Transcendental Brauer-Manin obstruction on a surface with a fibration in degree four curves

In $2004$ Olivier Wittenberg described an example of a transcendental Brauer-Manin obstruction to weak approximation on an elliptic surface over $\mathbb{Q}$ .

The Hasse principle is the idea that if $X/\mathbb{Q}$ has a $\mathbb{Q}_v$ -rational point for every completion $\mathbb{Q}_v$ , then it should have a $\mathbb{Q}$ -rational point.

Unfortunately this doesn’t always hold. The Brauer-Manin obstruction is a way to explain some failures of the Hasse principle.

Each variety $X/K$ has an associated Brauer group $\text{Br}(X)$ .

One way to think of $\text{Br}(X)$ is as $\{ \text{unramified central simple algebras over} \ K(X) \} / \{ \text{matrix algebras} \}$ where ``unramified’’ means that at every point $x\in X(\overline{K})$ we could evaluate and get an algebra over the residue field $k(x)$ .

The adele ring $\mathbb{A}_\mathbb{Q}=\widehat{\prod}_v\mathbb{Q}_v$ is the restricted product of the completions $\mathbb{Q}_v$ of $\mathbb{Q}$ . A failure of the Hasse principle is then when $X(\mathbb{Q})=\emptyset \quad \text{but} \quad X(\mathbb{A}_\mathbb{Q})\ne\emptyset.$

Each element $\alpha\in\text{Br}(X)$ defines an associated set $X(\mathbb{A}_{\mathbb{Q}})^\alpha$ satisfying $X(\mathbb{Q})\subseteq X(\mathbb{A}_{\mathbb{Q}})^\alpha \subseteq X(\mathbb{A}_{\mathbb{Q}}).$

If $X(\mathbb{A}_{\mathbb{Q}})^\alpha=\emptyset$ but $X(\mathbb{A}_\mathbb{Q})\ne\emptyset$ we say that $\alpha$ provides a Brauer-Manin obstruction to the Hasse principle.

If $X(\mathbb{A}_{\mathbb{Q}})^\alpha$ is a proper subset of $X(\mathbb{A}_{\mathbb{Q}})$ then we say $\alpha$ provides an obstruction to weak approximation.

The algebraic Brauer group $\text{Br}_1(X)$ of $X/K$ is the kernel of the base change to $\overline{K}$ map $\text{Br}(X)\to\text{Br}(\overline{X}).$

The transcendental elements of the Brauer group are then those that don’t belong to $\text{Br}_1(X)$ .

The Brauer group of a curve is always entirely algebraic.

If $E/K$ is a split elliptic curve with $K$ -rational $2$ -torsion there is an explicit description of the $2$ -torsion Brauer group $\text{Br}(E)[2]$ .

Wittenberg looks at an elliptic surface over $\mathcal{E}/\mathbb{Q}$ , i.e. a surface with a fibration $\pi:\mathcal{E}\to\mathbb{P}^1_\mathbb{Q}$ and generic fiber isomorphic to an elliptic curve $E/\mathbb{Q}(t)$ .

There is an inclusion $\text{Br}(\mathcal{E}/\mathbb{Q})[2]\subset\text{Br}(E/\mathbb{Q}(t))[2].$ So as long as an element in $\text{Br}(\mathcal{E})[2]$ doesn’t become trivial in $\text{Br}(E/\overline{\mathbb{Q}}(t))$ it will be a transcendental element of $\text{Br}(\mathcal{E})[2]$ .

Moreover, to check whether an element of $\text{Br}(E/\mathbb{Q}(t))[2]$ lies in $\text{Br}(\mathcal{E})$ it is enough to check ramification along the finite number of special fibers.

In Wittenberg’s paper he finds a particular elliptic surface where this all works out to find a transcendental element of $\text{Br}(\mathcal{E})$ that gives an obstruction to weak approximation.

References

Wittenberg, Olivier, Transcendental Brauer-Manin obstruction on a pencil of elliptic curves. Arithmetic of higher-dimensional algebraic varieties (Palo Alto, CA, 2002), 259–267, Progr. Math., 226, Birkhäuser Boston, Boston, MA, 2004. Available: https://arxiv.org/abs/1502.04386