TITLE: Why does $GL_n(F)\backslash GL_n(\mathbb{A}_F)/\prod_xGL_n(O_x)$ classify vector bundles over $X$?
QUESTION [6 upvotes]: 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 $X$?

REPLY [5 votes]: 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 $GL_n(A_F)$.