Suppose $X$ is an algebraic curve, $F$ its function field, $\mathbb{A}\_F$ its adele, $O\_x$ the ring of integers at local field of $x$. Why does $GL\_n(F)\backslash GL\_n(\mathbb{A}\_F)/\prod\_xGL\_n(O\_x)$ classify vector bundles over $X$?



Why does $GL_n(F)\backslash GL_n(\mathbb{A}_F)/\prod_xGL_n(O_x)$ classify vector bundles over $X$?
This is a good exercise. Here's a big hint for one direction. Given a rank $n$ vector bundle, write down an isomorphism between it and the trivial rank $n$ vector bundle on the generic point of the curve (note: different choices will differ by an element of $GL\_n(F)$). Now write down an isomorphism between it and trivial rank $n$ vector bundle at the completed local ring at every closed point~$x$ of the curve (note: different choices will differ by an element of $GL\_n(O\_x)$). Now at the generic point of each completed local ring we have an isomorphism between a trivialised bundle and itself, which is represented by an element of $GL\_n(Frac(O\_x))$ and now just put everything together to get the element of $GL\_n(A\_F)$. 


