Harari 2: Weak approximation on algebraic varieties (cohomology)

Let $X$ be a smooth, geometrically integral variety over $k$ (a number field), and suppose that $X$ is projective. We denote by $\overline{X(k)}$ the closure of $X(k)$ in $\prod_{v \in \Omega_k} X(k_v)=X(\mathbf A_k)$.

Here our aim is to: (i) explain the counterexamples to weak approximation; (ii) find `intermediate' sets $E$ between $\overline{X(k)}$ and $X(\mathbf A_k)$; (iii) in some cases, prove that $E=\overline{X(k)}$.

General setting

Let $G/k$ be an algebraic group (usually linear, but not necessarily connected, e.g. $G$ finite). If $G$ is commutative: define $H^i(X,G)$ the étale cohomology groups ($i=1,2$; the cohomological dimension of a number field forgetting real places makes the higher cohomology groups uninteresting). In general, we have only the pointed set $H^1(X,G)$ (defined by Cech cocycles for the étale topology). If $X={\mathrm{Spec}} k$, $H^1(X,G)=H^1(\Gamma,G(\overline{k}))$, where $\Gamma={\mathrm{Gal}}(\overline{k}/k)$. If $G$ is linear, $H^1(X,G)$ corresponds to $G$-torsors over $X$ up to isomorphism.

Take $f \in H^i(X,G)$, define

$$X(\mathbf A_k)^f=\left\{(M_v) \in X(\mathbf A_k):(f(M_v)) \in {\mathrm{img}}(H^i(k,G) \to \textstyle{\prod}_v H^i(k_v,G))\right\}.$$

Obviously $X(k) \subset X(\mathbf A_k)^f$. We will see that in many cases $\overline{X(k)} \subset X(\mathbf A_k)^f$

Example.

1. ${\mathrm{Br}}X=H^2(X,\mathbb{G}_m)$;
   $$\overline{X(k)} \subset X(\mathbf A_k)^{{\mathrm{Br}}} = \bigcap_{f \in {\mathrm{Br}}X} X(\mathbf A_k)^f.$$

   (Indeed the Brauer group of the ring of integers of $k_v$ is zero). $X(\mathbf A_k)^{{\mathrm{Br}}}$ is the Brauer-Manin set of $X$. Manin showed in 1970 that for a genus one curve with finite Tate-Shafarevich group, the condition $X(\mathbf A_k)^{{\mathrm{Br}}} \neq \emptyset$ implies the existence of a rational point.

2. Let $f:Y \to X$ be a Galois, geometrically connected, nontrivial étale covering with group $G$. Then $f \in H^1(X,G)$, where $G$ is considered as a constant group scheme. Then $\overline{X(k)} \subset X(\mathbf A_k)^f$ (via Hermite's Theorem). It is possible to find $(M_v) \not\in X(\mathbf A_k)^f$, which implies Minchev's result that $X$ does not satisfy weak approximation.

Remark. If $X$ is rational, then ${\mathrm{Br}}X/{\mathrm{Br}}k=H^1(k,{\mathrm{Pic}}\overline{X})$ is finite, where $\overline{X} = X \times_k \overline{k}$. Then $X(\mathbf A_k)^{{\mathrm{Br}}}$ is `computable'.

Theorem. [H, Skorobogatov] If $G$ is linear and $f \in H^1(X,G)$, then $\overline{X(k)} \subset X(\mathbf A_k)^f$ (and $X(\mathbf A_k)^f$ is "computable").

Abelian descent theory

This was developed by Colliot-Thélène and Sansuc, and recently completed by Skorobogatov.

Theorem. Define

$$X(\mathbf A_k)^{{\mathrm{Br}}_1}=\bigcap_{f \in {\mathrm{Br}}_1 X} X(\mathbf A_k)^f$$

where ${\mathrm{Br}}_1 X = \ker({\mathrm{Br}}X \to {\mathrm{Br}}\overline{X})$. Assume that $X(\mathbf A_k)^{{\mathrm{Br}}_1} \neq \emptyset$. Then:

1. We have
   $$X(\mathbf A_k)^{{\mathrm{Br}}_1} = \bigcap_{\substack{ f \in H^1(X,S) \\ S\text{\textup{ of multiplicative type}}}} X(\mathbf A_k)^f.$$

2. Assume further that ${\mathrm{Pic}}\overline{X}$ is of finite type, set $S_0$ such that $\widehat{S_0}={\mathrm{Pic}}\overline{X}$; then there exists a torsor $f_0:Y \to X$ under $S_0$ (a universal torsor, i.e. "as nontrivial as possible") such that
   $$X(\mathbf A_k)^{{\mathrm{Br}}_1} = X(\mathbf A_k)^{f_0}.$$

This Theorem is difficult, see Skorobogatov's book for a complete account on the subject. One of the ideas is to recover the Brauer group of $X$ (mod. ${\mathrm{Br}}k$) making cup-products $[Y] \cup a$, where $a \in H^1(k,\widehat S_0)$ and $[Y]$ is the class of $Y$ in $H^1(X,S_0)$.

Now assume that $X$ is a rational variety, so $X(\mathbf A_k)^{{\mathrm{Br}}}=X(\mathbf A_k)^{{\mathrm{Br}}_1}$ (since ${\mathrm{Br}}\overline{X}=0$). Assume $X(\mathbf A_k)^{{\mathrm{Br}}} \neq \emptyset$. Consider a universal torsor $f:Y \xrightarrow{S_0} X$. If $\sigma \in H^1(k,S_0)$, can define $f^\sigma:Y^\sigma \to X$ where

$$[Y^\sigma]=[Y]-\sigma \in H^1(X,S_0).$$

Then

$$X(\mathbf A_k)^f = \bigcup_{\sigma \in H^1(k,S_0)} f^\sigma(Y^{\sigma}(\mathbf A_k)).$$

If you can prove that the torsors $Y^\sigma$ satisfy weak approximation, then $\overline{X(k)}=X(\mathbf A_k)^f=X(\mathbf A_k)^{{\mathrm{Br}}}$, so the Brauer-Manin obstruction is the only one.

Example. There are many examples of this:

1. Châtelet surface: $y^2-az^2=P(x)$, $a \in k^*/k^{*2}$, $\deg P=4$. Colliot-Thélène, Sansuc, Swinnerton-Dyer showed that $\overline{X(k)}=X(\mathbf A_k)^{{\mathrm{Br}}}$, so the Brauer-Manin obstruction is the only one. If $P$ is irreducible, then ${\mathrm{Br}}X/{\mathrm{Br}}k=0$, so $X$ satisfies weak approximation.

   If $P$ is reducible, we can have a counterexample to weak approximation, e.g. $y^2-az^2=f_1(x)f_2(x)$, where $\deg f_1=\deg f_2=2$, $\gcd(f_1,f_2)=1$, in some cases there is an obstruction given by the Hilbert symbol $f=(a,f_1)$.

2. Conic bundles over $\mathbb{P}^1$ with at most $5$ degenerate fibres. Results of Colliot-Thélène, Salberger, Skorobogatov covered at most $4$. In the case of $5$, the existence of a global rational point is easy to show, so the only problem is weak approximation, and which is due to Salberger, Skorobogatov 1993 (using descent and $K$-theory).

Theorem. [Sansuc 1981] Let $G$ be a linear connected algebraic group over $k$, $X$ a smooth compactification of $G$, then the Brauer-Manin obstruction is the only one:

$$\overline{X(k)} = X(\mathbf A_k)^{{\mathrm{Br}}}.$$

Back to fibration methods

If $p:X \to B$ is a fibration, we saw that if the base and the fibres satisfy weak approximation, under certain circumstances then $X$ satisfies weak approximation.

Here we consider $p:X \to \mathbb{P}^1$, a projective, surjective morphism (and the generic fibre $X_\eta$ is smooth). Assume also that all fibres are geometrically integral (can do with all but one because of strong approximation on the affine line).

Theorem. [H 1993, 1996] Yes, $\overline{X(k)}=X(\mathbf A_k)^{{\mathrm{Br}}}$ if you assume that:

1. ${\mathrm{Pic}}\overline{X_\eta}$ is torsion-free, where $\overline{X_\eta}=X_\eta \times_{K} \overline{K}$, $K=k(\eta)$; e.g. $X_\eta$ rational, or smooth complete intersection of dimension at least three.
2. ${\mathrm{Br}}\overline{X_\eta}$ is finite.

Two ideas:

1. ${\mathrm{Br}}X_\eta/{\mathrm{Br}}K \to {\mathrm{Br}}X_P/{\mathrm{Br}}k$ is an isomorphism for many $k$-fibres $X_P$ (`many' in the sense of Hilbert's irreducibility theorem).
2. If $\alpha_1,\dots,\alpha_r$ are elements of ${\mathrm{Br}}X_\eta$, assume $\alpha_i \in {\mathrm{Br}}U$, $U \subset X$ an open subset. Use the `formal lemma': Take $(M_v) \in X(\mathbf A_k)^{{\mathrm{Br}}}$, $M_v \in U$, $\Sigma_0$ a finite set of places; then there exists $(P_v) \in X(\mathbf A_k)$, $P_v \in U$, $\Sigma \supset \Sigma_0$ finite such that:
   1. $P_v=M_v$ for $v \in \Sigma_0$;
   2. $\sum_{v \in \Sigma} j_v(\alpha_i(P_v))=0$ for $1 \leq i \leq r$, where $j_v:{\mathrm{Br}}k_v \to \mathbb{Q}/\mathbb{Z}$ is the local invariant.

Applications: (i) Recover Sansuc's result just knowing the case of a torus; (ii) If you know that $\overline{X(k)}=X(\mathbf A_k)^{{\mathrm{Br}}}$ for $X$ a smooth cubic surface, then by induction the same holds for hypersurfaces, so if $\dim X \geq 3$, then $X$ satisfies weak approximation.

Nonabelian descent

If $G/k$ is a finite but not commutative $k$-group, it is possible that for $f \in H^1(X,G)$, $X(\mathbf A_k)^f \not\supset X(\mathbf A_k)^{{\mathrm{Br}}}$.

Theorem. [Skorobogatov 1997] There exists $X/\mathbb{Q}$ a bi-elliptic surface such that $X(\mathbb{Q})=\emptyset$, $X(\mathbf A_\mathbb{Q})^{{\mathrm{Br}}} \neq \emptyset$.

Actually: $X(\mathbf A_\mathbb{Q})^f=\emptyset$ for some $f \in H^1(X,G)$, $G(\overline{\mathbb{Q}})=(\mathbb{Z}/4\mathbb{Z})^2 \rtimes \mathbb{Z}/2\mathbb{Z}$.

There are similar statements for weak approximation (H 1998), e.g. take $X/k$ any bi-elliptic surface, $X(k) \neq \emptyset$, then $\overline{X(k)} \subsetneq X(\mathbf A_k)^{{\mathrm{Br}}}$.

Nevertheless the Brauer-Manin condition is quite strong, as shows the following result :

Theorem. [H 2001] We have:

1. If $G/k$ is a linear connected $k$-group, $f \in H^1(X,G)$, then
   $$X(\mathbf A_k)^{{\mathrm{Br}}} \subset X(\mathbf A_k)^f.$$

2. If $G$ is any commutative $k$-group, $f \in H^2(X,G)$, then
   $$X(\mathbf A_k)^{{\mathrm{Br}}} \subset X(\mathbf A_k)^f.$$

Open question : is the first part of this theorem still true for a $G$ which is an extension of a finite abelian group by a connected linear group ? My guess is "no".

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