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 $X$?

2$\begingroup$ This is more or less a standard result, see for instance the references in ncatlab.org/nlab/show/moduli+space+of+bundles . Although I would like to see the paper where Weil first published this. $\endgroup$ – user40276 Sep 11 '15 at 3:05

3$\begingroup$ I think it should be adeles, not idele (for $\mathbb A_F$). $\endgroup$ – Amritanshu Prasad Sep 11 '15 at 4:14

2$\begingroup$ An aside: I think it's actually more natural to prove this bijection when the adele ring and $O_x$ are replaced by their uncompleted versions ("preadeles" or "ring of repartitions" and the usual local ring at $x$) and then note that it doesn't matter whether one completes or not. The uncompleted version follows quite readily by classifying for a given open covering of $X$ all vector bundles trivialized by it using Cech cocycles. $\endgroup$ – Nicolas Schmidt Sep 11 '15 at 11:22
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 $GL_n(A_F)$.