What question did this study set out to answer?

The research aims to explore the validity of Grothendieck’s section conjecture for hyperbolic curves of index less than one.

April 26, 2026Open Access

On Grothendieck’s section conjecture for curves of index 1

Key Points

The research aims to explore the validity of Grothendieck’s section conjecture for hyperbolic curves of index less than one.
Provided examples of curves satisfying the conjecture with index different from one.
Proved that hyperbolic curves with non-cyclic p-group actions have twisted forms of index one satisfying the conjecture.
Established connections between finite extension fields and finite étale covers for hyperbolic curves.
Demonstrated examples of hyperbolic curves satisfying Grothendieck’s section conjecture with index one are common.
Verified that for every hyperbolic curve over a finitely generated field, a corresponding finite extension and finite étale cover exists.
Found that every hyperbolic curve with a faithful action of a non-cyclic p-group has a twisted form that satisfies the conjecture.

Abstract

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.

On Grothendieck’s section conjecture for curves of index 1

Key Points

Abstract

Cite This Study

Also Consider

Also Consider