$\DeclareMathOperator\SL{SL}$Let $B$ be a Borel subgroup (upper triangular matrices), and let $\Gamma := \langle \sigma\rangle$ be the group generated by a (hyperbolic) element of $\SL_2/\mathbb{Q}_p$ of infinite order. Let $\mathcal{C}$ be the big Zariski site of $\mathbb{Q}_p$, and let $\Gamma\setminus \SL_2/B$ be the functor which sends a $\mathbb{Q}_p$ scheme $X$ to the set of double cosets $\Gamma\setminus \SL_2(\mathcal{O}(X))/B(\mathcal{O}(X))$, where $\mathcal{O}(X)$ is the ring of global sections on $X$.

Denote by $(\Gamma\setminus \SL_2/B)^+$ the sheafification of the functor $\Gamma\setminus \SL_2/B$.

My question is: is $(\Gamma\setminus \SL_2/B)^+$ representable by a scheme/algebraic space?

My trivial thoughts are: the sheafification of $\SL_2/B$ is the functor $\mathbb{P}^1_{\mathbb{Q}_p}$. If there is a way of realizing $(\Gamma\setminus \SL_2/B)^+$ as the sheafification of $(\Gamma\setminus \mathbb{P}^1_{\mathbb{Q}_p})$, then perhaps it is possible to realize this as a corollary to one of the theorems regarding the representability of the quotient of a scheme by a group.

$\DeclareMathOperator\SL{SL}$Let $B$ be a Borel subgroup (upper triangular matrices), and let $\Gamma := \langle \sigma\rangle$ be the group generated by a (hyperbolic) element of $\SL_2/\mathbb{Q}_p$ of infinite order. Let $\mathcal{C}$ be the big Zariski site of $\mathbb{Q}_p$, and let $\Gamma\backslash{\SL_2}/B$ be the functor which sends a $\mathbb{Q}_p$ scheme $X$ to the set of double cosets $\Gamma\backslash{\SL_2(\mathcal{O}(X))}/B(\mathcal{O}(X))$, where $\mathcal{O}(X)$ is the ring of global sections on $X$.

Denote by $(\Gamma\backslash{\SL_2}/B)^+$ the sheafification of the functor $\Gamma\backslash{\SL_2}/B$.

My question is: is $(\Gamma\backslash{\SL_2}/B)^+$ representable by a scheme/algebraic space?

My trivial thoughts are: the sheafification of $\SL_2/B$ is the functor $\mathbb{P}^1_{\mathbb{Q}_p}$. If there is a way of realizing $(\Gamma\backslash{\SL_2}/B)^+$ as the sheafification of $(\Gamma\backslash\mathbb{P}^1_{\mathbb{Q}_p})$, then perhaps it is possible to realize this as a corollary to one of the theorems regarding the representability of the quotient of a scheme by a group.

Let $B$ be a Borel subgroup (upper triangular matrices), and let $\Gamma := \langle \sigma\rangle$ be the group generated by a (hyperbolic) element of $SL_2/\mathbb{Q}_p$ of infinite order. Let $\mathcal{C}$ be the big Zariski site of $\mathbb{Q}_p$, and let $\Gamma\setminus SL_2/B$ be the functor which sends a $\mathbb{Q}_p$ scheme $X$ to the set of double cosets $\Gamma\setminus SL_2(\mathcal{O}(X))/B(\mathcal{O}(X))$, where $\mathcal{O}(X)$ is the ring of global sections on $X$.

Denote by $(\Gamma\setminus SL_2/B)^+$ the sheafification of the functor $\Gamma\setminus SL_2/B$.

My question is: is $(\Gamma\setminus SL_2/B)^+$ representable by a scheme/algebraic space?

My trivial thoughts are: the sheafification of $SL_2/B$ is the functor $\mathbb{P}^1_{\mathbb{Q}_p}$. If there is a way of realizing $(\Gamma\setminus SL_2/B)^+$ as the sheafification of $(\Gamma\setminus \mathbb{P}^1_{\mathbb{Q}_p})$, then perhaps it is possible to realize this as a corollary to one of the theorems regarding the representability of the quotient of a scheme by a group.

$\DeclareMathOperator\SL{SL}$Let $B$ be a Borel subgroup (upper triangular matrices), and let $\Gamma := \langle \sigma\rangle$ be the group generated by a (hyperbolic) element of $\SL_2/\mathbb{Q}_p$ of infinite order. Let $\mathcal{C}$ be the big Zariski site of $\mathbb{Q}_p$, and let $\Gamma\setminus \SL_2/B$ be the functor which sends a $\mathbb{Q}_p$ scheme $X$ to the set of double cosets $\Gamma\setminus \SL_2(\mathcal{O}(X))/B(\mathcal{O}(X))$, where $\mathcal{O}(X)$ is the ring of global sections on $X$.

Denote by $(\Gamma\setminus \SL_2/B)^+$ the sheafification of the functor $\Gamma\setminus \SL_2/B$.

My question is: is $(\Gamma\setminus \SL_2/B)^+$ representable by a scheme/algebraic space?

My trivial thoughts are: the sheafification of $\SL_2/B$ is the functor $\mathbb{P}^1_{\mathbb{Q}_p}$. If there is a way of realizing $(\Gamma\setminus \SL_2/B)^+$ as the sheafification of $(\Gamma\setminus \mathbb{P}^1_{\mathbb{Q}_p})$, then perhaps it is possible to realize this as a corollary to one of the theorems regarding the representability of the quotient of a scheme by a group.

Let $B$ be a Borel subgroup (upper triangular matrices), and let $\Gamma := \langle \sigma\rangle$ be the group generated by a (hyperbolic) element of $SL^2$ of infinite order. Let $\mathcal{C}$ be the big Zariski site of $\mathbb{Q}_p$, and let $\Gamma\setminus SL_2/B$ be the functor which sends a $\mathbb{Q}_p$ scheme $X$ to the set of double cosets $\Gamma\setminus SL_2(\mathcal{O}(X))/B(\mathcal{O}(X))$, where $\mathcal{O}(X)$ is the ring of global sections on $X$.

Denote by $(\Gamma\setminus SL_2/B)^+$ the sheafification of the functor $\Gamma\setminus SL_2/B$.

My question is: is $(\Gamma\setminus SL_2/B)^+$ representable by a scheme/algebraic space?

My trivial thoughts are: the sheafification of $SL_2/B$ is the functor $\mathbb{P}^1_{\mathbb{Q}_p}$. If there is a way of realizing $(\Gamma\setminus SL_2/B)^+$ as the sheafification of $(\Gamma\setminus \mathbb{P}^1_{\mathbb{Q}_p})$, then perhaps it is possible to realize this as a corollary to one of the theorems regarding the representability of the quotient of a scheme by a group.

Let $B$ be a Borel subgroup (upper triangular matrices), and let $\Gamma := \langle \sigma\rangle$ be the group generated by a (hyperbolic) element of $SL_2/\mathbb{Q}_p$ of infinite order. Let $\mathcal{C}$ be the big Zariski site of $\mathbb{Q}_p$, and let $\Gamma\setminus SL_2/B$ be the functor which sends a $\mathbb{Q}_p$ scheme $X$ to the set of double cosets $\Gamma\setminus SL_2(\mathcal{O}(X))/B(\mathcal{O}(X))$, where $\mathcal{O}(X)$ is the ring of global sections on $X$.

Denote by $(\Gamma\setminus SL_2/B)^+$ the sheafification of the functor $\Gamma\setminus SL_2/B$.

My question is: is $(\Gamma\setminus SL_2/B)^+$ representable by a scheme/algebraic space?

My trivial thoughts are: the sheafification of $SL_2/B$ is the functor $\mathbb{P}^1_{\mathbb{Q}_p}$. If there is a way of realizing $(\Gamma\setminus SL_2/B)^+$ as the sheafification of $(\Gamma\setminus \mathbb{P}^1_{\mathbb{Q}_p})$, then perhaps it is possible to realize this as a corollary to one of the theorems regarding the representability of the quotient of a scheme by a group.