Local-to-global principles for random Diophantine equations

本文主要研究多项式方程的积分解和有理解。用几何术语来说，这意味着研究代数变量上的积分点和有理点。我们将考虑无穷几何族，并证明在给定族中有正比例的变种有有理点。我们还将研究具有Brauer-Manin障碍的局域到全局原理所适用的变异的比例，即当类场论允许时，局域处处有理点的存在意味着全局点的存在。主要的工具是我们的定理，即Schinzel假设(H)以1的概率成立。假设(H)预测具有满足自然必要条件的整系数多项式具有无穷多个素数。我们计划证明这个定理的更一般的版本，这将导致关于随机振动成二次曲线和二次曲线的局部到全局原理的新结果。我们将尝试将我们的方法推广到更一般的纤维中，特别是，我们的目标是将有关纤维族的结果证明为属1的曲线，包括对有理点知之甚少的K3曲面。

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.