4 – The Riemann Surface of an Algebraic Function
297
k, f transforms [ζ
, ζ
+ n] into a necessarily closed lifting of t → z
e(nt). So
f (ζ
) = f (ζ
).
It follows that f is a homeomorphism of the quotient P/kZ onto Y , which
is obviously compatible with the complex analytic structures of both spaces
and with the projections q : Y −→ D
∗ and p k : P/kZ −→ D
∗ , so that the
given covering (Y, D
∗ , q) is isomorphic, including from a complex analytic
point of view, to the canonical covering (P/kZ, D
∗ , p k ) of example 2 of this n
◦ .
Hence it remains to construct the function ϕ of the theorem for this particular
covering. However, the map z → e(z/k) from P onto D
∗ is invariant under
kZ ; so there is one and only one map ϕ : P/kZ −→ D
∗ which, for all z ∈ P ,
transforms the class of z mod kZ into e(z/k). The reader is left to check that
ϕ is holomorphic and satisfies conditions (i) and (ii) of the theorem.
It applies to all disc centered at a ∈ ˆ
C ; replace relation ϕ(η)
k = q(η) by
ϕ(η)
k = q(η) − a if a = ∞ , ϕ(η)
k = 1/q(η) if a = ∞ .
It applies also to coverings of C
∗ : replace P by C in the preceding arguments.
4 – The Riemann Surface of an Algebraic Function
(i) Global uniform branches. Once again, consider an irreducible equation
P (z, ζ) = P 0 (z)ζ
n + . . . + P n (z) = 0
(4.1)
of degree n in ζ and, as in n
◦ 2, remove from C the finite set of the values of
z where (1) does not have n distinct roots. This gives an open subset B. As
shown, the subset X ⊂ C
2 of the curve P = 0 which projects onto B is both
a Riemann surface and a covering of order n of B . If (a, α) ∈ X and if D
is a sufficiently small open disc centered at a, there is a unique holomorphic
function f (z) in D satisfying P [z, f (z)] = 0 and f (a) = α, and the couple
(f (D), p) is a local holomorphic chart of X in the neighbourhood of (a, α),
defined in the “ disc ” D(α) = f (D). As (z, ζ) ∈ f (D) means that ζ = f (z),
the functions p : (z, ζ) → z and F : (z, ζ) → ζ are holomorphic in this chart,
and hence in X in the sense of n
◦ 1. More generally, if f and g are polynomial,
the functions (z, ζ) → f (z, ζ) and (z, ζ) → g(z, ζ) are holomorphic on X, so
that (z, ζ) → f (z, ζ)/g(z, ζ) is meromorphic if g does not identically vanish
on X (i.e. is not a multiple of the polynomial P ).
A first immediate consequence of these results concerns global uniform
branches of an algebraic function. Their existence is governed by the following result, where we keep the above notation :
Th 6. Let U be a simply connected open set contained in B. There are n
holomorphic functions f k in U whose values at each z ∈ U are the n roots of
the equation P (z, ζ) = 0.
297
k, f transforms [ζ
, ζ
+ n] into a necessarily closed lifting of t → z
e(nt). So
f (ζ
) = f (ζ
).
It follows that f is a homeomorphism of the quotient P/kZ onto Y , which
is obviously compatible with the complex analytic structures of both spaces
and with the projections q : Y −→ D
∗ and p k : P/kZ −→ D
∗ , so that the
given covering (Y, D
∗ , q) is isomorphic, including from a complex analytic
point of view, to the canonical covering (P/kZ, D
∗ , p k ) of example 2 of this n
◦ .
Hence it remains to construct the function ϕ of the theorem for this particular
covering. However, the map z → e(z/k) from P onto D
∗ is invariant under
kZ ; so there is one and only one map ϕ : P/kZ −→ D
∗ which, for all z ∈ P ,
transforms the class of z mod kZ into e(z/k). The reader is left to check that
ϕ is holomorphic and satisfies conditions (i) and (ii) of the theorem.
It applies to all disc centered at a ∈ ˆ
C ; replace relation ϕ(η)
k = q(η) by
ϕ(η)
k = q(η) − a if a = ∞ , ϕ(η)
k = 1/q(η) if a = ∞ .
It applies also to coverings of C
∗ : replace P by C in the preceding arguments.
4 – The Riemann Surface of an Algebraic Function
(i) Global uniform branches. Once again, consider an irreducible equation
P (z, ζ) = P 0 (z)ζ
n + . . . + P n (z) = 0
(4.1)
of degree n in ζ and, as in n
◦ 2, remove from C the finite set of the values of
z where (1) does not have n distinct roots. This gives an open subset B. As
shown, the subset X ⊂ C
2 of the curve P = 0 which projects onto B is both
a Riemann surface and a covering of order n of B . If (a, α) ∈ X and if D
is a sufficiently small open disc centered at a, there is a unique holomorphic
function f (z) in D satisfying P [z, f (z)] = 0 and f (a) = α, and the couple
(f (D), p) is a local holomorphic chart of X in the neighbourhood of (a, α),
defined in the “ disc ” D(α) = f (D). As (z, ζ) ∈ f (D) means that ζ = f (z),
the functions p : (z, ζ) → z and F : (z, ζ) → ζ are holomorphic in this chart,
and hence in X in the sense of n
◦ 1. More generally, if f and g are polynomial,
the functions (z, ζ) → f (z, ζ) and (z, ζ) → g(z, ζ) are holomorphic on X, so
that (z, ζ) → f (z, ζ)/g(z, ζ) is meromorphic if g does not identically vanish
on X (i.e. is not a multiple of the polynomial P ).
A first immediate consequence of these results concerns global uniform
branches of an algebraic function. Their existence is governed by the following result, where we keep the above notation :
Th 6. Let U be a simply connected open set contained in B. There are n
holomorphic functions f k in U whose values at each z ∈ U are the n roots of
the equation P (z, ζ) = 0.