Poincaré institute

Reinventing rational points

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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.