ABSTRACT

X2 −X is G-invariant and gives a map Y = P1 −→ P1/G. To check genericity, observe that any extension L/K with cyclic Galois group of order 3 defines a homomorphism φ : GK −→ G −→ Aut Y which can be viewed as a 1-cocycle with values in Aut Y . The extension L/K is given by a rational point on P1/G if and only if the twist of Y by this cocycle has a rational point not invariant by σ. This is a general property of Galois twists. But this twist has a rational point over a cubic extension of K, and every curve

of genus 0 which has a point over an odd-degree extension is a projective line, and hence has at least one rational point distinct from the ones fixed by σ.