}^{+}(\mathbb{R})$ and a counting measure on $\mathrm{PSL}_{2}(\mathbb{Z})$.

We proceed to explain our main result on geodesics. As we have seen, in the complex case, geodesics appear when dealing with real quadratic fields or in other words when the infinite place is split in the extension. Following this point of view, we exchange the role of $\infty$ and $p$ and we focus on quadratic extensions such that $p$ splits. The role played by $\mathrm{PSL}_{2}(\mathbb{Z})$ and $\mathrm{PGL}_{2}^{+}(\mathbb{R})$ will be held by $\Gamma^{+}\colonequals\mathrm{PGL}^{+}_{2}(\mathbb{Z}[1/p])$ and $\mathrm{PGL}_{2}(\mathbb{R}\times\mathbb{Q}_{p})$, respectively. Let $\mathcal{H}^{\prime}$ be the set of points $\tau$ in $\mathcal{H}$ such that $\tau$ generates an imaginary quadratic extension over $\mathbb{Q}$ in which $p$ splits. The quotient $\mathchoice{\text{\lower 2.15277pt\hbox{$\Gamma^{+}$}\big{\backslash}\raise 2.15277pt\hbox{$\mathcal{H}^{\prime}$}}}{\Gamma^{+}\,\backslash\,\mathcal{H}^{\prime}}{\Gamma^{+}\,\backslash\,\mathcal{H}^{\prime}}{\Gamma^{+}\,\backslash\,\mathcal{H}^{\prime}}$ was studied by Ihara who showed that it is related to the special fiber of the modular curve $X_{0}(p)$.

The $p$-adic covering of $Y_{0}(\mathbb{C})$ that we consider is the locally compact space

The natural projection $T^{(p)}(Y_{0}(\mathbb{C}))\to Y_{0}(\mathbb{C})$ is given by taking a right quotient by $\mathrm{PGL}_{2}(\mathbb{Z}_{p})$ (see section 5). Let $\mathcal{T}_{p}$ denote the Bruhat-Tits tree of $\mathrm{PGL}_{2}(\mathbb{Q}_{p})$. After identifying the boundary of the tree $\mathcal{T}_{p}$ with $\mathbb{P}^{1}(\mathbb{Q}_{p})$, the splitness condition on $p$ allows us to see $\mathcal{H}^{\prime}$ as boundary elements of $\mathcal{T}_{p}$. For $\tau$ in $\mathcal{H}^{\prime}$, we define in Section 5 a cycle $\Delta_{\tau}$ inside the space $T^{(p)}(Y_{0}(\mathbb{C}))$ than depends only on the $\Gamma^{+}$-orbit of $\tau$. The projection of $\Delta_{\tau}$ on $\Gamma^{+}\backslash\mathcal{H}$ is just $\Gamma^{+}\tau$. The fiber over $\Gamma^{+}\tau$ under the natural projection $\mathchoice{\text{\lower 2.15277pt\hbox{$\Gamma^{+}$}\big{\backslash}\raise 2.15277pt\hbox{$(\mathcal{H}^{\prime}\times\mathcal{T}_{p})$}}}{\Gamma^{+}\,\backslash\,(\mathcal{H}^{\prime}\times\mathcal{T}_{p})}{\Gamma^{+}\,\backslash\,(\mathcal{H}^{\prime}\times\mathcal{T}_{p})}{\Gamma^{+}\,\backslash\,(\mathcal{H}^{\prime}\times\mathcal{T}_{p})}\to\mathchoice{\text{\lower 2.15277pt\hbox{$\Gamma^{+}$}\big{\backslash}\raise 2.15277pt\hbox{$\mathcal{H}^{\prime}$}}}{\Gamma^{+}\,\backslash\,\mathcal{H}^{\prime}}{\Gamma^{+}\,\backslash\,\mathcal{H}^{\prime}}{\Gamma^{+}\,\backslash\,\mathcal{H}^{\prime}}$ is $\mathchoice{\text{\lower 2.15277pt\hbox{$\Gamma^{+}$}\big{\backslash}\raise 2.15277pt\hbox{$(\Gamma^{+}\tau\times\mathcal{T}_{p})$}}}{\Gamma^{+}\,\backslash\,(\Gamma^{+}\tau\times\mathcal{T}_{p})}{\Gamma^{+}\,\backslash\,(\Gamma^{+}\tau\times\mathcal{T}_{p})}{\Gamma^{+}\,\backslash\,(\Gamma^{+}\tau\times\mathcal{T}_{p})}$. This is a graph having the structure of a $(p+1)$-volcano, that can be identified with the $p$-isogeny graph attached to any elliptic curve coming from a $\mathrm{PSL}_{2}(\mathbb{Z})$-orbit in $\Gamma^{+}\tau$. The projection of $\Delta_{\tau}$ to this graph is its unique closed loop (known as its rim). Observe that there’s a bijection between $\mathchoice{\text{\lower 2.15277pt\hbox{$\Gamma^{+}$}\big{\backslash}\raise 2.15277pt\hbox{$(\Gamma^{+}\tau\times\mathcal{T}_{p})$}}}{\Gamma^{+}\,\backslash\,(\Gamma^{+}\tau\times\mathcal{T}_{p})}{\Gamma^{+}\,\backslash\,(\Gamma^{+}\tau\times\mathcal{T}_{p})}{\Gamma^{+}\,\backslash\,(\Gamma^{+}\tau\times\mathcal{T}_{p})}$ and $\{\tau\}\times\mathchoice{\text{\lower 2.15277pt\hbox{$\Gamma^{+}_{\tau}$}\big{\backslash}\raise 2.15277pt\hbox{$\mathcal{T}_{p}$}}}{\Gamma^{+}_{\tau}\,\backslash\,\mathcal{T}_{p}}{\Gamma^{+}_{\tau}\,\backslash\,\mathcal{T}_{p}}{\Gamma^{+}_{\tau}\,\backslash\,\mathcal{T}_{p}}$, where $\Gamma^{+}_{\tau}$ denotes the stabilizer of $\tau$ in $\Gamma^{+}$. The aforementioned loop is formed by taking the unique geodesic in $\mathcal{T}_{p}$ joining $\tau$ and its Galois conjugate and projecting it to $\mathchoice{\text{\lower 2.15277pt\hbox{$\Gamma^{+}_{\tau}$}\big{\backslash}\raise 2.15277pt\hbox{$\mathcal{T}_{p}$}}}{\Gamma^{+}_{\tau}\,\backslash\,\mathcal{T}_{p}}{\Gamma^{+}_{\tau}\,\backslash\,\mathcal{T}_{p}}{\Gamma^{+}_{\tau}\,\backslash\,\mathcal{T}_{p}}$. This resembles the complex case where the closed geodesics on $Y_{0}(\mathbb{C})$ are projections of geodesics in $\mathcal{H}$ joining two real quadratic numbers that are conjugated. In [BDIS02], such loops in $\mathchoice{\text{\lower 2.15277pt\hbox{$\Gamma^{+}_{\tau}$}\big{\backslash}\raise 2.15277pt\hbox{$\mathcal{T}_{p}$}}}{\Gamma^{+}_{\tau}\,\backslash\,\mathcal{T}_{p}}{\Gamma^{+}_{\tau}\,\backslash\,\mathcal{T}_{p}}{\Gamma^{+}_{\tau}\,\backslash\,\mathcal{T}_{p}}$ are called Shintani-cycles, for that reason we refer to $\Delta_{\tau}$ as the Ihara-Shintani cycle attached to $\Gamma^{+}\tau$. In this way, we think of the cycles $\Delta_{\tau}$ as the $p$-adic analogues to the compact geodesic orbits on the modular surface and the space $T^{(p)}(Y_{0}(\mathbb{C}))$ as a $p$-adic unit tangent bundle of $Y_{0}(\mathbb{C})$.

For $\tau\in\mathcal{H}^{\prime}$, we denote by $\mathcal{O}_{\tau}$ the ring formed by the elements in $M_{2}(\mathbb{Z}[1/p])$ having the column $(\tau\,\,\,1)^{t}$ as an eigenvector. Let $K$ be a quadratic imaginary field at which $p$ splits and let $\mathcal{O}$ be a $\mathbb{Z}[1/p]$-order inside $K$. We denote by $IS(\mathcal{O})$ the collection of $\Gamma^{+}$-orbits of those $\tau\in\mathcal{H}^{\prime}$ for which $\mathbb{Q}(\tau)=K$ and $\mathcal{O}_{\tau}$ is isomorphic to the order $\mathcal{O}$. The group $Pic(\mathcal{O})$ act simply transitive on $IS(\mathcal{O})$ and so this set is finite. Let $A$ be the diagonal group of $\mathrm{PGL}_{2}$. The cycle $\Delta_{\tau}$ comes equipped with a unique $A(\mathbb{Q}_{p})$-right invariant probability measure $\nu_{\tau}$. Denote by $\mu_{\mathcal{O}}$ the unique probability measure proportional to $\sum_{\Gamma^{+}\tau\in IS(\mathcal{O})}\nu_{\tau}$.

Equip $T^{(p)}(Y_{0}(\mathbb{C}))$ with the unique probability measure coming from a Haar measure on $\mathrm{PGL}_{2}(\mathbb{R}\times\mathbb{Q}_{p})$ and the counting measure in $\Gamma$. Denote this measure by $\mu$. Our $p$-adic analogue to Duke’s theorem on the equidistribution of closed geodesics is the following statement.

Now we discuss our result on Heegner points on Shimura curves. Fix $\mathcal{B}$ a non-split indefinite quaternion algebra defined over $\mathbb{Q}$ whose discriminant $N^{-}$ is divisible by $p$. Also fix $\mathcal{R}$ an Eichler-order in $\mathcal{B}$ of level $N^{+}$. Let $X$ be the Shimura curve attached to $(\mathcal{B},\mathcal{R})$ as in section 6. Let $\mathbb{Q}_{p^{2}}$ be the unique quadratic unramified extension of $\mathbb{Q}_{p}$. The compact space $X(\mathbb{Q}_{p^{2}})$ admits a uniformization by $\mathrm{PGL}_{2}(\mathbb{Q}_{p})$ due to Cerednik and Drinfeld. Pushing a Haar measure of $\mathrm{PGL}_{2}(\mathbb{Q}_{p})$ we can endow $X(\mathbb{Q}_{p^{2}})$ with a natural probability measure $\mu_{X}$. Let $\mathcal{O}$ be an order in a quadratic imaginary field in which $p$ is inert and whose discriminant is coprime to $pN^{-}N^{+}$. Let $Heeg(\mathcal{O})$ denote the collection of Heegner points in $X(\mathbb{Q}_{p^{2}})$ with endomorphism ring isomorphic to $\mathcal{O}$ defined in Section 6. We obtain the following equidistribution theorem.

A related result by Disegni [Dis22] concerns the case where the power of $p$ in the conductor of the Heegner points tends to infinity. In such a situation there is no accumulation measure on $X(\mathbb{C}_{p})$ describing the asymptotic distribution of their Galois orbits

The main tool for our purposes is Theorem 4.6 in [ELMV11]. In consequence, our strategy is to describe the previous objects and ambient spaces in an $S$-arithmetic and adelic context. In doing so we will appeal to the language of (oriented) optimal embeddings to parametrize our packets as orbits under certain Pic groups which will lead to a description of these as toric orbits inside a homogeneous space. In section 2 we cover the basics about the Bruhat-Tits tree of $\mathrm{PGL}_{2}$ making special emphasis on how quadratic algebras act on it. In section 3 we recall the language of adelic and $S$-arithmetic homogeneous spaces as presented for instance in [ELMV11]. In section 4 we elaborate on the definition of cycles $\Delta_{\psi}$ in a general setting, where $\psi$ runs over embeddings of quadratic extension to a quaternion algebra. More specifically, in section 4.1 and 4.2 we discuss the structure of Eichler orders and oriented optimal embeddings respectively. In section 4.3 we show how the group $Pic(\mathcal{O})$ acts simply and transitively on the set of oriented optimal embeddings (with respect to $\mathcal{O}$), showing that the cycles $\Delta_{\psi}$ defined in section 4.4 are finite and they can be recovered from a single oriented optimal embedding. Section 5 is a specialization of the previous sections and it is where we properly define the cycles $\Delta_{\tau}$ for $\tau$ in $\Gamma^{+}\backslash\mathcal{H}^{\prime}$ and discuss the analogies with the geodesics in Duke’s theorem. Section 6 deals with the different specialization related to Heegner points on the $p$-adic points of indefinite Shimura curves. Finally, section 7 presents Theorem 4.6 in [ELMV11]. We explain how to apply it to show the equidistribution of the objects defined in sections 5 and 6, proving in particular Theorem A and Theorem B.

I would like to thank my advisor Ricardo Menares for trusting me with this project and for his support throughout this work. I also would like to thank Philippe Michel and the Analytic Number Theory group at EPFL. I was very lucky to enjoy their hospitality during the academic period 2022-2023. This work was supported by ANID Doctorado Nacional No 21200911 and by the Bourse d’excellence de la Confédération suisse No. 2022.0414.

Our main references for this section are [Ser03], [Voi21] and [Cas14]. Let $\ell$ be a rational prime. Let $\mathcal{T}_{\ell}$ be the Bruhat-Tits tree of $\mathrm{PGL}_{2}(\mathbb{Q}_{\ell})$. Its set of vertices $\mathcal{V}(\mathcal{T}_{\ell})$ corresponds to homothety classes of $\mathbb{Z}_{\ell}$-lattices in $\mathbb{Q}_{\ell}^{2}$. The class of a lattice $\Lambda$ is denoted by $[\Lambda]$ and for $n\in\mathbb{Z}$ we set $v_{n}\colonequals[\ell^{n}\mathbb{Z}_{\ell}\times\mathbb{Z}_{\ell}]$. Two vertices are connected by an edge if they admit representatives $\Lambda$ and $\Lambda^{\prime}$ such that $\ell\Lambda\subseteq\Lambda^{\prime}\subseteq\Lambda$ and $\Lambda/\Lambda^{\prime}$ is cyclic of order $\ell$. With these definitions, $\mathcal{T}_{\ell}$ is a $(\ell+1)$-regular tree.

If $e$ is an edge connecting two vertices, the choice of one of them as its source (denoted $s(e)$) and the other one as its target (denoted $t(e)$) make $e$ an oriented edge. We denote by $\overrightarrow{\mathcal{E}}(\mathcal{T}_{\ell})$ the set of oriented edges. Let $e_{\infty}$ be the oriented edge such that $s(e_{\infty})=v_{0}$ and $t(e_{\infty})=v_{-1}$. The group $\mathrm{PGL}_{2}(\mathbb{Q}_{\ell})$ acts transitively on $\mathcal{V}(\mathcal{T}_{\ell})$ and $\overrightarrow{\mathcal{E}}(\mathcal{T}_{\ell})$. The subgroups $\mathrm{PGL}_{2}(\mathbb{Z}_{\ell})$ and

are the stabilizer of $v_{0}$ and $e_{\infty}$ respectively. Consequently, we have identifications

A path of length $n\geq 0$ in $\mathcal{T}_{\ell}$ is sequence of $n$ adjacent vertices without backtracking. Paths of length $n$ starting from $v_{0}$ are identified with $\mathbb{P}^{1}(\mathbb{Z}/\ell^{n}\mathbb{Z})$: To give such a path is equivalent to give a vertex $v$ at distance $n$ from $v_{0}$. The group $\mathrm{PGL}_{2}(\mathbb{Z}_{\ell})$ acts transitively on such vertices and one such vertex is $v_{-n}$. Its stabilizer in $\mathrm{PGL}_{2}(\mathbb{Z}_{\ell})$ is

Therefore we have $\mathrm{PGL}_{2}(\mathbb{Z}_{\ell})$-equivariant identifications between

and paths of length $n$ starting from $v_{0}$ such that the path ending at $v_{-n}$ corresponds to $\infty$. The action of $\mathrm{PGL}_{2}(\mathbb{Z}_{\ell})$ on the right hand side of (2.1) is by fractional linear transformations.

A branch in $\mathcal{T}_{\ell}$ is a sequence of adjacent vertices without backtracking. Two branches are equivalent if they differ by a finite initial sequence. An equivalence class of branches is called an end of $\mathcal{T}_{\ell}$ and we denote the set of ends by $\partial\mathcal{T}_{\ell}$. We can see $\partial\mathcal{T}_{\ell}$ as the boundary of the tree and identify it with $\mathbb{P}^{1}(\mathbb{Q}_{\ell})$: The group $PGL_{2}(\mathbb{Q}_{\ell})$ acts transitively on the set of ends and the stabilizer of the end coming from the branch $\{v_{-n}\}_{n\geq 0}$ is the subgroup $B(\mathbb{Q}_{\ell})\subseteq\mathrm{PGL}_{2}(\mathbb{Q}_{\ell})$ of upper triangular elements. We have $\mathrm{PGL}_{2}(\mathbb{Q}_{\ell})$-equivariant identifications

such that that the branch coming from $\{v_{-n}\}_{n\geq 0}$ corresponds to $\infty$. As before, under this identification the action of $\mathrm{PGL}_{2}(\mathbb{Q}_{\ell})$ on $\mathbb{P}^{1}(\mathbb{Q}_{\ell})$ corresponds to the action by fractional linear transformations.

A geodesic is the path without backtracking in the tree connecting two different points in the boundary. The geodesic joining the points $0$ and $\infty$ is given by the vertices $\{v_{n}\}_{n\in\mathbb{Z}}$ and we denote it by $\mathcal{G}$. In general $\mathcal{G}_{x}^{y}$ denotes the geodesic joining $x$ and $y$.

Let $v$ and $v^{\prime}$ be two vertices in $\mathcal{T}_{\ell}$. Being $\mathcal{T}_{\ell}$ a tree, there exists a unique finite path connecting $v$ and $v^{\prime}$. The distance $d(v,v^{\prime})$ between $v$ and $v^{\prime}$ is the length of this path. For instance, two vertices are at distance $1$ if they are connected by an edge. Let $e$ an edge and $\eta$ a path (no necessarily of finite length) in the tree. The distance $d(v,\eta)$ between $v$ and $\eta$ is $\min_{w\in\eta}d(v,w)$ and the distance $d(e,\eta)$ between $e$ and $\eta$ is $\max_{w\in e}{d(w,\eta)}$.

We denote by $A\subseteq PGL_{2}(\mathbb{Q}_{\ell})$ the subgroup of diagonal elements.

Let $\mathcal{K}\subseteq M_{2}(\mathbb{Q}_{\ell})$ be a quadratic $\mathbb{Q}_{\ell}$-algebra. We denote by $\mathcal{O}_{\mathcal{K}}$ the maximal compact and open subring of $\mathcal{K}$. The order of conductor $\ell^{n}$ is $\mathbb{Z}_{\ell}+\ell^{n}\mathcal{O}_{\mathcal{K}}$. We set $\mathfrak{m}_{\mathcal{K}}\colonequals\ell\mathcal{O}_{\mathcal{K}}$, $\mathcal{O}_{\mathcal{K}}^{(0)}=\mathcal{O}_{\mathcal{K}}^{\times}$ and for $n\geq 1$ we define $\mathcal{O}_{\mathcal{K}}^{(n)}=1+\mathfrak{m}_{\mathcal{K}}^{n}$, the $n$-th principal subgroup of units.

Fix $p$ be a rational prime. Let $\mathbb{A}$ be the ring of adeles over $\mathbb{Q}$. Let $B$ be a quaternion algebra over $\mathbb{Q}$ that splits over $p$. Denote by $\boldsymbol{G}$ the algebraic group $PB^{\times}\colonequals B^{\times}/\mathbb{G}_{m}$ of projectivized units of $B$. We use $\widetilde{\boldsymbol{G}}$ for the algebraic group $B^{1}$ of units of reduced norm $1$ in $B$.

Let $A$ be a ring. If $v$ a place of $\mathbb{Q}$, we denote by $A_{v}$ the algebra $A\otimes\mathbb{Z}_{v}$ if $v$ is finite or $A\otimes\mathbb{R}$ if $v=\infty$. Let $S$ be a finite set of places of $\mathbb{Q}$ containing the place $p$. We use the notations $\mathbb{A}^{(S)}=\prod^{\prime}_{w\not\in S}\mathbb{Q}_{w}$, $\mathbb{A}_{f}=\mathbb{A}^{(\infty)}$, $\mathbb{A}_{\#}=\mathbb{A}^{(p,\infty)}$ and $\mathbb{Q}_{S}=\prod_{v\in S}\mathbb{Q}_{v}$. Anagolously , we use $\widehat{\mathbb{Z}}$, $\widehat{\mathbb{Z}}_{\#}$ to denote the closure of $\mathbb{Z}$ in $\mathbb{A}_{f}$ and in $\mathbb{A}^{(p)}\subset\mathbb{A}_{\#}$ respectively. In the same direction, $\widehat{A}$ and $\widehat{A}_{\#}$ will denote $A\otimes\widehat{\mathbb{Z}}$ and $A\otimes\widehat{\mathbb{Z}}_{\#}$ respectively. Finally, if $t$ is an adelic element in $\mathbb{A}$, $t_{f}$, $t_{\#}$, $t_{S}$ and $t_{v}$ will denote the respective projection of $t$ in $\mathbb{A}_{f}$, $\mathbb{A}_{\#}$, $\mathbb{Q}_{S}$ and $\mathbb{Q}_{v}$. Similarly for $\mathbb{A}$-points over $\boldsymbol{G}$ and $\widetilde{\boldsymbol{G}}$.

The group $\widetilde{\boldsymbol{G}}$ satisfies the following strong approximation theorem.

In particular, if $W\subseteq\widetilde{\boldsymbol{G}}(\mathbb{A}^{(S)})$ is an open subgroup, then $\widetilde{\boldsymbol{G}}(\mathbb{A})=\widetilde{\boldsymbol{G}}(\mathbb{Q})\widetilde{\boldsymbol{G}}(\mathbb{Q}_{S})W$.

Since $\boldsymbol{G}$ has no non-trivial $\mathbb{Q}$-characters, the subgroup $\boldsymbol{G}(\mathbb{Q})$ is a lattice in $\boldsymbol{G}(\mathbb{A})$ by Theorem 5.5 in [PR94]. We will use the notation $[\boldsymbol{G}]$ for the locally compact space of finite measure $\mathchoice{\text{\lower 2.15277pt\hbox{$\boldsymbol{G}(\mathbb{Q})$}\big{\backslash}\raise 2.15277pt\hbox{$\boldsymbol{G}(\mathbb{A})$}}}{\boldsymbol{G}(\mathbb{Q})\,\backslash\,\boldsymbol{G}(\mathbb{A})}{\boldsymbol{G}(\mathbb{Q})\,\backslash\,\boldsymbol{G}(\mathbb{A})}{\boldsymbol{G}(\mathbb{Q})\,\backslash\,\boldsymbol{G}(\mathbb{A})}$. If $\boldsymbol{K}$ is a compact subgroup of $\boldsymbol{G}(\mathbb{A})$, we use $[\boldsymbol{G}]_{\boldsymbol{K}}$ to denote the locally compact space $\mathchoice{\text{\raise 2.15277pt\hbox{$[\boldsymbol{G}]$}\big{/}\lower 2.15277pt\hbox{$\boldsymbol{K}$}}}{[\boldsymbol{G}]\,/\,\boldsymbol{K}}{[\boldsymbol{G}]\,/\,\boldsymbol{K}}{[\boldsymbol{G}]\,/\,\boldsymbol{K}}$. The symbol $[\cdot]_{\boldsymbol{K}}$ denotes the natural projection $\boldsymbol{G}(\mathbb{A})\to[\boldsymbol{G}]_{\boldsymbol{K}}$ so that the class of $g\in\boldsymbol{G}(\mathbb{A})$ in $[\boldsymbol{G}]_{\boldsymbol{K}}$ is $[g]_{\boldsymbol{K}}$.

We denote by G the group $\boldsymbol{G}(\mathbb{Q}_{S})$. Assume that $\boldsymbol{K}$ is the projection in $\boldsymbol{G}(\mathbb{A})$ from an open subgroup $H\subseteq B^{\times}(\mathbb{A}^{(S)})$ satisfying the hypothesis of Corollary 3.2 and set $\Gamma=\boldsymbol{G}(\mathbb{Q})\cap\boldsymbol{K}\subseteq G$. By the conclusion of the cited corollary, given $g\in\boldsymbol{G}(\mathbb{A})$ there exists $x\in\boldsymbol{G}(\mathbb{Q})$ such that $xg_{\#}\in\boldsymbol{K}$. Like this we obtain an identification

sending $[g]_{\boldsymbol{K}}$ to the $\Gamma$-orbit of $xg_{S}$ with $x$ as before.