Rational Points and Arithmetic of Fundamental Groups

Abstract

We recall the theory of non-abelian first cohomology as presented in [Se97] I 5 and emphasize the role played by conjugacy classes of sections and by equivariant torsors. The difference between two sections can be described by either a cocycle or an equivariant torsor.The analogue in the non-abelian setup of the familiar abelian long exact sequences for short exact sequences of coefficients or of low degree terms in the Hochschild-Serre spectral sequence turn out to exist but in a restricted form depending on the amount of commutativity that one is willing to spend as an assumption.

Anabelian geometry and descent obstructions on moduli spaces

“…This is widely believed in the case of hyperbolic curves over number fields and is usually referred as the section conjecture. For a similar statement in the non-projective case, one needs to consider the so-called cuspidal sections, see [48], Section 18. Although we will discuss non-projective varieties in what follows, we will not need to specify the notion of cuspidal sections.…”

“…It's not unreasonable to conjecture the same for all anabelian varieties. The relationship between the finite descent obstruction and the section conjecture in anabelian geometry has been discussed by Harari and Stix [20], Stix [48], Section 11 and others. We will review the relevant definitions below, although our point of view will be slightly different.…”

Galois sections for abelian varieties over number fields

“…The section conjecture of anabelian geometry [2] predicts, in the case of proper smooth curves over number fields of genus at least 2, that the profinite Kummer map is a bijection. We refer to [6] for details of the above construction and the section conjecture in general. The main result of this note states that the profinite Kummer map is never surjective for abelian varieties.…”

“…(2) Knowing the prospects of the profinite Kummer map being a bijection for abelian varieties has some significance for attemps to prove the section conjecture. Although the section conjecture avoids a prediction for abelian varieties, the abelianization of sections by composition with the Albanese map X → Alb X may help to understand the case of arbitrary smooth projective curves X, see [6] §3.1.…”

“…under which the profinite Kummer map κ : A(k) → S π 1 (A/k) factors over the limit of the maps δ n : A(k) → H 1 (k, A n ) of the multiplication by n sequence for A, see [6] Corollary 71. The long exact sequence of Galois cohomology leads to a diagram with an exact row (1.2)…”