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Over a function field of a curve $K = k(C)$, there is the Weil uniformization

$$\mathrm{Bun}_{GL_n}(C) = GL_n(K) \backslash GL_n(\mathbb{A}_K) / GL_n(\mathcal{O}_K).$$

This equality is (for example) an equality of sets, where $\mathrm{Bun}_{GL_n}(C)$ is the set of rank $n$ vector bundles over $C$.

If $K$ is a number field, the natural replacement of the right-hand side (when we put the appropriate thing at the archimedean places) of the above equation is an arithmetic symmetric space, e.g., for $K = \mathbb{Q}$ and $n = 2$ it is the modular curve $Y(1)$. One might hope that $Y(1)$ have an interpretation as "vector bundles of rank $2$ over $\mathrm{Spec }\mathbb{Z} \cup \{\infty\}$." I think the usual way to do this is through the theory of adelic metrized line bundles. Do these arithmetic symmetric spaces have interpretations as moduli spaces of adelic metrized vector bundles?

The $\mathbb{C}$-points of the moduli space of elliptic curves is $Y(1)$. Is it a coincidence that elliptic curves over $\mathbb{C}$ are related to Arakelov vector bundles of rank $2$ over $\mathrm{Spec } \mathbb{Z}$?

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    $\begingroup$ I don't know what an Arakelov vector bundle is, but one useful keyword (for symmetric spaces defined via GL_n) is "Shimura variety" (which you probably already know). $\endgroup$
    – skd
    Apr 5, 2020 at 18:00

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