Local-to-global principles for random Diophantine equations

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The proposed research concerns integral and rational solutions of polynomial equations. In geometric terms this means studying integral and rational points on algebraic varieties. We shall consider infinite, geometric families and prove that a positive proportion of varieties in a given family have rational points. We shall also study the proportion of varieties for which the local-to-global principle with the Brauer-Manin obstruction holds, that is, the existence of rational points everywhere locally implies the existence of global points when this is allowed by class field theory. The main tool is our theorem that Schinzel's Hypothesis (H) holds with probability 1. Hypothesis (H) predicts that polynomials with integral coefficients satisfying natural necessary conditions have infinitely many prime values. We plan to prove more general versions of this theorem that would lead to new results on the local-to-global principle for random fibrations into conics and quadrics. We shall try to generalise our method for more general fibres, in particular, we aim at proving results about families of fibrations into curves of genus 1, including K3 surfaces where very little is known about rational points.