1. Algebraic Methods in the Study of Compact Riemann surface All the Riemann surfaces are assumed to be connected.
Let X be a Riemann surface. A meromorphic function on X is a holomorphic mapping f : X → P1. The set of all meromorphic functions on X denoted by C(X) forms a field.
Theorem 1.1. Every compact Riemann surface has a nonconstant meromorphic function.
Since all nonconstant holomorphic mapping f : X → Y is a branched covering of Y, Theorem 1.1 implies that every compact Riemann surface can be made into a branched covering of P1.
If f : X → Y is a nonconstant holomorphic mapping between compact Riemann surfaces, then f induces a C-algebra homomorphism
f∗ : C(Y ) → C(X), ψ 7→ ψ ◦ f.
Since both C(Y ) and C(X) are fields and f is nonconstant, f∗ is a monomorphism. Hence we can identify C(Y ) with a subfield of C(X); hence C(X) is an extension field of C(Y ).
Lemma 1.1. Let π : X → P1 be a degree n branched covering of P1. There exists a meromorphic function f on X which separates the sheets of X over P1 in the following sense: there exists a point z0 on P1 such that f takes n-distinct values at points of X over z0.
Theorem 1.2. Let π : X → P1 be a deg n-branched covering.
(1) [C(X), C(z)] = n, and hence C(X) is an n-dimensional field extension of C(z).
(2) Any f ∈ C(X) satisfies a polynomial equation of degree ≤ n with coefficients in C(z), i.e. there exist a0(z), · · · , an−1(z) in C(z) so that
fn+ an−1(z)fn−1+ · · · + a0(z) = 0.
(3) Let f be a meromorphic function on X which separates the sheets of X over P1. Then C(X) is generated by f over C(z).
(4) Assume that fn+ an−1(z)fn−1 + · · · + a0(z) = 0. Then X is isomorphic to the nonsingular compactified Riemann surface defined by
wn+ an−1(z)wn−1+ · · · + a1(z)w + a0(z) = 0.
Conversely, if F is an n-dimensional extension of C(z), then there exists a compact Rie- mann surface X together with a degree n branched covering π : X → P1 so that π induces an isomorphism C(X) ∼= F of field extensions of C(z). Notice that isomorphic Riemann surfaces induce isomorphic field extensions of C(z).
We say that a morphism from a branched covering π : X → P1 to a branched covering π0 : X0 → P1 of P1 is a holomorphic map f : X → X0 so that π = π0◦ f. The set of all branched coverings of P1 together with morphisms between them form a category.
We conclude that:
Proposition 1.1. There is a bijection between the set of isomorphism classes of n-dimensional extension field of C(z) and the set of isomorphism classes of deg n-branched covering of P1.
Recall that a finite dimensional field extension E/F is called Galois with group G if G acts on E by automorphisms of K and KG = F (the G-invariant subsets of E is exactly F.) We denote G by Gal(E/F ) called the Galois group of E/F.
Proposition 1.2. The automorphisms of X that commute with f : X → Y are precisely the automorphisms of C(X) which fix C(Y ).
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Corollary 1.1. A map f : X → Y defines a Galois extension on the fields of meromorphic functions if and only if there exists a group G of automorphisms of X commuting with f and acting simply transitively on the fibers π−1(y) for general y ∈ Y. Such a map is called a Galois cover with group G.
2. Hyperelliptic Riemann surface
Definition 2.1. A compact Riemann surface is called hyperelliptic if it carries a meromor- phic function of degree 2.
Equivalently, X is hyperbolic if there is a degree 2 branched covering π : X → P1. From the previous section, C(X) is a two dimensional field extension of C(z). Let u ∈ C(X) \ C(z) be a generator of C(X) over C(z). Then we can find a(z) and b(z) in C(z) so that
u2+ a(z)u + b(z) = 0.
Completing square leads to v2+ c(z) = 0. Multiplying the square of the denominator of c(z) gives w2= f (z) where f (z) is a polynomial. Any repeated factors of f (z) can be divide our and incorporated in w. We obtain a square free polynomial f (z).
Proposition 2.1. Any hyperelliptic Riemann surface is isomorphic to the compactification of the Riemann surface of
w2 = f (z) when f is a polynomial over C with simple roots.
