On the birational p-adic section conjecture

Abstract

In this article we introduce and prove a Z/p meta-abelian form of the birational p-adic section conjecture for curves. This is a much stronger result than the usual p-adic birational section conjecture for curves, and makes an effective p-adic section conjecture for curves quite plausible.

Conditional results on the birational section conjecture over small number fields

“…On the other hand, this follows immediately from [12], Theorem A, (2). This completes the proof of Proposition 1.7.…”

“…Then it follows from a result obtained in [8], as well as [12], that, for each p ∈ P f k , a birational Galois section s of X/k uniquely determines a k p -valued point x p of X such that, for any open subscheme U ⊆ X of X, the image of the homomorphism G p → π 1 (U ⊗ k k p ) naturally determined by the isomorphism π 1 …”

“…Moreover, in [12], Pop obtained a result concerning birational Galois sections over p-adic local fields [cf. [12], Theorem A], which leads naturally to a proof of the geometrically prop version of Koenigsmann's result over p-adic local fields [cf. Proposition 1.7].…”

The Brauer–Manin obstruction for sections of the fundamental group

“…The most convincing evidence consists perhaps in Koenigsmann's proof in [17] of a birational analogue for function fields in one variable over a local p-adic field. A minimalist metabelian pro-p approach to Koenigsmann's theorem was successfully developed by Pop in [33], whereas Esnault and Wittenberg,[8,Prop. 3.1], were able to reprove algebraically the abelian part via the cycle class of a section.…”

On the -adic section conjecture

“…If one only assumes that this condition holds, without assuming the liftability condition, then the proof of the main theorem of Pop is still valid in this case (cf. the details in the proof of the main results in [3]). Now, going back to the proof of Theorem A, assertion (A3) implies in our situation that the natural map p Br k → p Br L; where L is as above, is injective.…”