The Weil uniformization theorem says that if $k$ is an algebraically closed field, $G$ a reductive group, and $C$ a curve, we have an isomorphism of stacks $Bun_G(C)\cong G(F_C)\backslash G(\mathbb{A}_C)/G(\mathbb{O}_C)$, where $\mathbb{A}$ and $\mathbb{O}$ are the rings of adeles and integral adeles respectively. This is the version of the theorem stated in this paper. However, I have seen other versions of this theorem which require e.g. that $G$ is affine, that $C$ is smooth and projective, and so on. I can't seem to find the original or definitive paper on this result, and I'm interested in generalizing it to non-algebraically-closed fields, so I would like to read that. If anyone knows either the original paper or one which describes this generalization, I would appreciate the reference.
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asked Apr 10, 2023 at 22:15
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$\begingroup$ Have you looked at Corollary 3.39 and the following discussion in the paper you cite? $\endgroup$– S. Carnahan ♦Commented Apr 11, 2023 at 2:13
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$\begingroup$ I took a look at it in a bit more detail, and it does seem to prove the result over arbitrary fields. I'm still interested in finding Weil's original paper, however. $\endgroup$– Doron Grossman-NaplesCommented Apr 11, 2023 at 2:54
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