2-Nilpotent Real Section Conjecture

Abstract

We show a 2-nilpotent section conjecture over R: for a geometrically connected curve X over R such that each irreducible component of its normalization has R-points, pi_0(X(R)) is determined by the maximal 2-nilpotent quotient of the fundamental group with its Galois action, as the kernel of an obstruction of Jordan Ellenberg. This implies that for X smooth and proper, X(R)^{+/-} is determined by the maximal 2-nilpotent quotient of Gal(C(X)) with its Gal(R)-action, where X(R)^{+/-} denotes the set of real points equipped with a real tangent direction, showing a 2-nilpotent birational real section conjecture.

24 pages, 1 figure. To appear Mathematiche Annalen

References in corpus (3)

• Introduction to birational anabelian geometry
• The real section conjecture and Smith's fixed point theorem for pro-spaces
• Reconstruction of function fields

Cited by in corpus (4)

• An elementary approach to Stix's proof of the real section conjecture
• Algebraic part of motivic cohomology with compact supports
• On the nilpotent section conjecture for finite group actions on curves
• 3-nilpotent obstructions to pi_1 sections for P^1_Q - {0,1,infty}