Nick Rome

The Loughran-Smeets Conjectures and Conic Bundles

This talk aims to survey the surge of recent interest in the study of rational points in families of varieties, specifically how many varieties in a family have a rational point. Special attention will be paid to the case of conic bundles and the connection with the solubility of ternary quadratic forms. Don't be put off by the geometric language, at the heart of the results are analytic number theory techniques, including the circle method and character sum estimates in the spirit of Fouvry--Klüners.