4 – The Riemann Surface of an Algebraic Function
307
fractions in z and ζ, where ζ is an algebraic function of z ; hence, if the base
field is C, this is not a generalization. This point of view is different because
we do not a priori choose some ζ ∈ L such that L is the set of polynomials in
ζ with coefficients in C(z). That being the case, how can a Riemann surface
ˆ
X associated to L be constructed?
The basic idea is very simply and with some technical changes applies to
far more general fields than C. The x ∈ L must be meromorphic functions on
ˆ
X. If that is the case, a value x(P ) ∈ ˆ
C can be assigned to x at each point
P ∈ ˆ
X; it must satisfy the following conditions :
(P 1) the set of x such that x(P ) = ∞ is a subring o(P ) of L, and the map
P → x(P ) is a homomorphism from o(P ) onto C ;
(P 2) the relation x(P ) = ∞ implies that x
−1 (P ) = 0 ;
(P 3) x(P ) = x for all x ∈ C.
Any map P : x → x(P ) from L to ˆ
C satisfying the previous conditions is,
by definition, a place of the field L. Having said that, the Riemann surface
sought is the set of these places, a set on which a compact Riemann structure
is then defined.
Some authors define the places de L by using the associated rings o(P ),
which can be characterized directly: they must contain C and satisfy
x /
∈ o =⇒ x
−1
∈ o .
A subring of a field K with this property is a valuation ring of K. An example
in the field Q of rational numbers is the set of fractions whose denominator
does not contain a given prime p. An example in C(X) is the set of holomorphic rational fractions in some given a ∈ ˆ
C.
Exercise 3. We thus obtain all the valuation rings of Q, and of C(X)
containing C, which corresponds to the fact that the Riemann surface of
C(X) is ˆ
C.
Other authors prefer to define the places of a field L of algebraic function
by using valuations . For them, a point of the Riemann surface is a map
v : L −→ Z
satisfying the following conditions:
v(xy) = v(x) + v(y) , v(x + y) ≥ min (v(x), v(y))
and v(x) = 0 if x ∈ C. This definition corresponds to the fact that at every
point P of the Riemann surface of L, its order v P (x) at P can be associated
to each point x ∈ L since in the neighbourhood of P , the function x is a
Laurent series in a local uniformizer. The reader will have no difficulty in
determining all the valuations of C(X), or of Q.
307
fractions in z and ζ, where ζ is an algebraic function of z ; hence, if the base
field is C, this is not a generalization. This point of view is different because
we do not a priori choose some ζ ∈ L such that L is the set of polynomials in
ζ with coefficients in C(z). That being the case, how can a Riemann surface
ˆ
X associated to L be constructed?
The basic idea is very simply and with some technical changes applies to
far more general fields than C. The x ∈ L must be meromorphic functions on
ˆ
X. If that is the case, a value x(P ) ∈ ˆ
C can be assigned to x at each point
P ∈ ˆ
X; it must satisfy the following conditions :
(P 1) the set of x such that x(P ) = ∞ is a subring o(P ) of L, and the map
P → x(P ) is a homomorphism from o(P ) onto C ;
(P 2) the relation x(P ) = ∞ implies that x
−1 (P ) = 0 ;
(P 3) x(P ) = x for all x ∈ C.
Any map P : x → x(P ) from L to ˆ
C satisfying the previous conditions is,
by definition, a place of the field L. Having said that, the Riemann surface
sought is the set of these places, a set on which a compact Riemann structure
is then defined.
Some authors define the places de L by using the associated rings o(P ),
which can be characterized directly: they must contain C and satisfy
x /
∈ o =⇒ x
−1
∈ o .
A subring of a field K with this property is a valuation ring of K. An example
in the field Q of rational numbers is the set of fractions whose denominator
does not contain a given prime p. An example in C(X) is the set of holomorphic rational fractions in some given a ∈ ˆ
C.
Exercise 3. We thus obtain all the valuation rings of Q, and of C(X)
containing C, which corresponds to the fact that the Riemann surface of
C(X) is ˆ
C.
Other authors prefer to define the places of a field L of algebraic function
by using valuations . For them, a point of the Riemann surface is a map
v : L −→ Z
satisfying the following conditions:
v(xy) = v(x) + v(y) , v(x + y) ≥ min (v(x), v(y))
and v(x) = 0 if x ∈ C. This definition corresponds to the fact that at every
point P of the Riemann surface of L, its order v P (x) at P can be associated
to each point x ∈ L since in the neighbourhood of P , the function x is a
Laurent series in a local uniformizer. The reader will have no difficulty in
determining all the valuations of C(X), or of Q.