298
X – The Riemann Surface of an Algebraic Function
As X is a covering space of B, p
−1 (U ) is a covering space of U , hence
is trivial (theorem 3). So it has n sections z → (z, f k (z)), with functions f k
defined on U and locally, hence globally, holomorphic, qed.
Our task is now going to be to complete X to obtain a compact Riemann surface ˆ
X for which the map p : X −→ B = C − S can be extended
to a map ˆ
X −→ ˆ
C which will be holomorphic without however satisfying
axiom (R) of coverings in the neighbourhoods of the points z ∈ ˆ
C − B. For
this, finitely many points need to be adjoined to X “ over ” the z ∈ ˆ
C − B
and the holomorphic charts in the neighbourhood of these points need to be
defined.
12
This supposes that the structure of X over a neighbourhood of any a ∈ ˆ
C−
B is known ; theorem 5 provides the answer. It will nonetheless be somewhat
long, but there is no quick method, especially if we want to explain everything.
(ii) Definition of a Riemann surface ˆ
X. Let us return to the Riemann
surface (X, B, p) of the equation P (z, ζ) = 0 over the open subset B of
C . For any given a ∈ ˆ
C − B, choose an open disc D(a) centered at a not
containing any other point of ˆ
C − B apart from a, and let Y be a connected
component of p
−1 (D
∗ (a)) ; it is a connected covering space of order k ≤ n
of D
∗ (a). Let ϕ Y be a holomorphic function on Y satisfying the properties of
theorem 5 with respect to D
∗ (a) ; since
ϕ Y (z, ζ)
k = z − a resp. 1/z
(4.2)
for all η = (z, ζ) ∈ Y , ϕ Y (η) tends to 0 as p(η) = z tends to a. An “ ideal
point ” (so Forster says) then needs to be added to Y as is done to go from C
to ˆ
C. Denote this point by η Y , set ˆ
Y = Y ∪{η Y } and assume that ϕ Y (η Y ) = 0.
This gives a bijection from ˆ
Y onto the open disc centered at 0 in C ; thus the
holomorphic structure of this disc can be transferred to ˆ
Y . The connected
component Y = ˆ
Y − |η Y | becomes an open subset of ˆ
Y and the holomorphic
functions on an open subset U of ˆ
Y are those that can be expressed holomorphically by using ϕ Y : equip ˆ
Y with the holomorphic structure for which
( ˆ
Y , ϕ Y ) is a holomorphic chart of ˆ
Y and ϕ Y is a local uniformizer at η Y .
We show that the holomorphic functions on every open subset U of Y ,
that we are already acquainted with from the end of n
◦ 2 (an open subset
of Y is also open in X) are just the holomorphic functions of ϕ Y . First of
all, U is the union of “ discs ” of X. The notion of holomorphy being a local
one, we can confine ourselves to the case of a disc U , hence suppose that
U = D
(β) for some (b, β) ∈ Y , where D
⊂ D
∗ (a) ⊂ B is a sufficiently
small open disc centered at b = a over which the covering X, and hence
Y , is trivial. D
(β) is the image of D
under z → (z, f (z)), where f is the
12 The following developments, and even some of the preceding ones, are strongly
influenced by Chap. I of Otto Forster’s Lectures on Riemann Surfaces (Springer,
1981). Let us also mention Hershel M. Farkas and Irwin Kra, Riemann Surfaces
(Springer, 1980).
X – The Riemann Surface of an Algebraic Function
As X is a covering space of B, p
−1 (U ) is a covering space of U , hence
is trivial (theorem 3). So it has n sections z → (z, f k (z)), with functions f k
defined on U and locally, hence globally, holomorphic, qed.
Our task is now going to be to complete X to obtain a compact Riemann surface ˆ
X for which the map p : X −→ B = C − S can be extended
to a map ˆ
X −→ ˆ
C which will be holomorphic without however satisfying
axiom (R) of coverings in the neighbourhoods of the points z ∈ ˆ
C − B. For
this, finitely many points need to be adjoined to X “ over ” the z ∈ ˆ
C − B
and the holomorphic charts in the neighbourhood of these points need to be
defined.
12
This supposes that the structure of X over a neighbourhood of any a ∈ ˆ
C−
B is known ; theorem 5 provides the answer. It will nonetheless be somewhat
long, but there is no quick method, especially if we want to explain everything.
(ii) Definition of a Riemann surface ˆ
X. Let us return to the Riemann
surface (X, B, p) of the equation P (z, ζ) = 0 over the open subset B of
C . For any given a ∈ ˆ
C − B, choose an open disc D(a) centered at a not
containing any other point of ˆ
C − B apart from a, and let Y be a connected
component of p
−1 (D
∗ (a)) ; it is a connected covering space of order k ≤ n
of D
∗ (a). Let ϕ Y be a holomorphic function on Y satisfying the properties of
theorem 5 with respect to D
∗ (a) ; since
ϕ Y (z, ζ)
k = z − a resp. 1/z
(4.2)
for all η = (z, ζ) ∈ Y , ϕ Y (η) tends to 0 as p(η) = z tends to a. An “ ideal
point ” (so Forster says) then needs to be added to Y as is done to go from C
to ˆ
C. Denote this point by η Y , set ˆ
Y = Y ∪{η Y } and assume that ϕ Y (η Y ) = 0.
This gives a bijection from ˆ
Y onto the open disc centered at 0 in C ; thus the
holomorphic structure of this disc can be transferred to ˆ
Y . The connected
component Y = ˆ
Y − |η Y | becomes an open subset of ˆ
Y and the holomorphic
functions on an open subset U of ˆ
Y are those that can be expressed holomorphically by using ϕ Y : equip ˆ
Y with the holomorphic structure for which
( ˆ
Y , ϕ Y ) is a holomorphic chart of ˆ
Y and ϕ Y is a local uniformizer at η Y .
We show that the holomorphic functions on every open subset U of Y ,
that we are already acquainted with from the end of n
◦ 2 (an open subset
of Y is also open in X) are just the holomorphic functions of ϕ Y . First of
all, U is the union of “ discs ” of X. The notion of holomorphy being a local
one, we can confine ourselves to the case of a disc U , hence suppose that
U = D
(β) for some (b, β) ∈ Y , where D
⊂ D
∗ (a) ⊂ B is a sufficiently
small open disc centered at b = a over which the covering X, and hence
Y , is trivial. D
(β) is the image of D
under z → (z, f (z)), where f is the
12 The following developments, and even some of the preceding ones, are strongly
influenced by Chap. I of Otto Forster’s Lectures on Riemann Surfaces (Springer,
1981). Let us also mention Hershel M. Farkas and Irwin Kra, Riemann Surfaces
(Springer, 1980).