Let $k$ be an algebraically closed field, $G$ a reductive group, and $C$ a curve. The algebraic version of the Weil uniformization theorem (see e.g. arXiv:1511.06271v2) says that groupoid of $G$-torsors over $C$ is naturally equivalent to the groupoid of $C$-adelic $G$-automorphic forms. Notice that, once we pass to the quotient, the latter of these is naturally a group. In the case that $k=\mathbb{C}$, for instance, this is just the ordinary vector space structure on automorphic forms. This means that there is correspondingly a natural addition on (isomorphism classes of) $G$-torsors over $C$. Does this addition admit a nice description?


(Incidentally, I'm also interested in the dual question: in the case that $G=GL\_n$, how do the sum and tensor product of vector bundles transfer over to automorphic forms? This must, of course, include all $n$ at once since the sum of rank $n$ and $m$ vector bundles is of rank $n+m$.)



What is the sum operation on torsors induced by Weil uniformization?
As pointed out by Jesse Silliman in the comments, this question is ill-posed. In fact, adelic automorphic forms are global sections of the structure sheaf of the moduli of $G$-torsors, not elements of this stack.


