278
X – The Riemann Surface of an Algebraic Function
at the point ϕ(a). If for example h is a meromorphic function on X and if
h is is represented by a meromorphic function h ϕ (ζ) in (U, ϕ) then, by the
multivariate chain rule, its differential dh is represented by h
ϕ (ζ)dζ; therefore,
dh is a meromorphic differential form with the same poles as h. The product
of a differential meromorphic form ω and of a meromorphic function is defined
in an obvious way. For example, for any meromorphic function h, consider
the form dh/h whose poles, like in C, are clearly the poles and zeros of h.
For a differential form ω = h ϕ (ζ)dζ, where h ϕ is meromorphic on ϕ(U ),
there is an expansion h ϕ (ζ) =
c n [ζ − ϕ(a)]
n in the neighbourhood of all
a ∈ U ; by definition, the coefficient c −1 is the residue of ω at a, written
Res(ω, a). It also is independent of the choice of the chart [Chap. VIII, n
◦ 5,
(v) : invariance of the residue under conformal representation]. For example,
take ω = dh/h, where h is meromorphic on X; if h is represented in the chart
(U, ϕ) by a meromorphic function h ϕ (ζ), then ω is clearly represented by the
form h
ϕ (ζ)dζ/h ϕ (ζ) ; ω is, therefore, a meromorphic differential form, and
Res a (dh/h) = v a (h)
(1.4)
as in C.
We now prove a result generalizing what has been seen in Chap. 8, n
◦ 5
about functions defined on the Riemann sphere:
Theorem 1. Let X be a compact Riemann surface.
(a) Any function defined and holomorphic on X is a constant.
(b) For any meromorphic function f on X,
v(f ) =
v a (f ) = 0 .
(1.5)
(c) For any meromorphic differential form ω on X,
Res a (ω) = 0 .
(1.6)
(a) is obvious: a function h everywhere holomorphic reaches its maximum
somewhere, and so is constant in the neighbourhood of its maximum. As X
is by definition connected, classical arguments apply verbatim. (Corollary :
the only entire elliptic functions on C are the constants).
To obtain (b), it suffices by (4), to prove (c). It will follow from Stokes’ theorem. The latter can be applied since the Jacobians of holomorphic changes
of charts are > 0, making it possible for X to be oriented by these charts.
Having said this, X being compact and the poles of ω being isolated, the
latter are finite in number; let us denote them by a k (1 ≤ k ≤ n). For each
k, choose a local chart (U k , ϕ k ) such that ϕ k (a k ) = 0 and, for sufficiently
small r > 0, denote by D k (r) the set of x ∈ U k such that |ϕ k (x)| ≤ r. These
“ discs ” are closed in X and if r is sufficiently small, also pairwise disjoint;
then ω has a unique pole at a k in D k (r) and even in D k (r
) for some r
> r.
X – The Riemann Surface of an Algebraic Function
at the point ϕ(a). If for example h is a meromorphic function on X and if
h is is represented by a meromorphic function h ϕ (ζ) in (U, ϕ) then, by the
multivariate chain rule, its differential dh is represented by h
ϕ (ζ)dζ; therefore,
dh is a meromorphic differential form with the same poles as h. The product
of a differential meromorphic form ω and of a meromorphic function is defined
in an obvious way. For example, for any meromorphic function h, consider
the form dh/h whose poles, like in C, are clearly the poles and zeros of h.
For a differential form ω = h ϕ (ζ)dζ, where h ϕ is meromorphic on ϕ(U ),
there is an expansion h ϕ (ζ) =
c n [ζ − ϕ(a)]
n in the neighbourhood of all
a ∈ U ; by definition, the coefficient c −1 is the residue of ω at a, written
Res(ω, a). It also is independent of the choice of the chart [Chap. VIII, n
◦ 5,
(v) : invariance of the residue under conformal representation]. For example,
take ω = dh/h, where h is meromorphic on X; if h is represented in the chart
(U, ϕ) by a meromorphic function h ϕ (ζ), then ω is clearly represented by the
form h
ϕ (ζ)dζ/h ϕ (ζ) ; ω is, therefore, a meromorphic differential form, and
Res a (dh/h) = v a (h)
(1.4)
as in C.
We now prove a result generalizing what has been seen in Chap. 8, n
◦ 5
about functions defined on the Riemann sphere:
Theorem 1. Let X be a compact Riemann surface.
(a) Any function defined and holomorphic on X is a constant.
(b) For any meromorphic function f on X,
v(f ) =
v a (f ) = 0 .
(1.5)
(c) For any meromorphic differential form ω on X,
Res a (ω) = 0 .
(1.6)
(a) is obvious: a function h everywhere holomorphic reaches its maximum
somewhere, and so is constant in the neighbourhood of its maximum. As X
is by definition connected, classical arguments apply verbatim. (Corollary :
the only entire elliptic functions on C are the constants).
To obtain (b), it suffices by (4), to prove (c). It will follow from Stokes’ theorem. The latter can be applied since the Jacobians of holomorphic changes
of charts are > 0, making it possible for X to be oriented by these charts.
Having said this, X being compact and the poles of ω being isolated, the
latter are finite in number; let us denote them by a k (1 ≤ k ≤ n). For each
k, choose a local chart (U k , ϕ k ) such that ϕ k (a k ) = 0 and, for sufficiently
small r > 0, denote by D k (r) the set of x ∈ U k such that |ϕ k (x)| ≤ r. These
“ discs ” are closed in X and if r is sufficiently small, also pairwise disjoint;
then ω has a unique pole at a k in D k (r) and even in D k (r
) for some r
> r.