1 – Riemann Surfaces
277
an open subset O of X are those which, composed with p : C −→ C/L,
are holomorphic or meromorphic on the open subset p
−1 (O) of C. They are
invariant under translations z → z + ω, ω ∈ L. Any global existence proof
of meromorphic functions on C/L, therefore, shows, without the slightest
calculation, the existence of elliptic functions , i.e. of meromorphic functions
on C invariant under translations z → z + ω. This result cannot be made
trivial: either you prove the theorem holds for all Riemann surfaces, or else,
like Weierstrass, you write series
ω∈L
1/(z − ω)
k , k = 4, 6, 8, . . .
answering the question (Chap. II, n
◦ 23).
Let us now define the order v a (h) of a meromorphic function h at a
point a. For this choose a local uniformizer q a at a . This gives a Laurent
series expansion h(x) =
c n q
n
a (x) in the neighbourhood of a. v a (h) is then
the smallest integer for which c n = 0. This definition does not depend on the
choice of the chart.
Let ω be a complex-valued C
∞ differential form of degree 1 on X (or
more generally on an open subset O of X: replace X by O). Let (U, ϕ) be a
local chart of X, understood to be henceforth always holomorphic. Setting
ϕ(x) = ζ, (U, ϕ) transforms ω into a form ω ϕ on ϕ(U ) which can be written
ω ϕ = h ϕ (ζ)dζ + k ϕ (ζ)d ¯
ζ ,
(1.2)
with C
∞ functions h ϕ and k ϕ depending on the chart considered. ω will
be said to be holomorphic if, for every chart (U, ϕ), ω ϕ = h ϕ (ζ)dζ with a
holomorphic function h ϕ on ϕ(U ) or, equivalently, if ω is the inverse image
under ϕ of a holomorphic differential form on ϕ(U ). If (V, ψ) is another chart,
changes of charts
θ : ϕ(U ∩ V ) −→ ψ(U ∩ V ) , ρ : ψ(U ∩ V ) −→ ϕ(U ∩ V )
clearly (transitivity of inverse images) transform ω ϕ into ω ψ and conversely;
hence
ω ϕ = h ϕ (ζ)dζ =⇒ ω ψ = h ϕ [ρ(ζ)] d [ρ(ζ)] = h ϕ [ρ(ζ)] ρ
(ζ)dζ ,
so that the coefficient h ϕ of ω in the local chart (U, ϕ) is transformed by
h ψ (ζ) = h ϕ [ρ(ζ)] ρ
(ζ) .
(1.3)
This is a very particular case of tensor calculus formulas. Like in C, dω = 0.
More generally, meromorphic differential forms can be defined on X : these
are holomorphic forms on X − D, where D is a discrete subset of X and such
that, for all a ∈ D ω ϕ = h ϕ (ζ)dζ in a sufficiently small (hence in any) local
chart (U, ϕ) at a, where h ϕ is meromorphic on ϕ(U ) and has a unique pole
277
an open subset O of X are those which, composed with p : C −→ C/L,
are holomorphic or meromorphic on the open subset p
−1 (O) of C. They are
invariant under translations z → z + ω, ω ∈ L. Any global existence proof
of meromorphic functions on C/L, therefore, shows, without the slightest
calculation, the existence of elliptic functions , i.e. of meromorphic functions
on C invariant under translations z → z + ω. This result cannot be made
trivial: either you prove the theorem holds for all Riemann surfaces, or else,
like Weierstrass, you write series
ω∈L
1/(z − ω)
k , k = 4, 6, 8, . . .
answering the question (Chap. II, n
◦ 23).
Let us now define the order v a (h) of a meromorphic function h at a
point a. For this choose a local uniformizer q a at a . This gives a Laurent
series expansion h(x) =
c n q
n
a (x) in the neighbourhood of a. v a (h) is then
the smallest integer for which c n = 0. This definition does not depend on the
choice of the chart.
Let ω be a complex-valued C
∞ differential form of degree 1 on X (or
more generally on an open subset O of X: replace X by O). Let (U, ϕ) be a
local chart of X, understood to be henceforth always holomorphic. Setting
ϕ(x) = ζ, (U, ϕ) transforms ω into a form ω ϕ on ϕ(U ) which can be written
ω ϕ = h ϕ (ζ)dζ + k ϕ (ζ)d ¯
ζ ,
(1.2)
with C
∞ functions h ϕ and k ϕ depending on the chart considered. ω will
be said to be holomorphic if, for every chart (U, ϕ), ω ϕ = h ϕ (ζ)dζ with a
holomorphic function h ϕ on ϕ(U ) or, equivalently, if ω is the inverse image
under ϕ of a holomorphic differential form on ϕ(U ). If (V, ψ) is another chart,
changes of charts
θ : ϕ(U ∩ V ) −→ ψ(U ∩ V ) , ρ : ψ(U ∩ V ) −→ ϕ(U ∩ V )
clearly (transitivity of inverse images) transform ω ϕ into ω ψ and conversely;
hence
ω ϕ = h ϕ (ζ)dζ =⇒ ω ψ = h ϕ [ρ(ζ)] d [ρ(ζ)] = h ϕ [ρ(ζ)] ρ
(ζ)dζ ,
so that the coefficient h ϕ of ω in the local chart (U, ϕ) is transformed by
h ψ (ζ) = h ϕ [ρ(ζ)] ρ
(ζ) .
(1.3)
This is a very particular case of tensor calculus formulas. Like in C, dω = 0.
More generally, meromorphic differential forms can be defined on X : these
are holomorphic forms on X − D, where D is a discrete subset of X and such
that, for all a ∈ D ω ϕ = h ϕ (ζ)dζ in a sufficiently small (hence in any) local
chart (U, ϕ) at a, where h ϕ is meromorphic on ϕ(U ) and has a unique pole