4 – The Riemann Surface of an Algebraic Function
299
uniform branch in D
of the algebraic function P (z, ζ) = 0 which takes the
value β at b. If U is regarded as an open subset of X, the end of n
◦ 2
tells us that the holomorphic functions of (z, ζ) on U are the holomorphic
functions of z on D
. So it amount to showing that the holomorphic functions
of ϕ Y (z, ζ) on D
(β) are identical to the holomorphic functions of z. However,
ϕ Y (z, ζ) = ϕ Y (z, f (z)) in D
(β) . As ϕ Y is holomorphic on Y in the sense of
n
◦ 2 (theorem 5) and as z → (z, f (z)) is a holomorphic map of z from D
to X, the left hand side is a holomorphic function of z on D
. Oppositely,
relation ϕ Y (z, f (z))
k = z −a resp. 1/z shows that z is a holomorphic function
of ϕ Y (z, f (z)) on D
(β), qed.
The complete Riemann surface ˆ
X sought is then obtained as follows: for
each a ∈ ˆ
C − B, the points η Y corresponding to the connected components
of X over D
∗ (a) are adjoined to X. These η Y are considered to be pairwise
distinct. If the covering space Y of D
∗ (a) is of order k, ( ˆ
X, ˆ
C, p), which is a
covering only over B and perhaps over a neighbourhood of ∞, is said to be
a branched covering of ˆ
C at the branch point η Y of order k. This is not a
property of the Riemann surface ˆ
X, which is as smooth as possible a manifold;
it is a property of the map p : ˆ
X −→ ˆ
C.
For example, if we take the algebraic equation ζ
2
− z = 0, then X is the
set of (z, ζ) ∈ C
2 such that ζ
2 = z, z = 0 . Over a disc D which does not
contain 0, the surface has two disjoint connected components corresponding
to the two uniform branches on D of the pseudo-function z
1/2 . This is no
longer the case over a disc D centered at 0, since, when z circles once around
the point 0, the determination chosen at the start for z
1/2 becomes the opposite determination at termination; this means that two points of the surface
projecting onto z can be connected by a curve, as in the case of the logarithm
(Chap. IV, § 4). The surface X is, therefore, connected over D
∗ = D − {0} ;
this is also the case over D since as z tends to 0, the two possible values
of z
1/2 also tend to 0, and only one point, namely (0, 0), remains over the
origin. There is a “ branch point ” in this case only because we are trying to
express ζ by using z in the neighbourhood of 0; if we tried to express z as a
function of ζ, all would become normal again. But, even in the real domain,
a general algebraic curve can have singularities (multiple points, cusp points,
etc.) otherwise more complicated than a vertical tangent.
In the general case, ˆ
X being the union of X and of the ˆ
Y , we define the
topology of ˆ
X by declaring U ⊂ ˆ
X to be open if U ∩X is open in X and if, for
all Y such that η Y ∈ U , the set U ∩ ˆ
Y is open in ˆ
Y . The Hausdorff axiom and
the fact that X and the ˆ
Y are open in ˆ
X are immediately verified. Finally, the
complex analytic structure of ˆ
X is obtained by adjoining the charts ( ˆ
Y , ϕ Y )
to the charts already available in X. As shown, these are compatible with
the complex analytic structure of the open subsets Y , hence of X, defined in
n
◦ 2 ; they are also compatible with each other as they are pairwise disjoint.
This provides ˆ
X with the structure of a Riemann surface, which coincides
with that of n
◦ 2 in the open set X.
299
uniform branch in D
of the algebraic function P (z, ζ) = 0 which takes the
value β at b. If U is regarded as an open subset of X, the end of n
◦ 2
tells us that the holomorphic functions of (z, ζ) on U are the holomorphic
functions of z on D
. So it amount to showing that the holomorphic functions
of ϕ Y (z, ζ) on D
(β) are identical to the holomorphic functions of z. However,
ϕ Y (z, ζ) = ϕ Y (z, f (z)) in D
(β) . As ϕ Y is holomorphic on Y in the sense of
n
◦ 2 (theorem 5) and as z → (z, f (z)) is a holomorphic map of z from D
to X, the left hand side is a holomorphic function of z on D
. Oppositely,
relation ϕ Y (z, f (z))
k = z −a resp. 1/z shows that z is a holomorphic function
of ϕ Y (z, f (z)) on D
(β), qed.
The complete Riemann surface ˆ
X sought is then obtained as follows: for
each a ∈ ˆ
C − B, the points η Y corresponding to the connected components
of X over D
∗ (a) are adjoined to X. These η Y are considered to be pairwise
distinct. If the covering space Y of D
∗ (a) is of order k, ( ˆ
X, ˆ
C, p), which is a
covering only over B and perhaps over a neighbourhood of ∞, is said to be
a branched covering of ˆ
C at the branch point η Y of order k. This is not a
property of the Riemann surface ˆ
X, which is as smooth as possible a manifold;
it is a property of the map p : ˆ
X −→ ˆ
C.
For example, if we take the algebraic equation ζ
2
− z = 0, then X is the
set of (z, ζ) ∈ C
2 such that ζ
2 = z, z = 0 . Over a disc D which does not
contain 0, the surface has two disjoint connected components corresponding
to the two uniform branches on D of the pseudo-function z
1/2 . This is no
longer the case over a disc D centered at 0, since, when z circles once around
the point 0, the determination chosen at the start for z
1/2 becomes the opposite determination at termination; this means that two points of the surface
projecting onto z can be connected by a curve, as in the case of the logarithm
(Chap. IV, § 4). The surface X is, therefore, connected over D
∗ = D − {0} ;
this is also the case over D since as z tends to 0, the two possible values
of z
1/2 also tend to 0, and only one point, namely (0, 0), remains over the
origin. There is a “ branch point ” in this case only because we are trying to
express ζ by using z in the neighbourhood of 0; if we tried to express z as a
function of ζ, all would become normal again. But, even in the real domain,
a general algebraic curve can have singularities (multiple points, cusp points,
etc.) otherwise more complicated than a vertical tangent.
In the general case, ˆ
X being the union of X and of the ˆ
Y , we define the
topology of ˆ
X by declaring U ⊂ ˆ
X to be open if U ∩X is open in X and if, for
all Y such that η Y ∈ U , the set U ∩ ˆ
Y is open in ˆ
Y . The Hausdorff axiom and
the fact that X and the ˆ
Y are open in ˆ
X are immediately verified. Finally, the
complex analytic structure of ˆ
X is obtained by adjoining the charts ( ˆ
Y , ϕ Y )
to the charts already available in X. As shown, these are compatible with
the complex analytic structure of the open subsets Y , hence of X, defined in
n
◦ 2 ; they are also compatible with each other as they are pairwise disjoint.
This provides ˆ
X with the structure of a Riemann surface, which coincides
with that of n
◦ 2 in the open set X.