Harari 1: Weak approximation on algebraic varieties (introduction)

Let $k$ be a number field, and let $k_v$ be the completion of $k$ at $v$. Let $\Omega_k$ be the set of all places of $k$.

Basic Facts

Theorem. [Weak Approximation] Let $\Sigma \subset \Omega_k$ be a finite set of places of $k$. Let $\alpha_v \in k_v$ for $v \in \Sigma$. Then there is an $\alpha \in k$ which is arbitrarily close to $\alpha_v$ for $v \in \Sigma$.

This is a refinement of the Chinese remainder theorem. One reformulation of it is as follows: the diagonal embedding $k \hookrightarrow \prod_{v \in \Omega_k} k_v$ is dense, the product equipped with the product of the $v$-adic topologies.

We have the slight refinement: $\mathbb{P}^1(k)$ is dense in $\prod_v \mathbb{P}^1(k_v)$.

Definition. Let $X/k$ be a geometrically integral algebraic variety. Then $X$ satisfies weak approximation if given $\Sigma \subset \Omega_k$ a finite set of places and $M_v \in X(k_v)$ for $v \in \Sigma$, there exists a $k$-rational point $M \in X(k)$ which is arbitrarily close to $M_v$ for $v \in \Sigma$.

Care must be taken if $\prod_{v \in \Omega_k} k_v$ is empty; by convention, we will say that in this case $X$ satisfies weak approximation even if $X(k)$ is empty.

We see weak approximation is equivalent to the statement that $X(k)$ is dense in $\prod_v X(k_v)$.

Remark. If $X$ is projective, $X(\mathbf A_k)=\prod_v X(k_v)$ and weak approximation is equivalent to strong approximation, namely, $X(k)$ is dense in $X(\mathbf A_k)$ for the adelic topology. (Here, $X(k_v)=\mathscr{X}(\mathcal{O}_v)$, $\mathscr{X}\to {\mathrm{Spec}} \mathcal{O}_k$ a flat and proper model of $X$.)

Let $X,X'$ be smooth. Assume that $X$ is $k$-birational to $X'$. Then $X$ satisfies weak approximation if and only if $X'$ satisfies weak approximation (a consequence of the implicit function theorem for $k_v$).

We can speak about weak approximation for a function field $k(X)$: this means that weak approximation holds for any smooth (projective) model of $X$.

Example. The spaces $\mathbb{A}_k^1, \mathbb{P}_k^1$, and more generally, $\mathbb{A}_k^n,\mathbb{P}_k^n$, satisfy weak approximation, as does any $k$-rational variety, e.g. a smooth quadric with a $k$-point.

More Examples

Theorem. Let $Q \subset \mathbb{P}_k^n$ a (smooth) projective quadric. Then $Q$ satisfies weak approximation.

Here, we do not assume that there is a $k$-rational point. This is the difficult part, the Hasse-Minkowski theorem: if $Q(k_v) \neq \emptyset$ for all $v$, then $Q(k) \neq \emptyset$.

There are several results for complete intersections:

1. A smooth intersection of $2$ quadrics $X \subset \mathbb{P}_k^n$ (Colliot-Thélène, Sansuc, Swinnerton-Dyer 1987) satisfies weak approximation if $n \geq 8$ or if $n \geq 4$ and there exists a pair of skew-conjugate lines on $X$.

2. Châtelet surfaces: $y^2-az^2=P(x)$, where $\deg P=4$, $a \in k^*-k^{*2}$. If $P$ is irreducible, then $X$ (a smooth projective model) satisfies weak approximation. (Uses descent method.)

3. The circle method: $X \subset \mathbb{P}_k^n$ a smooth cubic hypersurface, then weak approximation holds for $n \geq 16$ (Skinner 1997).

There are also results for linear algebraic groups:

1. If $T$ is a $k$-torus, and $\dim T \leq 2$, then $T$ satisfies weak approximation beacuse $T$ is $k$-rational (Voskreseskii).
2. If $G$ is a semi-simple, simply connected linear $k$-group, then $G$ satisfies weak approximation (Kneser-Platonov, around 1969).

The Fibration Method

Theorem. Let $p:X \to B$ be a projective, flat surjective morphism (with $X$ smooth, to simplify). Assume that

1. $B$ is projective and satisfies weak approximation;
2. Almost all $k$-fibers of $p$ satisfy weak approximation; and
3. All fibers of $p$ are geometrically integral.

Then $X$ satisfies weak approximation.

(Here almost all means on a Zariski-dense open subset).

There are refinements when $B$ is the projective space : you can accept degenerate fibers on one hyperplane (using the strong approximation theorem for the affine space).

Applications: (i) Hasse-Minkowski theorem, from four variables to five; (ii) intersection of $2$ quadrics in $\mathbb{P}^n$ for $n \geq 8$ (here one uses a fibration in Châtelet surfaces) and $n \geq 5$ with a pair of skew conjugate lines (to go from $n=4$ to $n \geq 5$ by induction); (iii) cubic hypersurfaces of dimension $\geq 4$ with 3 conjugate singular points (Colliot-Thélène, Salberger).

Proof. Start with $M_v$ a smooth $k_v$-point for any $v$ on $X$. Project $p(M_v)=P_v \in B(k_v)$. Use weak approximation on $B$, so can approximate $P_v$ by $P \in B(k)$ for $v \in S$. Consider the fiber $p^{-1}(P)=X_P \subset X$; $X_P$ has a $k_v$-point $M_v'$ close to $M_v$ for $v \in \Sigma$ by the implicit function theorem. To apply weak approximation on $X_P$, we check that $X_P(k_v) \neq \emptyset$ for $v \not\in \Sigma$; this is OK if $\Sigma$ is sufficiently large by the Weil estimates : here we use that all $k$-fibers are geometrically irreducible, which implies that the reduction mod. $v$ of $X_P$ also is for a sufficiently large $v$ (independent of $P$). $\qedsymbol$

Some Counterexamples

Cubic surfaces: the surface $5x^3+9y^3+10z^3+12w^3=0$ fails the Hasse Principle (Cassels, Guy).

Certain intersections of two quadrics in $\mathbb{P}_k^4$ (see above).

Looking (over the rationals) at $y^2+z^2=f(x)g(x)$, $\deg(f)=\deg(g)=2$, $\gcd(f,g)=1$, it is possible to construct counterexamples to weak approximation. The idea: $K=\mathbb{Q}(i)$, $K_v=K \otimes_\mathbb{Q}\mathbb{Q}_v$; there exists a finite set $\Sigma_0$ such that if $v \not\in \Sigma_0$ and $M_v \in X(\mathbb{Q}_v)$, then $f(M_v)$ is a norm of $K_v/\mathbb{Q}_v$ (use a computation with valuations). If you find $\Sigma \supset \Sigma_0$ and $v_0 \in \Sigma$ such that there exists $M_{v_0}$ such that $f(M_{v_0})$ is not a local norm and $v \neq v_0$ there exists $M_v$ such that $f(M_v)$ is a local norm, then there is no weak approximation. (Think: global reciprocity of class field theory.)

For tori, let $K/k$ be a biquadratic extension, then there are counterexamples like $T:N_{K/k}(x_1w_1+\dots+x_4w_4)=1$, where $w_1,\dots,w_4$ is a basis of $K/k$; this holds e.g. for $k=\mathbb{Q}$, $K=\mathbb{Q}(i,\sqrt{5})$.

Theorem. [Minchev] Let $X$ be a projective, smooth $k$-variety, assume that $\pi_1(\overline{X}) \neq 0$, where $\overline{X}=X \otimes \overline{k}$, $\overline{k}$ an algebraic closure. Assume $X(k) \neq \emptyset$, then $X$ does not satisfy weak approximation.

Proof. [Sketch of proof] Enlarge the situation over ${\mathrm{Spec}}\mathscr{O}_{k,\Sigma_0}$ where $\Sigma_0$ is a finite set of places. By assumption, there is a nontrivial geometrically connected covering $Y \to X$, which for models gives $\mathscr{Y}\to \mathscr{X}$. Take an arbitrary $M \in X(k)$, the fibre $Y_M={\mathrm{Spec}}L$ where $L$ is an étale algebra $L=k_1 \times \dots \times k_r$; each $k_i$ is unramified outside $\Sigma_0$. Only finitely many $k_i$ are possible (by Hermite's Theorem). Find $v \not\in \Sigma_0$ such that $v$ is totally split for each $k_i$ (by Cebotarev's Theorem); find $M_v$ such that the fiber of $Y$ at $X$ for $M_v$ is not (this is possible because $Y$ is geometrically connected, via a "geometric" Cebotarev-like Theorem). Then $M_v$ cannot be approximated by a rational point $M$. $\qedsymbol$

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