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X – The Riemann Surface of an Algebraic Function
quotient space X = C/L of classes mod L. Equip it with the obvious topology : U ⊂ X is open if and only if its inverse image is open in C or, what
amounts to the same in these circumstances, if it is the image of an open
subset of C ; X becomes a compact space homeomorphic to the torus T
2 . To
define a complex structure on X, first note that if D ⊂ C is a sufficiently
small open disc centered at a, its images under the translations z → z + ω,
with ω ∈ L, are pairwise disjoint, so that p : C −→ C/L maps D homeomorphically onto an open subset U of S . If ϕ denotes the inverse map U −→ D
to p, the couple (U, ϕ) is a chart of X, and to find the analytic structure
sought, it suffices to show that the charts thus defined satisfy the condition
imposed above, which is obvious. The function ϕ(x) − a, where a is the centre
of D, is then a local uniformizer at the point p(a).
If X is a Riemann surface, for any open subset O ⊂ S, it is possible to
define functions h, which will be said to be holomorphic (resp. meromorphic ) on O, by requiring that they satisfy the following condition: for any
holomorphic chart (U, ϕ), there exists a holomorphic (resp. meromorphic)
function h ϕ on the open subset ϕ(U ∩ O) of C such that h(x) = h ϕ [ϕ(x)]
for all x ∈ U ∩ O; checking it for the charts in an atlas would be enough.
An equivalent definition: in the neighbourhood of every a ∈ U ∩ O, the function h is the sum of a power series (resp. Laurent series with finitely many
terms of degree < 0) in ϕ(x) − ϕ(a) :
h(x) =
c n [ϕ(x) − ϕ(a)]
n =
c n q a (x)
n .
(1.1)
Hence, like in C, a meromorphic function on O is not defined everywhere,
unless it is assigned the value ∞ at all points of a discrete subset of O.
Denote by H(O) (resp. M(O)) the set of holomorphic (resp. meromorphic)
functions on O. Their defining property is clearly of a local nature. Like on
C, the usual algebraic operations, including division, can be performed on
meromorphic functions on X: if f is not the function 0, its zeros are isolated
since X is connected, which removes all difficulties. To be correct, the result
should also be defined at points, where, apparently this is not the case: if for
example f and g have poles at a ∈ X and if their polar parts in their Laurent
series (1) cancel mutually, a value is assigned to f + g at a, namely the sum
of the constant terms of their Laurent series. Therefore, the set M(X) of
meromorphic functions on X can also be considered a commutative field.
An essential difference between C
∞ theory and that of Riemann surfaces
is the existence of holomorphic or meromorphic functions on a given open set
O, while at the same time it being clear that if O is contained in the domain
of a chart, then this is not obvious in the other cases. Showing – which we will
not do here – that there are many meromorphic functions defined globally on
a Riemann surface is the first major difficulty encountered as we progress in
the theory.
This is not surprising. If, as above, we consider a lattice L in C and
the Riemann surface X = C/L, holomorphic or meromorphic functions on
X – The Riemann Surface of an Algebraic Function
quotient space X = C/L of classes mod L. Equip it with the obvious topology : U ⊂ X is open if and only if its inverse image is open in C or, what
amounts to the same in these circumstances, if it is the image of an open
subset of C ; X becomes a compact space homeomorphic to the torus T
2 . To
define a complex structure on X, first note that if D ⊂ C is a sufficiently
small open disc centered at a, its images under the translations z → z + ω,
with ω ∈ L, are pairwise disjoint, so that p : C −→ C/L maps D homeomorphically onto an open subset U of S . If ϕ denotes the inverse map U −→ D
to p, the couple (U, ϕ) is a chart of X, and to find the analytic structure
sought, it suffices to show that the charts thus defined satisfy the condition
imposed above, which is obvious. The function ϕ(x) − a, where a is the centre
of D, is then a local uniformizer at the point p(a).
If X is a Riemann surface, for any open subset O ⊂ S, it is possible to
define functions h, which will be said to be holomorphic (resp. meromorphic ) on O, by requiring that they satisfy the following condition: for any
holomorphic chart (U, ϕ), there exists a holomorphic (resp. meromorphic)
function h ϕ on the open subset ϕ(U ∩ O) of C such that h(x) = h ϕ [ϕ(x)]
for all x ∈ U ∩ O; checking it for the charts in an atlas would be enough.
An equivalent definition: in the neighbourhood of every a ∈ U ∩ O, the function h is the sum of a power series (resp. Laurent series with finitely many
terms of degree < 0) in ϕ(x) − ϕ(a) :
h(x) =
c n [ϕ(x) − ϕ(a)]
n =
c n q a (x)
n .
(1.1)
Hence, like in C, a meromorphic function on O is not defined everywhere,
unless it is assigned the value ∞ at all points of a discrete subset of O.
Denote by H(O) (resp. M(O)) the set of holomorphic (resp. meromorphic)
functions on O. Their defining property is clearly of a local nature. Like on
C, the usual algebraic operations, including division, can be performed on
meromorphic functions on X: if f is not the function 0, its zeros are isolated
since X is connected, which removes all difficulties. To be correct, the result
should also be defined at points, where, apparently this is not the case: if for
example f and g have poles at a ∈ X and if their polar parts in their Laurent
series (1) cancel mutually, a value is assigned to f + g at a, namely the sum
of the constant terms of their Laurent series. Therefore, the set M(X) of
meromorphic functions on X can also be considered a commutative field.
An essential difference between C
∞ theory and that of Riemann surfaces
is the existence of holomorphic or meromorphic functions on a given open set
O, while at the same time it being clear that if O is contained in the domain
of a chart, then this is not obvious in the other cases. Showing – which we will
not do here – that there are many meromorphic functions defined globally on
a Riemann surface is the first major difficulty encountered as we progress in
the theory.
This is not surprising. If, as above, we consider a lattice L in C and
the Riemann surface X = C/L, holomorphic or meromorphic functions on