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

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).

A smooth intersection of $2$\ quadrics in $\mathbb{P}^n$\ for $n \g...
...ypersurface (of dimension at least $3$) satisfies weak

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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