The Space of Sections in the Arithmetic Case and the Section Conjecture in Covers

Abstract

We resume the discussion of the space of sections from Chap. 4 under arithmetic assumptions. The space

$${\mathcal{S}}_{{\pi }_{{}_{ 1}}(X/k)}$$

turns out to be a compact profinite space if X ∕ k is a smooth projective curve over an algebraic number field, see Proposition 97 or for any hyperbolic curve if k is a finite extension of ${\mathbb{Q}}_{p}$. We also recall the known relation between a weak form of the section conjecture, see Conjecture 100, and the genuine version, see Conjecture 2, and between the claim for affine versus proper curves.In the arithmetic situation, the centraliser of a section is always trivial, see Proposition 104. It follows that, just like rational points, sections obey Galois descent, see Corollary 107. Furthermore, we examine going up and going down for the section conjecture with respect to a finite étale map. The discussion relies on Chap. 3.