Rational parametrisations
Theorem (Duval)
There exists a globally Gal(K/K)-invariant set of
parametrisations Xj (t), Yj (t)


, with Xj (t) = bj tej for each j,
such that the roots of f (x)(y) = 0 in K{{x}} are the
Yj

ζej
ej
q
b−1
j x1/ej

for ζ
ej
ej = 1. In particular,
P
j ej = n.
Suppose the Xj (t), Yj (t)


for 1 ⩽ j ⩽ g form a system of
representatives of Galois orbits. For each j, let Kj be
K(bj , coefs of Yj ), and fj = [Kj : K]. Then
Pg
i=1 ej fj = n, and
f (x)(y) =
g
Y
j=1
Y
σ:Kj ,→K
Y
β
ej =b−1
j
y − Yj (βx1/ei
)


| {z }
irr. factors over K((x))
| {z }
irr. factors over K((x))
.
Nicolas Mascot Algebraic curves