Transcendental Brauer-Manin obstruction on a surface with a fibration in degree four curvesPosted on August 27, 2019
In Olivier Wittenberg described an example of a transcendental Brauer-Manin obstruction to weak approximation on an elliptic surface over .
The Hasse principle is the idea that if has a -rational point for every completion , then it should have a -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 has an associated Brauer group .
One way to think of is as where ``unramified’’ means that at every point we could evaluate and get an algebra over the residue field .
The adele ring is the restricted product of the completions of . A failure of the Hasse principle is then when
Each element defines an associated set satisfying
If but we say that provides a Brauer-Manin obstruction to the Hasse principle.
If is a proper subset of then we say provides an obstruction to weak approximation.
The algebraic Brauer group of is the kernel of the base change to map
The transcendental elements of the Brauer group are then those that don’t belong to .
The Brauer group of a curve is always entirely algebraic.
If is a split elliptic curve with -rational -torsion there is an explicit description of the -torsion Brauer group .
Wittenberg looks at an elliptic surface over , i.e. a surface with a fibration and generic fiber isomorphic to an elliptic curve .
There is an inclusion So as long as an element in doesn’t become trivial in it will be a transcendental element of .
Moreover, to check whether an element of lies in 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 that gives an obstruction to weak approximation.
- 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