Algebraic groups and rational points.
(Groupes algébriques et points rationnels.)

*(French)*Zbl 1042.14004Let \(k\) be an algebraic number field, and denote by \(\overline k\) a fixed algebraic closure of \(k\), with \(\Gamma= \text{Gal}(\overline k/k)\); the set of places of \(k\) is denoted by \(\Omega_k\). Let \(X\) be an algebraic variety defined over \(k\), and suppose that \(X\) possesses points in every completion of \(k\), which (by results of Lang and Weil and of Hensel) is equivalent to saying that the set \(X(\mathbb{A}_k)\) of adelic points of \(X\) is non-empty.

For every algebraic \(k\)-group, \(G\), the cohomology group \(H^1_{\text{et}}(X,G)\) (defined by Čech cocycles with respect to the étale cohomology) is denoted by \(H^1(X, G)\). For \(G\) commutative, \(H^1(X, G)\) is the usual Abelian cohomology group and the groups \(H^i(X,G)\), \(i\geq 2\), may be defined similarly. For \(X= \text{Spec\,}k\), \(H^1(k, G)\) denotes the usual Galois cohomology (after J.-P. Serre [Cohomologie Galoisienne. 5ème éd., Lect. Notes Math. 5. Springer-Verlag, Berlin (1994; Zbl 0812.12002)]).

The set \(X(k)\) of rational points of \(X\) may be mapped into the set \(X(\mathbb{A}_k)\). Let \(f\in H^i(X, G)\), \(i= 1,2\), where one supposes \(G\) commutative if \(i= 2\). Then for every completion \(k_v\) of \(k\) and each point \(P_v\in X(k_v)\), one has \(f(P_v)\in H^i(k_v, G)\), and if \((P_v)\in X(\mathbb{A}_k)\), the evaluation \((f(P_v))\) is an element of \(\prod_{v\in\Omega_k} H^i(k_v, G)\). The subset \(X(\mathbb{A}_k)^f\) is defined by \[ X(\mathbb{A}_k)^f= \Biggl\{(P_v)\in X(\mathbb{A}_k): (f(P_v))\in \text{Im}\Biggl[H^i(k, G)\to \prod_{v\in\Omega_k} H^i(k_v, G)\Biggr]\Biggr\}. \] One has \(X(k)\subset X(\mathbb{A}_k)^f\) and if \(X(\mathbb{A}_k)^f= \varnothing\) then \(X\) is a counterexample to the Hasse principle. Now write \(X(\mathbb{A}_k)^{\text{Br}}= \bigcap_{f\in \text{Br\,}X} X(\mathbb{A}_k)^f\), where \(\text{Br\,}X\) denotes the Brauer group, \(H^2(X, G_m)\). Then one has \(X(k)\subset X(\mathbb{A}_k)^{\text{Br}}\). The condition \(X(\mathbb{A}_k)^{\text{Br}}= \varnothing\) is the Manin obstruction to the Hasse principle, and if one sets \[ X(\mathbb{A}_k)^{\text{Br}_1}= \bigcap_{f\in \text{Br}_1 X}X(\mathbb{A}_k)^f, \] then \(X(\mathbb{A}_k)^{\text{Br}_1}= \varnothing\) is the algebraic Manin obstruction to the Hasse principle.

The paper explores the relationship between those several obstructions to the Hasse principle; for example if \(G\) is commutative and \(X\) is smooth with \(\overline k[X]^*=\overline k^*\), then the algebraic Manin obstruction is finer than those depending on \(H^1(X, G)\), and if \(G\) is linear and connected (but the geometric variety \(X\) is arbitrary) then the Manin obstruction is finer than those depending on \(H^1(X, G)\).

For every algebraic \(k\)-group, \(G\), the cohomology group \(H^1_{\text{et}}(X,G)\) (defined by Čech cocycles with respect to the étale cohomology) is denoted by \(H^1(X, G)\). For \(G\) commutative, \(H^1(X, G)\) is the usual Abelian cohomology group and the groups \(H^i(X,G)\), \(i\geq 2\), may be defined similarly. For \(X= \text{Spec\,}k\), \(H^1(k, G)\) denotes the usual Galois cohomology (after J.-P. Serre [Cohomologie Galoisienne. 5ème éd., Lect. Notes Math. 5. Springer-Verlag, Berlin (1994; Zbl 0812.12002)]).

The set \(X(k)\) of rational points of \(X\) may be mapped into the set \(X(\mathbb{A}_k)\). Let \(f\in H^i(X, G)\), \(i= 1,2\), where one supposes \(G\) commutative if \(i= 2\). Then for every completion \(k_v\) of \(k\) and each point \(P_v\in X(k_v)\), one has \(f(P_v)\in H^i(k_v, G)\), and if \((P_v)\in X(\mathbb{A}_k)\), the evaluation \((f(P_v))\) is an element of \(\prod_{v\in\Omega_k} H^i(k_v, G)\). The subset \(X(\mathbb{A}_k)^f\) is defined by \[ X(\mathbb{A}_k)^f= \Biggl\{(P_v)\in X(\mathbb{A}_k): (f(P_v))\in \text{Im}\Biggl[H^i(k, G)\to \prod_{v\in\Omega_k} H^i(k_v, G)\Biggr]\Biggr\}. \] One has \(X(k)\subset X(\mathbb{A}_k)^f\) and if \(X(\mathbb{A}_k)^f= \varnothing\) then \(X\) is a counterexample to the Hasse principle. Now write \(X(\mathbb{A}_k)^{\text{Br}}= \bigcap_{f\in \text{Br\,}X} X(\mathbb{A}_k)^f\), where \(\text{Br\,}X\) denotes the Brauer group, \(H^2(X, G_m)\). Then one has \(X(k)\subset X(\mathbb{A}_k)^{\text{Br}}\). The condition \(X(\mathbb{A}_k)^{\text{Br}}= \varnothing\) is the Manin obstruction to the Hasse principle, and if one sets \[ X(\mathbb{A}_k)^{\text{Br}_1}= \bigcap_{f\in \text{Br}_1 X}X(\mathbb{A}_k)^f, \] then \(X(\mathbb{A}_k)^{\text{Br}_1}= \varnothing\) is the algebraic Manin obstruction to the Hasse principle.

The paper explores the relationship between those several obstructions to the Hasse principle; for example if \(G\) is commutative and \(X\) is smooth with \(\overline k[X]^*=\overline k^*\), then the algebraic Manin obstruction is finer than those depending on \(H^1(X, G)\), and if \(G\) is linear and connected (but the geometric variety \(X\) is arbitrary) then the Manin obstruction is finer than those depending on \(H^1(X, G)\).

Reviewer: J. V. Armitage (Durham)

##### MSC:

14G25 | Global ground fields in algebraic geometry |

14L10 | Group varieties |

14L30 | Group actions on varieties or schemes (quotients) |

11G35 | Varieties over global fields |

11E72 | Galois cohomology of linear algebraic groups |