Weil uniformization theorem



Complex geometry



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

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

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

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

The theorem is based on the fact that GG-bundles on XX trivialize on the complement of finitely many points and that this double quotient then expresses the GG-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.


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

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

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


Review is for instance

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

See also

For more references see at moduli stack of bundles.

Last revised on April 12, 2018 at 08:24:55. See the history of this page for a list of all contributions to it.