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

Posted on August 27, 2019

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.


  1. 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: