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Hello.

Let $K = F_q(x)$ be the rational function field and let $G = \textbf{PGL}_{2,K}$. For any finite and non empty set $S$ of valuations of $K$, we refer to the subgroup of the adelic group $G(\Bbb{A})$:

$$ G(A_S) = \prod_{p \in S} G_p(K_p) \times \prod_{p \notin S} \underline{G}_p(\mathcal{O}_p) $$

on which for any prime $p$, $K_p$ is the completion of $K$ at $p$, $G_p = G \otimes K_p$, $\mathcal{O}_p$ is the ring of integers of $K_p$ and $\underline{G}_p$ is some matrix realization of $G_p$ in $\textbf{GL} \otimes K_p$.
In particular, if $S$ contains only the single point at infinity we denote $G(\Bbb{A}_S)$ by $G(\Bbb{A}(\infty))$.
The adelic group is decomposed into double cosets:

$$ G(\Bbb{A}) = \bigcup_{i=1}^h G(\Bbb{A}(\infty)) x_i G(K) $$

where $h=h(G)$ is finite and is called the class number of $G$.

As the universal covering of $G$ -- namely $\textbf{SL}_{2,K}$ -- admits the strong approximation property, the subgroup $G(\Bbb{A}(\infty))G(K)$ is normal in $G(\Bbb{A})$ and: $h(G) = (G(\Bbb{A}):G(\Bbb{A}(\infty)) G(K))$.

Could someone please tell me what is $h(G)$ in this case ?

Thank you, rony.

Hello.

Let $K = F_q(x)$ be the rational function field and let $G = \textbf{PGL}_{2,K}$. For any finite and non empty set $S$ of valuations of $K$, we refer to the subgroup of the adelic group $G(\Bbb{A})$:

$$ G(A_S) = \prod_{p \in S} G_p(K_p) \times \prod_{p \notin S} \underline{G}_p(\mathcal{O}_p) $$

on which for any prime $p$, $K_p$ is the completion of $K$ at $p$, $G_p = G \otimes K_p$, $\mathcal{O}_p$ is the ring of integers of $K_p$ and $\underline{G}_p$ is some matrix realization of $G_p$ in $\textbf{GL} \otimes \mathcal{O}_p$.
In particular, if $S$ contains only the single point at infinity we denote $G(\Bbb{A}_S)$ by $G(\Bbb{A}(\infty))$.
The adelic group is decomposed into double cosets:

$$ G(\Bbb{A}) = \bigcup_{i=1}^h G(\Bbb{A}(\infty)) x_i G(K) $$

where $h=h(G)$ is finite and is called the class number of $G$.

As the universal covering of $G$ -- namely $\textbf{SL}_{2,K}$ -- admits the strong approximation property, the subgroup $G(\Bbb{A}(\infty))G(K)$ is normal in $G(\Bbb{A})$ and: $h(G) = (G(\Bbb{A}):G(\Bbb{A}(\infty)) G(K))$.

Could someone please tell me what is $h(G)$ in this case ?

Thank you, rony.

Hello.

Let $K = F_q(x)$ be the rational function field and let $G = \textbf{PGL}_{2,K}$. For any finite and non empty set $S$ of valuations of $K$, we refer to the subgroup of the adelic group $G(\Bbb{A})$:

$ G(\Bbb{A}_S) $ $$ = \prod_{p \in S} G_p(K_p) \times \prod_{p \notin S} \underline{G}_p(\mathcal{O}_p) $$

on which for any prime $p$, $K_p$ is the completion of $K$ at $p$, $G_p = G \otimes K_p$, $\mathcal{O}_p$ is the ring of integers of $K_p$ and $\underline{G}_p$ is some matrix realization of $G_p$ in $\textbf{GL} \otimes K_p$.
In particular, if $S$ contains only the single point at infinity we denote $G(\Bbb{A}_S)$ by $G(\Bbb{A}(\infty))$.
The adelic group is decomposed into double cosets:

$$ G(\Bbb{A}) = \bigcup_{i=1}^h G(\Bbb{A}(\infty)) x_i G(K) $$

where $h=h(G)$ is finite and is called the class number of $G$.

As the universal covering of $G$ -- namely $\textbf{SL}_{2,K}$ -- admits the strong approximation property, the subgroup $G(\Bbb{A}(\infty))G(K)$ is normal in $G(\Bbb{A})$ and: $h(G) = (G(\Bbb{A}):G(\Bbb{A}(\infty)) G(K))$.

Could someone please tell me what is $h(G)$ in this case ?

Thank you, rony.

Hello.

Let $K = F_q(x)$ be the rational function field and let $G = \textbf{PGL}_{2,K}$. For any finite and non empty set $S$ of valuations of $K$, we refer to the subgroup of the adelic group $G(\Bbb{A})$:

$$ G(A_S) = \prod_{p \in S} G_p(K_p) \times \prod_{p \notin S} \underline{G}_p(\mathcal{O}_p) $$

on which for any prime $p$, $K_p$ is the completion of $K$ at $p$, $G_p = G \otimes K_p$, $\mathcal{O}_p$ is the ring of integers of $K_p$ and $\underline{G}_p$ is some matrix realization of $G_p$ in $\textbf{GL} \otimes K_p$.
In particular, if $S$ contains only the single point at infinity we denote $G(\Bbb{A}_S)$ by $G(\Bbb{A}(\infty))$.
The adelic group is decomposed into double cosets:

$$ G(\Bbb{A}) = \bigcup_{i=1}^h G(\Bbb{A}(\infty)) x_i G(K) $$

where $h=h(G)$ is finite and is called the class number of $G$.

As the universal covering of $G$ -- namely $\textbf{SL}_{2,K}$ -- admits the strong approximation property, the subgroup $G(\Bbb{A}(\infty))G(K)$ is normal in $G(\Bbb{A})$ and: $h(G) = (G(\Bbb{A}):G(\Bbb{A}(\infty)) G(K))$.

Could someone please tell me what is $h(G)$ in this case ?

Thank you, rony.

Hello.

Let $K = F_q(x)$ be the rational function field and let $G = \textbf{PGL}_{2,K}$. For any finite and non empty set $S$ of valuations of $K$, we refer to the subgroup of the adelic group $G(\Bbb{A})$:

$$ G(\Bbb{A}_S) = \prod_{p \in S} G_p(K_p) \times \prod_{p \notin S} \underline{G}_p(\mathcal{O}_p) $$

on which for any prime $p$, $K_p$ is the completion of $K$ at $p$, $G_p = G \otimes K_p$, $\mathcal{O}_p$ is the ring of integers of $K_p$ and $\underline{G}_p$ is some matrix realization of $G_p$ in $\textbf{GL} \otimes K_p$.
In particular, if $S$ contains only the single point at infinity we denote $G(\Bbb{A}_S)$ by $G(\Bbb{A}(\infty))$.
The adelic group is decomposed into double cosets:

$$ G(\Bbb{A}) = \bigcup_{i=1}^h G(\Bbb{A}(\infty)) x_i G(K) $$

where $h=h(G)$ is finite and is called the class number of $G$.

As the universal covering of $G$ -- namely $\textbf{SL}_{2,K}$ -- admits the strong approximation property, the subgroup $G(\Bbb{A}(\infty))G(K)$ is normal in $G(\Bbb{A})$ and: $h(G) = (G(\Bbb{A}):G(\Bbb{A}(\infty)) G(K))$.

Could someone please tell me what is $h(G)$ in this case ?

Thank you, rony.

Hello.

Let $K = F_q(x)$ be the rational function field and let $G = \textbf{PGL}_{2,K}$. For any finite and non empty set $S$ of valuations of $K$, we refer to the subgroup of the adelic group $G(\Bbb{A})$:

$ G(\Bbb{A}_S) $ $$ = \prod_{p \in S} G_p(K_p) \times \prod_{p \notin S} \underline{G}_p(\mathcal{O}_p) $$

on which for any prime $p$, $K_p$ is the completion of $K$ at $p$, $G_p = G \otimes K_p$, $\mathcal{O}_p$ is the ring of integers of $K_p$ and $\underline{G}_p$ is some matrix realization of $G_p$ in $\textbf{GL} \otimes K_p$.
In particular, if $S$ contains only the single point at infinity we denote $G(\Bbb{A}_S)$ by $G(\Bbb{A}(\infty))$.
The adelic group is decomposed into double cosets:

$$ G(\Bbb{A}) = \bigcup_{i=1}^h G(\Bbb{A}(\infty)) x_i G(K) $$

where $h=h(G)$ is finite and is called the class number of $G$.

As the universal covering of $G$ -- namely $\textbf{SL}_{2,K}$ -- admits the strong approximation property, the subgroup $G(\Bbb{A}(\infty))G(K)$ is normal in $G(\Bbb{A})$ and: $h(G) = (G(\Bbb{A}):G(\Bbb{A}(\infty)) G(K))$.

Could someone please tell me what is $h(G)$ in this case ?

Thank you, rony.