296
X – The Riemann Surface of an Algebraic Function
discs and the compatibility clause is immediate. The following result will play
an essential role in the construction of the compact Riemann surface of an
algebraic function :
Theorem 5. Let (Y, D
∗ , q) be a connected covering of order k < +∞ of
the pointed disc D
∗ : 0 < |z| < 1. Then there is a holomorphic function
ϕ in Y such that (i) ζ → ϕ(ζ) is a conformal representation of Y on D
∗ ,
(ii) ϕ(ζ)
k = q(ζ).
This means that (Y, D
∗ , q) is isomorphic to the covering of D
∗ obtained
by taking Y = D
∗ and q(z) = z
k [(i), Example 2], or to the covering obtained
by constructing the Riemann surface of the algebraic function z
1/k using the
method of n
◦ 2, or finally that the algebraic function z
1/k , which obviously
does not have uniform branches on the pointed disc, becomes uniform on Y :
a k
th root of z = q(ζ) can be associated to each ζ ∈ Y so that it only depends
holomorphically on ζ (but obviously not on z).
(a) Choose an arbitrary point a of D
∗ and some β ∈ Y such that q(β) = a
and associate to every closed path γ with initial point a in D
∗ the terminal
point of its lifting ν with initial point β in Y . As the terminal point of ν only
depends on the homotopy class of γ, by theorem 2 and its corollary 2, only
the paths γ n : t → ae(nt) need to be considered; let ν n be the lifting of γ n .
The path γ m .γ n −1 consisting in γ m followed by the opposite of γ n is clearly
the path γ m−n , up to parameterisation. If ν m and ν n have the same terminal
point, the lifting of γ m−n is obviously the closed path ν m .ν n −1 . If, conversely,
the lifting ν m−n of γ m−n is closed, the path ν m−n .ν n with initial point β is
a lifting of γ m−n .γ n , a path, identical to γ m , up to parameterisation . Then
ν m = ν m−n .ν n up to parametrization, so that the paths ν m and ν n have the
same terminal point.
This leads to two conclusions. First, if ν m and ν n are closed, so is ν m−n ;
the set of n such that γ n is lifted onto a closed path is, therefore, the subgroup
pZ of Z. Secondly, the terminal point of ν n only depends on the class of
n mod p. As any point of Y projecting onto a is the terminal point of such a
lifting, we conclude that there are an many classes mod p as points of Y over
a, i.e. that p = k.
(b) Having made this point, let us consider the universal covering (P, D
∗ , e)
of D
∗ , where P is the half-plane Im(ζ) > 0 and e the map ζ → e(ζ) from
P onto D
∗ . If we choose some α ∈ P such that e(α) = a, then theorem 4
shows the existence and uniqueness of a continuous map f : P −→ Y such
that f (α) = β and q[f (ζ)] = e(ζ) for all ζ ∈ P . Let ζ
and ζ
be two points
such that f (ζ
) = f (ζ
). So ζ
= ζ
+ n for some n ∈ Z . Setting e(ζ
) = z
,
e(ζ
) = z
, f (ζ
) = f (ζ
) = η ∈ Y , the map f transforms every path μ
connecting ζ
to ζ
into a closed path ν with initial point η in Y , which is a
lifting of the path γ, the image of μ under e . Choosing as μ the rectilinear
path [ζ
, ζ
] = [ζ
, ζ
+ n], clearly γ(t) = z
e(nt). Since γ can be lifted to a
closed path ν in Y , n = 0 mod k by part (a) of the proof, which can be applied
to all points of D
∗ and in particular to z
. Conversely, if n is a multiple of
X – The Riemann Surface of an Algebraic Function
discs and the compatibility clause is immediate. The following result will play
an essential role in the construction of the compact Riemann surface of an
algebraic function :
Theorem 5. Let (Y, D
∗ , q) be a connected covering of order k < +∞ of
the pointed disc D
∗ : 0 < |z| < 1. Then there is a holomorphic function
ϕ in Y such that (i) ζ → ϕ(ζ) is a conformal representation of Y on D
∗ ,
(ii) ϕ(ζ)
k = q(ζ).
This means that (Y, D
∗ , q) is isomorphic to the covering of D
∗ obtained
by taking Y = D
∗ and q(z) = z
k [(i), Example 2], or to the covering obtained
by constructing the Riemann surface of the algebraic function z
1/k using the
method of n
◦ 2, or finally that the algebraic function z
1/k , which obviously
does not have uniform branches on the pointed disc, becomes uniform on Y :
a k
th root of z = q(ζ) can be associated to each ζ ∈ Y so that it only depends
holomorphically on ζ (but obviously not on z).
(a) Choose an arbitrary point a of D
∗ and some β ∈ Y such that q(β) = a
and associate to every closed path γ with initial point a in D
∗ the terminal
point of its lifting ν with initial point β in Y . As the terminal point of ν only
depends on the homotopy class of γ, by theorem 2 and its corollary 2, only
the paths γ n : t → ae(nt) need to be considered; let ν n be the lifting of γ n .
The path γ m .γ n −1 consisting in γ m followed by the opposite of γ n is clearly
the path γ m−n , up to parameterisation. If ν m and ν n have the same terminal
point, the lifting of γ m−n is obviously the closed path ν m .ν n −1 . If, conversely,
the lifting ν m−n of γ m−n is closed, the path ν m−n .ν n with initial point β is
a lifting of γ m−n .γ n , a path, identical to γ m , up to parameterisation . Then
ν m = ν m−n .ν n up to parametrization, so that the paths ν m and ν n have the
same terminal point.
This leads to two conclusions. First, if ν m and ν n are closed, so is ν m−n ;
the set of n such that γ n is lifted onto a closed path is, therefore, the subgroup
pZ of Z. Secondly, the terminal point of ν n only depends on the class of
n mod p. As any point of Y projecting onto a is the terminal point of such a
lifting, we conclude that there are an many classes mod p as points of Y over
a, i.e. that p = k.
(b) Having made this point, let us consider the universal covering (P, D
∗ , e)
of D
∗ , where P is the half-plane Im(ζ) > 0 and e the map ζ → e(ζ) from
P onto D
∗ . If we choose some α ∈ P such that e(α) = a, then theorem 4
shows the existence and uniqueness of a continuous map f : P −→ Y such
that f (α) = β and q[f (ζ)] = e(ζ) for all ζ ∈ P . Let ζ
and ζ
be two points
such that f (ζ
) = f (ζ
). So ζ
= ζ
+ n for some n ∈ Z . Setting e(ζ
) = z
,
e(ζ
) = z
, f (ζ
) = f (ζ
) = η ∈ Y , the map f transforms every path μ
connecting ζ
to ζ
into a closed path ν with initial point η in Y , which is a
lifting of the path γ, the image of μ under e . Choosing as μ the rectilinear
path [ζ
, ζ
] = [ζ
, ζ
+ n], clearly γ(t) = z
e(nt). Since γ can be lifted to a
closed path ν in Y , n = 0 mod k by part (a) of the proof, which can be applied
to all points of D
∗ and in particular to z
. Conversely, if n is a multiple of