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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 $ℂ(x)$;

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

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

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

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

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