# Rational points in families
of varieties

## Dan Loughran

The first couple of lectures will consist of a general introduction to
some algebro-geometric techniques in arithmetic geometry, such as the
Lang-Weil estimates, Hensel's lemma and the Chebotarev density theorem
for finitely generated extensions of the rationals, with the common
theme being the importance of models and schemes. We will also cover
basic terminology and techniques from the fibraton method
(split/non-split fibres and local solubility).
We will then use these techniques to prove the main theorem from the paper:
Loughran, Smeets - *Fibrations with few rational points.*
This theorem says that for families of varieties satisfying certain
geometric conditions, \(100\%\)
of the varieties in the family have no
rational point.
The intended audience is PhD students in arithmetic geometry.