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$\DeclareMathOperator\SL{SL}$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.

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    $\begingroup$ It seems like, even if the answer is yes, it probably isn't what you want to ask. To take a simpler example, remove the two fixed points of $\sigma$ from $\mathbb{P}^1$, so what is left over is a copy of $\mathbb{G}_m$ on which $\sigma$ acts freely. The quotient $\mathbb{G}_m/\langle \sigma \rangle$ wants to be an elliptic curve. (For example, that's what it is as a complex manifold, if we are working over $\mathbb{C}$.) But I don't think that is what your sheafification description gives. $\endgroup$
    – David E Speyer
    Commented Mar 24, 2023 at 13:42
  • $\begingroup$ 1/2: What I am eventually interested in is understanding the ring of global sections on a stack of the form $\mathbb{A}^1\times [\Gamma\setminus SL_2/B]$. By definition, these are the global sections of the sheaf $Hom(\mathbb{A}^1\times [\Gamma\setminus SL_2/B], \mathbb{A}^1)$. Since the map between a presheaf and a sheaf factors through its sheafification, the former is isomorphic to $Hom(\mathbb{A}^1\times (\Gamma\setminus SL_2/B)^+, \mathbb{A}^1)$. $\endgroup$
    – kindasorta
    Commented Mar 24, 2023 at 14:01
  • $\begingroup$ 2/2: I want to show that $Hom(\mathbb{A}^1\times (\Gamma\setminus SL_2/B)^+, \mathbb{A}^1)$ is isomorphic to the tensor product $Hom(\mathbb{A}^1, \mathbb{A}^1)\otimes_{\mathbb{Q}_p} Hom((\Gamma\setminus SL_2/B)^+, \mathbb{A}^1)$, but this is not true in the generality of sheaves, so I was hoping to try and show that $(\Gamma\setminus SL_2/B)^+$ is an algebraic space, so that maybe the claim I am trying to prove would have a better chance of being true in this generality. $\endgroup$
    – kindasorta
    Commented Mar 24, 2023 at 14:04
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    $\begingroup$ What happened with that earlier "answer"? Was that produced by a bot (as I suspected)? $\endgroup$
    – Jason Starr
    Commented Mar 24, 2023 at 20:33
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    $\begingroup$ Yes. How common is that on the stack exchange forum now days? $\endgroup$
    – kindasorta
    Commented Mar 24, 2023 at 20:48

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