X – The Riemann Surface of an Algebraic
Function
1 –Riemann Surfaces
Let X be a 2-dimensional C
0 manifold in the sense of Chap. IX, n
◦ 11, (ii).
If (U, ϕ) is a chart for X, ϕ may be considered a homeomorphism from the
open set U onto an open subset of C : if ξ 1 (x), ξ 2 (x) are the coordinates of
ϕ(x) ∈ R
2 for x ∈ U , it suffices to agree that ϕ(x) = ξ 1 (x) + iξ 2 (x).
Let (U, ϕ) and (V, ψ) be two charts for S. The change of charts ϕ(U ∩ V )
−→ ψ(U ∩ V ) is a priori not C
0 , so that if, by miracle, a function defined on
U ∩ V can be expressed holomorphically in (U, ϕ), there is no reason this is
also possible in (V, ψ). For this, the change of charts would need to transform
holomorphic functions into holomorphic functions, in other words would need
to be a conformal representation of ϕ(U ∩ V ) on ψ(U ∩ V ).
This brings us to the notion of a Riemann surface (or of a complex analytic manifold with complex dimension 1) : It is a 2-dimensional connected
1
manifold X of class C
0 with an atlas (U i , ϕ i ) all of whose changes of charts
ϕ i (U i ∩ U j ) −→ ϕ j (U i ∩ U j )
are holomorphic, in which case these are conformal representations (permute
i and j). This leads to the more general definition of a holomorphic chart
(U, ϕ) of X by the condition that, for all i, coordinate changes ϕ i (x) → ϕ(x)
and ϕ(x) → ϕ i (x) be holomorphic on the open sets on which they are defined.
When (U, ϕ) is a holomorphic local chart at a ∈ U such that ϕ(a) = 0, the
function ϕ is said to be a local uniformizer at a ; for this case, some authors
adopt the notation q a instead of ϕ. We will also occasionally do so despite the
fact that it could give the wrong idea that q a is determined by a. Holomorphic
functions being C
∞ , a Riemann surface is first of all a C
∞ manifold.
The most obvious example, apart from that of an open subset of C, is the
Riemann sphere ˆ
C = C ∪ {∞} : it has an atlas with two charts (U, ϕ) and
(V, ψ), where U = C, ϕ(z) = z, V = ˆ
C − {0}, ψ(z) = 1/z.
A more useful case as it controls the theory of elliptic functions consists
in choosing a lattice L in C, i.e. a discrete subgroup generated by two nonproportional numbers ω 1 and ω 2 (Chap. II, n
◦ 23) and in considering the
1 Not assuming this would lead to ridiculous complications. To start with, theorem
1 below would become false.
© Springer International Publishing Switzerland 2015
275
R. Godement, Analysis III, Universitext, DOI 10.1007/978-3-319-16053-5_3
Function
1 –Riemann Surfaces
Let X be a 2-dimensional C
0 manifold in the sense of Chap. IX, n
◦ 11, (ii).
If (U, ϕ) is a chart for X, ϕ may be considered a homeomorphism from the
open set U onto an open subset of C : if ξ 1 (x), ξ 2 (x) are the coordinates of
ϕ(x) ∈ R
2 for x ∈ U , it suffices to agree that ϕ(x) = ξ 1 (x) + iξ 2 (x).
Let (U, ϕ) and (V, ψ) be two charts for S. The change of charts ϕ(U ∩ V )
−→ ψ(U ∩ V ) is a priori not C
0 , so that if, by miracle, a function defined on
U ∩ V can be expressed holomorphically in (U, ϕ), there is no reason this is
also possible in (V, ψ). For this, the change of charts would need to transform
holomorphic functions into holomorphic functions, in other words would need
to be a conformal representation of ϕ(U ∩ V ) on ψ(U ∩ V ).
This brings us to the notion of a Riemann surface (or of a complex analytic manifold with complex dimension 1) : It is a 2-dimensional connected
1
manifold X of class C
0 with an atlas (U i , ϕ i ) all of whose changes of charts
ϕ i (U i ∩ U j ) −→ ϕ j (U i ∩ U j )
are holomorphic, in which case these are conformal representations (permute
i and j). This leads to the more general definition of a holomorphic chart
(U, ϕ) of X by the condition that, for all i, coordinate changes ϕ i (x) → ϕ(x)
and ϕ(x) → ϕ i (x) be holomorphic on the open sets on which they are defined.
When (U, ϕ) is a holomorphic local chart at a ∈ U such that ϕ(a) = 0, the
function ϕ is said to be a local uniformizer at a ; for this case, some authors
adopt the notation q a instead of ϕ. We will also occasionally do so despite the
fact that it could give the wrong idea that q a is determined by a. Holomorphic
functions being C
∞ , a Riemann surface is first of all a C
∞ manifold.
The most obvious example, apart from that of an open subset of C, is the
Riemann sphere ˆ
C = C ∪ {∞} : it has an atlas with two charts (U, ϕ) and
(V, ψ), where U = C, ϕ(z) = z, V = ˆ
C − {0}, ψ(z) = 1/z.
A more useful case as it controls the theory of elliptic functions consists
in choosing a lattice L in C, i.e. a discrete subgroup generated by two nonproportional numbers ω 1 and ω 2 (Chap. II, n
◦ 23) and in considering the
1 Not assuming this would lead to ridiculous complications. To start with, theorem
1 below would become false.
© Springer International Publishing Switzerland 2015
275
R. Godement, Analysis III, Universitext, DOI 10.1007/978-3-319-16053-5_3