286
X – The Riemann Surface of an Algebraic Function
connected (i.e.there are arbitrarily small connected neighbourhoods of all
points z ∈ B, which is for example the case with manifolds). To encourage the
reader to compare the general theory to arguments already used for complex
variables in the previous n
◦ and in § 4 of Chap. IV, we will denote an arbitrary
point of B by z and an arbitrary point of X by ζ.
In order for the triplet (X, B, p) to be a covering, we compel it to be
locally trivial, more precisely :
(R) every point z ∈ B has an open neighbourhood D whose inverse
image p
−1 (D) is the union of a family (D i ) i∈I of pairwise disjoint open
sets mapped homeomorphically onto D by p.
The most obvious consequence of (R) is that p transforms every neighbourhood of a point ζ ∈ X onto a neighbourhood of p(ζ). In particular, p transforms every open subset of X onto an open subset of B. Moreover, for all
z ∈ B, the fiber p
−1 ({z}) of z in X is a discrete subset of X since its intersections with the open subsets D i reduce to a point.
Example 1. Choose B = C
∗ , X = C and p to be the map
e : ζ −→ exp(2πiζ)
from X onto B. As e
(ζ) = 0 everywhere, p is a local homeomorphism
(Chap. VIII, n
◦ 5, theorem 7 applied locally). If a ∈ B is the image of
some α ∈ X and if D ⊂ B is a sufficiently small disc centered at a, then
there is neighbourhood D
of α homeomorphically mapped onto D by p. The
periodicity of the exponential function shows that
p
−1 (D) =
Z
D
+ n =
D n ,
and if D
(i.e. D) is sufficiently small. the translates D n of D
are pairwise
disjoint and homeomorphically mapped onto D by p. Hence C can be regarded as a covering space, moreover simply connected, of C
∗ . If we choose a
lattice L of periods like in the theory of elliptic functions, then C becomes a
simply connected covering space of the torus C/L. Finally, the map t → e(t)
transforms R into a covering space of T = R/Z.
Example 2. For a given integer k > 0, consider the quotient P/kZ of P
by the group of horizontal translations ζ → ζ + nk, where n ∈ Z, with the
obvious topology. The function e(ζ/k) is invariant under these translations,
and so defines a continuous map p : P/kZ −→ D
∗ . It is more or less obvious
that we thus obtain a “ k-sheeted ” covering space of D
∗ , as it used to be
called earlier, i.e. the canonical covering of order k of D
∗ . It can also be
constructed, up to isomorphism, by using the map z → z
k from D
∗ onto D
∗ ;
axiom (R) holds by lemma 1 of n
◦ 2 applied to the polynomial Y
k
− X for
a = 0; in fact, the canonical covering of order k of D
∗ is just the subset of
the Riemann surface of the polynomial Y
k
− X located over D
∗ .
It may happen that condition (R) holds for D = B. If every ζ ∈ D i is
identified to the couple (z, i), where z = p(ζ), X is transformed into the
X – The Riemann Surface of an Algebraic Function
connected (i.e.there are arbitrarily small connected neighbourhoods of all
points z ∈ B, which is for example the case with manifolds). To encourage the
reader to compare the general theory to arguments already used for complex
variables in the previous n
◦ and in § 4 of Chap. IV, we will denote an arbitrary
point of B by z and an arbitrary point of X by ζ.
In order for the triplet (X, B, p) to be a covering, we compel it to be
locally trivial, more precisely :
(R) every point z ∈ B has an open neighbourhood D whose inverse
image p
−1 (D) is the union of a family (D i ) i∈I of pairwise disjoint open
sets mapped homeomorphically onto D by p.
The most obvious consequence of (R) is that p transforms every neighbourhood of a point ζ ∈ X onto a neighbourhood of p(ζ). In particular, p transforms every open subset of X onto an open subset of B. Moreover, for all
z ∈ B, the fiber p
−1 ({z}) of z in X is a discrete subset of X since its intersections with the open subsets D i reduce to a point.
Example 1. Choose B = C
∗ , X = C and p to be the map
e : ζ −→ exp(2πiζ)
from X onto B. As e
(ζ) = 0 everywhere, p is a local homeomorphism
(Chap. VIII, n
◦ 5, theorem 7 applied locally). If a ∈ B is the image of
some α ∈ X and if D ⊂ B is a sufficiently small disc centered at a, then
there is neighbourhood D
of α homeomorphically mapped onto D by p. The
periodicity of the exponential function shows that
p
−1 (D) =
Z
D
+ n =
D n ,
and if D
(i.e. D) is sufficiently small. the translates D n of D
are pairwise
disjoint and homeomorphically mapped onto D by p. Hence C can be regarded as a covering space, moreover simply connected, of C
∗ . If we choose a
lattice L of periods like in the theory of elliptic functions, then C becomes a
simply connected covering space of the torus C/L. Finally, the map t → e(t)
transforms R into a covering space of T = R/Z.
Example 2. For a given integer k > 0, consider the quotient P/kZ of P
by the group of horizontal translations ζ → ζ + nk, where n ∈ Z, with the
obvious topology. The function e(ζ/k) is invariant under these translations,
and so defines a continuous map p : P/kZ −→ D
∗ . It is more or less obvious
that we thus obtain a “ k-sheeted ” covering space of D
∗ , as it used to be
called earlier, i.e. the canonical covering of order k of D
∗ . It can also be
constructed, up to isomorphism, by using the map z → z
k from D
∗ onto D
∗ ;
axiom (R) holds by lemma 1 of n
◦ 2 applied to the polynomial Y
k
− X for
a = 0; in fact, the canonical covering of order k of D
∗ is just the subset of
the Riemann surface of the polynomial Y
k
− X located over D
∗ .
It may happen that condition (R) holds for D = B. If every ζ ∈ D i is
identified to the couple (z, i), where z = p(ζ), X is transformed into the