COMPLEX GEOMETRY Course notes

2.7 The Riemann surface of an algebraic function
Let Z 2 be a Riemann surface.
Proposition 2.7.1. Let
P (T ) = T n + c 1 T n−1 + · · · + c n
in M(Z 2 )[T ] be an irreducible polynomial. Then there exists a map of Riemann surfaces f : Z 1 −→ Z 2 of
degree n and a meromorphic function F ∈ M(Z 1 ) that satisfies:
F n + f ∗ (c 1 )F n−1 + · · · + f ∗ (c n ) = 0.
(∗)
Proof: Let ∆ ⊆ Z 2 be the discrete set containing the poles of the c i ’s and the points p where
P p (T ) := T n + c 1 (p)T n−1 + · · · + c n (p)
has multiple roots. Then U = {(p, z) ∈ (Z 2 − ∆) × C / P p (z) = 0} is a Riemann surface, and Z 2 − ∆ is
a finite étale cover. We claim that it is connected, i.e.,
Claim: Given f : Z 1 −→ Z 2 , then every F ∈ M(Z 1 ) is algebraic over M(Z 2 ) and satisfies an equality
of the form (∗) but with degree less or equal than the degf.
Corollary 2.7.1. If Z 1 −→ Z 2 is finite, then
is a finite field extension.
f ∗ : M(Z 2 ) −→ M(Z 1 )
Example 2.7.1. M(P 1 ) = C(Z), the field of rational functions in variable z.
Theorem 2.7.1. As soon as there exists a meromorphic function f on Z 1 (←− compact =⇒ f is finite),
M(Z 1 ) is finite algebraic over C(z) of extension deg = degf.
Z 1
f
−→ P 1
Corollary 2.7.2.
(1) If Z 1
f
−→ Z 2 is finite then M(Z 2 ) f ∗
↩→ M(Z 1 ) is a finite field extension of degree (f).
(2) Conversely, let M(Z 2 ) ϕ
↩→ M(Z 1 ) = K be a finite field extension of degree d. Then there exists a finite
map Z 1 −→ Z 2 of degree d whose field extension is isomorphic to ϕ.
(3) A field K of transcendental degree = 1 over C is isomorphic to M(Z) of some compact Riemann
surface. Such a Z is called a model of K.
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