Each curve for which Grothendieck’s section conjecture has been proved has no rational points, and additionally it has index different from 1 . We provide many new examples of curves satisfying the conjecture; in particular, we prove that examples of index 1 are very common. Given an odd prime p , we prove that every hyperbolic curve with a faithful action of a non-cyclic p -group has a twisted form of index 1 which satisfies Grothendieck’s section conjecture. Furthermore, we prove that for every hyperbolic curve S over a field k finitely generated over ℚ there exists a finite extension K / k and a finite étale cover C → S K such that C satisfies the conjecture.
Giulio Bresciani (Fri,) studied this question.