280
X – The Riemann Surface of an Algebraic Function
for any meromorphic form on X, v(hω) = v(h) + v(ω) = v(ω) for all meromorphic functions h. But if ω and ω
are two meromorphic forms, there is is
a meromorphic function h such that ω
= hω . This is obvious in any local
chart (h is the ratio between the coefficients of ω and ω
), and the functions h
obtained in the local charts can be “ glued ” to give a globally defined function because of the transformation formula (3) which is identical for ω and
ω
. The conclusion is that the integer v(ω) is the same for all meromorphic
differentials on S. Set
v(ω) = 2g − 2 ,
where g is the genus of the compact Riemann surface X.
We prove that g is an integer ≥ 0 and, what is far less obvious, that two
compact Riemann surfaces are homeomorphic
2 if and only if they have the
same genus. The Riemann sphere has genus 0. Indeed, the differential form
ω = dz has a double pole at infinity since it must be computed by using the
local uniformizer ζ = 1/z, and as, at all a ∈ C, ω = 1.d(z − a) with 1 = 0,
the pole at infinity is the only contribution to the calculation of v(ω) ; hence
v(ω) = −2 and g = 0. Conversely, any compact Riemann surface of genus 0 is
isomorphic (and not only homeomorphic) to the Riemann sphere. For g = 1,
we get the quotients C/L of the theory of elliptic functions; this classical case
will be studied in volume IV. For g ≥ 1, X is homeomorphic to a sphere with
g handles.
2 – Algebraic Functions
Riemann imagined his surfaces in order to study algebraic functions of one
variable and in particular, to make them uniform, though in his work they
were far less clearly defined than here. To understand this, what is meant
by an algebraic function ζ = F(z) of a complex variable z should be first
understood. The foremost characteristic of an algebraic “ function ” is that it
is not a function: like Log z, which is not algebraic, or like z
1/3 or
z
2 + 1
1/2 −
z
3
− 2z + 1
1/3 1/4
that are so, it can take several values (an infinite number in the first case, 3
in the second and 24 in the third) for a given value of z; the notation F(z)
can, therefore, only represent one set of complex numbers, the only notation
making sense being ζ ∈ F(z) and not ζ = F(z). By definition, the elements of
F(z) are the roots (possibly including ∞ as will be seen later) of an equation
P (z, ζ) = 0 ,
(2.1)
2 But not isomorphic as complex manifolds (an isomorphism being a holomorphic homeomorphism whose inverse is also holomorphic). The classification of
Riemann surfaces, up to isomorphism, is far more complicated.
X – The Riemann Surface of an Algebraic Function
for any meromorphic form on X, v(hω) = v(h) + v(ω) = v(ω) for all meromorphic functions h. But if ω and ω
are two meromorphic forms, there is is
a meromorphic function h such that ω
= hω . This is obvious in any local
chart (h is the ratio between the coefficients of ω and ω
), and the functions h
obtained in the local charts can be “ glued ” to give a globally defined function because of the transformation formula (3) which is identical for ω and
ω
. The conclusion is that the integer v(ω) is the same for all meromorphic
differentials on S. Set
v(ω) = 2g − 2 ,
where g is the genus of the compact Riemann surface X.
We prove that g is an integer ≥ 0 and, what is far less obvious, that two
compact Riemann surfaces are homeomorphic
2 if and only if they have the
same genus. The Riemann sphere has genus 0. Indeed, the differential form
ω = dz has a double pole at infinity since it must be computed by using the
local uniformizer ζ = 1/z, and as, at all a ∈ C, ω = 1.d(z − a) with 1 = 0,
the pole at infinity is the only contribution to the calculation of v(ω) ; hence
v(ω) = −2 and g = 0. Conversely, any compact Riemann surface of genus 0 is
isomorphic (and not only homeomorphic) to the Riemann sphere. For g = 1,
we get the quotients C/L of the theory of elliptic functions; this classical case
will be studied in volume IV. For g ≥ 1, X is homeomorphic to a sphere with
g handles.
2 – Algebraic Functions
Riemann imagined his surfaces in order to study algebraic functions of one
variable and in particular, to make them uniform, though in his work they
were far less clearly defined than here. To understand this, what is meant
by an algebraic function ζ = F(z) of a complex variable z should be first
understood. The foremost characteristic of an algebraic “ function ” is that it
is not a function: like Log z, which is not algebraic, or like z
1/3 or
z
2 + 1
1/2 −
z
3
− 2z + 1
1/3 1/4
that are so, it can take several values (an infinite number in the first case, 3
in the second and 24 in the third) for a given value of z; the notation F(z)
can, therefore, only represent one set of complex numbers, the only notation
making sense being ζ ∈ F(z) and not ζ = F(z). By definition, the elements of
F(z) are the roots (possibly including ∞ as will be seen later) of an equation
P (z, ζ) = 0 ,
(2.1)
2 But not isomorphic as complex manifolds (an isomorphism being a holomorphic homeomorphism whose inverse is also holomorphic). The classification of
Riemann surfaces, up to isomorphism, is far more complicated.