Troisi\`eme groupe de cohomologie non ramifi\'ee des torseurs universels sur les surfaces rationnelles

Abstract

Let $k$ a field of characteristic zero. Let $X$ be a smooth, projective, geometrically rational $k$-surface. Let $\mathcal{T}$ be a universal torsor over $X$ with a $k$-point et $\mathcal{T}^c$ a smooth compactification of $\mathcal{T}$. There is an open question: is $\mathcal{T}^c$ $k$-birationally equivalent to a projective space? We know that the unramified cohomology groups of degree 1 and 2 of $\mathcal{T}$ and $\mathcal{T}^c$ are reduced to their constant part. For the analogue of the third cohomology groups, we give a sufficient condition using the Galois structure of the geometrical Picard group of $X$. This enables us to show that $H^{3}_{nr}(\mathcal{T}^{c},\mathbb{Q}/\mathbb{Z}(2))/H^3(k,\mathbb{Q}/\mathbb{Z}(2))$ vanishes if $X$ is a generalised Ch\^atelet surface and that this group is reduced to its $2$-primary part if $X$ is a del Pezzo surface of degree at least 2. Comment: 27 page, in French

A Comparison of Cohomological Obstructions to the Hasse Principle and to Weak Approximation

“…Let X 2 (T ′ ) be the group of everywhere locally trivial elements of H 2 (K , T ′ ). Restricting to the subgroup X 2 (T ′ ) of H 2 (K , T ′ ) yields a map (1) , Y (A K ) ̸ = ∅ implies that the class [Y ] ∈ H 1 (K , T ) actually lies in X 1 (T ). Now we arrive at:…”

“…As for weak approximation, Harari, Scheiderer and Szamuely announced that the defect to weak approximation for tori can be described by X 2 ω (T ′ ), where X 2 ω (T ′ ) denotes the subgroup of H 2 (K , T ′ ) consisting of elements vanishing in H 2 (K v , T ′ ) for all but finitely many v ∈ X (1) . To this end, they constructed a pairing (see [7, §4, pp.…”

“…Let us make the above statements more precise. Let X (1) be the set of closed points on X . The local ring O X ,v at v ∈ X (1) is a discrete valuation ring, so we may denote by K v the completion of K with respect to v ∈ X (1) .…”

“…Let X (1) be the set of closed points on X . The local ring O X ,v at v ∈ X (1) is a discrete valuation ring, so we may denote by K v the completion of K with respect to v ∈ X (1) . Let Y be a K -torsor under a K -torus T .…”

Non rationalité stable sur les corps quelconques

“…-[6] Soit X une surface projective, lisse connexe, géométriquement rationnelle sur un corps k. Si X n'est pas k-birationnelle à une surface de del Pezzo k-minimale de degré 1, et si T est un torseur universel sur X avec un k-point, H3 nr (k(T )/k, Q/Z(2))/H 3 (k, Q/Z(2)) est un groupe de torsion 2-primaire. Soit k un corps de car.…”