an:05566608
Zbl 1247.11090
Demarche, Cyril
Descent obstruction and ??tale Brauer-Manin obstruction
FR
Algebra Number Theory 3, No. 2, 237-254 (2009).
00250441
2009
j
11G35 14G05 11E72 14G25
Brauer-Manin obstruction; descent obstruction; Hasse principle; torsors; Galois cohomology
Given an algebraic variety \(X\) over a number field \(k\) with ring of ad??les \(\mathbb{A}_k\), one has the obvious inclusion of the set \(X(k)\) of rational points into the set \(X(\mathbb{A}_k)\) of adelic points, and the smaller set may be empty while the larger one is not, as classical counterexamples to the Hasse principle show. Several kinds of intermediate subsets provide general theories of obstruction to the Hasse principle. The first one, introduced by Manin and called the \textit{Brauer-Manin obstruction}, concerns the subset \(X(\mathbb{A}_k)^{\text{Br}}\) of \(X(\mathbb{A}_k)\) consisting of adelic points that are orthogonal to the Brauer group \(\text{Br}(X):=H^2_{\text{??t}}(X,\mathbb{G}_m)\) with respect to the Brauer-Manin pairing. The \textit{descent obstruction} is constructed by using torsors under linear algebraic groups: If \(f: Y\overset{G}{\to} X\) is a torsor over \(X\) under a linear algebraic \(k\)-group \(G\), every cocycle class \([\sigma]\in H^1(k,G)\) defines a twisted torsor \(f^{\sigma}: Y^{\sigma}\to X\) over \(X\). Adelic points coming from a twist of \(f\) (i.e. lying in \(f^{\sigma}(Y^{\sigma}(\mathbb{A}_k))\) for some \([\sigma]\in H^1(k,G)\)) form the descent subset \(X(\mathbb{A}_k)^f\), and the intersection \(X(\mathbb{A}_k)^{\text{desc}}\) of all the descent subsets \(X(\mathbb{A}_k)^f\) defined by torsors \(f: Y\to X\) defines the descent obstruction. The \textit{??tale Brauer-Manin obstruction} is of some combinatorial nature. It is defined by the set
\[
X(\mathbb{A}_k)^{\text{??t},\text{Br}}:=\bigcap_{f: Y\overset{G}{\to} X,G\text{ finite}}\bigcup_{[\sigma]\in H^1(k,G)}f^{\sigma}(Y^{\sigma}(\mathbb{A}_k)^{\text{Br}}).
\]
An example constructed by \textit{A. N. Skorobogatov} [Invent. Math. 135, No. 2, 399--424 (1999; Zbl 0951.14013)] shows that both the descent obstruction and the ??tale Brauer-Manin obstruction are strictly finer than the Brauer-Manin obstruction. More recently, \textit{B. Poonen} [Ann. Math. (2) 171, No. 3, 2157--2169 (2010; Zbl 1284.11096)] has given examples of varieties \(X\) for which \(X(\mathbb{A}_k)^{\text{??t},\text{Br}}\neq \emptyset\) but \(X(k)=\emptyset\), showing that the ??tale Brauer-Manin obstruction may not be the only obstruction to the Hasse principle.
Comparison between the descent and the ??tale Brauer-Manin obstructions has been discussed earlier by several authors. In the paper under review, the author proves the inclusion \(X(\mathbb{A}_k)^{\text{??t},\text{Br}}\subseteq X(\mathbb{A}_k)^{\text{desc}}\) for smooth projective geometrically integral varieties \(X\) over a number field \(k\). This answers in the affirmative a question of Poonen [loc. cit.] as well as a similar question of \textit{M. Stoll} [Algebra Number Theory 1, No. 4, 349--391 (2007; Zbl 1167.11024)], and together with the opposite inclusion proved by \textit{A. Skorobogatov} [Math. Ann. 344, No. 3, 501--510 (2009; Zbl 1180.14017)], this implies that the descent obstruction coincides with the ??tale Brauer-Manin obstruction.
The main idea in the proof of the main theorem is to reduce the question of lifting adelic points for torsors under general groups to the special cases of torsors under finite or connected groups. This relies on a result about lifting 1-cococyles whose proof constitutes the most technical part.
Yong Hu (Paris)
Zbl 0951.14013; Zbl 1167.11024; Zbl 1180.14017; Zbl 1284.11096