nLab
Riemann surface via valuations

The following is the special case of the general notion of a Riemann surface over an arbitrary field due to algebraists in 19th century.

One of the first lessons from Mumford’s famous Red Book is the “amazing” correspondence between

  • Fields which arise as finite algebraic extensions of the field of rational functions (x)\mathbb{C}(x);

  • Compact Riemann surfaces (compact complex manifolds of complex dimension 1, or “curves”).

The correspondence goes roughly as follows: to each compact Riemann surface CC, one may associate the field of meromorphic functions Mer(C)Mer(C), or holomorphic functions C 1()C \to \mathbb{P}^1(\mathbb{C}). Moreover, for each such CC, there exists a finite branched covering

ϕ:C 1()\phi\colon C \to \mathbb{P}^1(\mathbb{C})

which contravariantly induces a field homomorphism (x)Mer(C)\mathbb{C}(x) \to Mer(C).

In the other direction, to each field KK of transcendence degree 1 over \mathbb{C}, there is a Riemann surface whose points may be identified with valuation rings in KK. (More precisely, with the discrete valuation rings in KK. All valuation rings of KK are discrete except for KK itself, which plays the role of a “generic point”.)

Revised on January 8, 2012 14:58:36 by Toby Bartels (216.96.8.189)