nLab Weil uniformization theorem

Idea

The uniformization theorem for principal bundles over algebraic curves $X$ (going back to André Weil) expresses the moduli stack of principal bundles on $X$ as a double quotient stack of the $G$-valued Laurent series around finitely many points by the product of the $G$-valued formal power series around these points and the $G$-valued functions on the complement of theses points.

If a single point $x$ is sufficient and if $D$ denotes the formal disk around that point and $X^\ast, D^\ast$ denote the complements of this point, respectively then the theorem says for suitable algebraic group $G$ that there is an equivalence of stacks

$$[X^\ast, G] \backslash [D^\ast, G] / [D,G] \simeq Bun_X(G) \,,$$

between the double quotient stack of $G$-valued functions (mapping stacks) as shown on the left and the moduli stack of G-principal bundles over $X$, as shown on the right.

The theorem is based on the fact that $G$-bundles on $X$ trivialize on the complement of finitely many points and that this double quotient then expresses the $G$-Cech cohomology with respect to the cover given by the complement of the points and the formal disks around them.

For details see at moduli stack of bundles – over curves.

Applications

The theorem is a key ingredient in the function field analogy where for $K$ a global field the nonabelian generalization of quotients of the idele class group by integral adeles

$$GL_n(K) \backslash GL_n(\mathbb{A}_K) / GL_n(\mathcal{O}_K)$$

are analogous to the moduli stack of $G$-bundles. This motivates notably the geometric Langlands correspondence as a geometric analog of the number-theoretic Langlands correspondence.

Related concepts

• differential cohesion and idelic structure

References

Review is for instance

• Christoph Sorger, Lectures on moduli of principal $G$-bundles over algebraic curves, 1999 (pdf)

See also

• Jochen Heinloth, Uniformization of $\mathcal{G}$-Bundles (pdf)

For more references see at moduli stack of bundles.