302
X – The Riemann Surface of an Algebraic Function
calculated by the Newton polygon method,
13 but the latter has fortunately
long disappeared from presentations on Riemann surfaces.
In all cases, the situation over the point a is then easily elucidated. Let
us return to the disc D(a) centered at a used above and let Y 1 , . . . , Y m be
the various connected components of X over D
∗ (a). Each Y i is a connected
covering space of order k i of D
∗ (a), and k 1 + . . . + k m = n since there
are n points of X over each point of B. If q i (z, ζ) denotes the function ϕ Y
corresponding to Y = Y i , so that (Y i , q i ) is a chart of ˆ
X, then
q i (z, ζ)
ki = z − a if a = ∞ , = 1/z if a = ∞ ;
(4.7)
(z, ζ) → ζ is a meromorphic function h i (q i ) on Y i , possibly with a pole at
q i = 0, hence a meromorphic Laurent series in the local uniformizer q i . The
points k i of Y i located over a given z ∈ D
∗ (a) correspond to the k i values of
q i satisfying (7) ; they can be deduced from each other by taking the product
of q i and of the k i -th roots of unity. For z ∈ D
∗ (a), equation P (z, ζ) = 0 has
exactly k i distinct roots such that (z, ζ) ∈ Y i ; they correspond to the distinct
k i points of Y i located over z ; these are the numbers obtained by replacing
q i , in the series h i , by the k i k i -th roots of z − a or of 1/z. As (z, ζ) ∈ Y i tends
to the point η Y = η i adjoined to Y = Y i , ζ tends either to the finite limit
ζ i = h i (0) satisfying P (a, ζ i ) = 0, or to infinity. If P 0 (a) = 0 and a = ∞,
we have seen that the h i are holomorphic at the origin and we deduce that,
for all r > 0, there exists ρ > 0 such that, for |z − a| < ρ, the equation
P (z, ζ) = 0 has at least k i simple roots satisfying |z − ζ i | < r ; as a result, by
lemma 1 of n
◦ 2, the order of multiplicity of the root ζ i of P (z, ζ) = 0 is at
least k i .
As
k i = n, this result suggests that the connected components Y i of
X over D
∗ (a) correspond bijectively to the various roots of the equation
P (a, ζ) = 0, each multiple root ζ i of order k i giving rise to a component Y i
of order k i .
False : ζ i = ζ j may hold for i = j, in which case the multiplicity of the
root ζ i is at least k i + k j .
Exercise 1. Consider the equation
ζ
2
− zζ − z
4 = 0 .
(4.8)
It has double roots in ζ for z = 0, i/2 and −i/2 . Therefore, the Riemann
surface X constructed in n
◦ 2 is the subset of the graph of relation (8) in C
2
located over the open subset
B = C − {0, i/2, −i/2}
13 The shortest presentation, but not necessarily the most accessible, is that of
Dieudonn´ e in Calcul infinit´ esimal, Appendix to Chap. III. The fact that he only
looks for real branches and limited expansions instead of Laurent series in q so
as not to traumatize his novice readers at the outset does not change anything.
Besides, the method applies to functions that are not necessarily algebraic.
X – The Riemann Surface of an Algebraic Function
calculated by the Newton polygon method,
13 but the latter has fortunately
long disappeared from presentations on Riemann surfaces.
In all cases, the situation over the point a is then easily elucidated. Let
us return to the disc D(a) centered at a used above and let Y 1 , . . . , Y m be
the various connected components of X over D
∗ (a). Each Y i is a connected
covering space of order k i of D
∗ (a), and k 1 + . . . + k m = n since there
are n points of X over each point of B. If q i (z, ζ) denotes the function ϕ Y
corresponding to Y = Y i , so that (Y i , q i ) is a chart of ˆ
X, then
q i (z, ζ)
ki = z − a if a = ∞ , = 1/z if a = ∞ ;
(4.7)
(z, ζ) → ζ is a meromorphic function h i (q i ) on Y i , possibly with a pole at
q i = 0, hence a meromorphic Laurent series in the local uniformizer q i . The
points k i of Y i located over a given z ∈ D
∗ (a) correspond to the k i values of
q i satisfying (7) ; they can be deduced from each other by taking the product
of q i and of the k i -th roots of unity. For z ∈ D
∗ (a), equation P (z, ζ) = 0 has
exactly k i distinct roots such that (z, ζ) ∈ Y i ; they correspond to the distinct
k i points of Y i located over z ; these are the numbers obtained by replacing
q i , in the series h i , by the k i k i -th roots of z − a or of 1/z. As (z, ζ) ∈ Y i tends
to the point η Y = η i adjoined to Y = Y i , ζ tends either to the finite limit
ζ i = h i (0) satisfying P (a, ζ i ) = 0, or to infinity. If P 0 (a) = 0 and a = ∞,
we have seen that the h i are holomorphic at the origin and we deduce that,
for all r > 0, there exists ρ > 0 such that, for |z − a| < ρ, the equation
P (z, ζ) = 0 has at least k i simple roots satisfying |z − ζ i | < r ; as a result, by
lemma 1 of n
◦ 2, the order of multiplicity of the root ζ i of P (z, ζ) = 0 is at
least k i .
As
k i = n, this result suggests that the connected components Y i of
X over D
∗ (a) correspond bijectively to the various roots of the equation
P (a, ζ) = 0, each multiple root ζ i of order k i giving rise to a component Y i
of order k i .
False : ζ i = ζ j may hold for i = j, in which case the multiplicity of the
root ζ i is at least k i + k j .
Exercise 1. Consider the equation
ζ
2
− zζ − z
4 = 0 .
(4.8)
It has double roots in ζ for z = 0, i/2 and −i/2 . Therefore, the Riemann
surface X constructed in n
◦ 2 is the subset of the graph of relation (8) in C
2
located over the open subset
B = C − {0, i/2, −i/2}
13 The shortest presentation, but not necessarily the most accessible, is that of
Dieudonn´ e in Calcul infinit´ esimal, Appendix to Chap. III. The fact that he only
looks for real branches and limited expansions instead of Laurent series in q so
as not to traumatize his novice readers at the outset does not change anything.
Besides, the method applies to functions that are not necessarily algebraic.