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 ;
Compact Riemann surfaces (compact complex manifolds of complex dimension 1, or “curves”).
The correspondence goes roughly as follows: to each compact Riemann surface , one may associate the field of meromorphic functions , or holomorphic functions . Moreover, for each such , there exists a finite branched covering
which contravariantly induces a field homomorphism .
In the other direction, to each field of transcendence degree 1 over , there is a Riemann surface whose points may be identified with valuation rings in . (More precisely, with the discrete valuation rings in . All valuation rings of are discrete except for itself, which plays the role of a “generic point”.)