Brauer-Manin obstruction, K3 surfaces and families of twists of abelian varieties

丢番图方程是纯数学中最古老的部分之一，也是数论发展的起点。勒让德和高斯开创了一种从局部到全局的方法来求解丢番图方程，最终形成了类域理论和数域上二次型的Minkowski-Hasse定理。本质上，这是关于从模为自然数的多项式同余到积分中多项式方程的解的问题。1970年，Manin发现了一种方法，将类场理论应用于整体域上任意代数簇上的有理点的存在性问题。由此产生的布劳尔-马宁阻塞理论已有很多应用。后来，它与费马、莫代尔、塞尔默、卡塞尔的降落法相结合，并与哈塞的纤维化法相结合。这些方法可以用来证明Brauer-Manin障碍控制着某些几何有理簇上有理点的存在和分布。与数论中的传统一样，代数技术的成功取决于解析数论的结果。最近，Green、Tao和Ziegler用加性组合学的方法得到了很强的解析结果。作为应用，解决了长期存在的关于二次曲线和二次曲线族的猜想的重要特例。另一方面，对于至少由一条亏格曲线参数表示的二次曲线族，已经找到了反例。K3曲面是下一类至关重要的代数簇，它在某种意义上占据了有理簇和更一般簇之间的中间地带，在有理簇中，人们期望有理点的行为由Brauer-Manin障碍控制，而在更一般的簇中，没有有效的局部到全局的方法。从另一个角度来看，K3曲面是几何上单连通的二维椭圆曲线，因此人们期待着K3曲面及其上的有理点有一个深刻而丰富的算术理论。目前已知的证明K3曲面上有理点存在的唯一方法是Swinnerton-Dyer。它适用于阿贝尔变种的二次或三次扭曲的族，例如椭圆曲线。椭圆曲线理论最近取得了巨大的突破(由于Bharagava等人)，我们希望能够利用这些结果来推进我们对K3曲面上有理点和更一般簇的理解。

Diophantine equations are one of the oldest parts of pure mathematics and the starting point in the development of number theory. Legendre and Gauss initiated a local-to-global approach to Diophantine equations culminating in class field theory and the Minkowski-Hasse theorem for quadratic forms over number fields. In essence this is the question about the passage from polynomial congruences modulo natural numbers to solutions of polynomial equations in integres. In 1970 Manin found a way to apply class field theory to the problem of existence of rational points on arbitrary algebraic varieties over global fields. The resulting theory of Brauer-Manin obstruction has had very many applications. It was later merged with the method of descent going back to Fermat, Mordell, Selmer, Cassels, and with the method of fibration going back to Hasse. These methods can be used to show that the Brauer-Manin obstruction controls the existence and distribution of rational points on certain geometrically rational varieties. As is traditional in number theory, the success of an algebraic technique depends on results from analytic number theory. Very strong analytic results have recently been obtained by Green, Tao and Ziegler by methods of additive combinatorics. As an application, important particular cases of long standing conjectures about rational families of conics and quadrics have been settled. On the other hand, for families of conics and quadrics parameterised by a curve of genus at least one, counterexamples have been found. K3 surfaces is athe next crucial class of algebraic varieties that in some sense occupies the middle ground between rational varieties, where one expects the behaviour of rational points to be controlled by the Brauer-Manin obstruction, and more general varieties where no efficient local-to-global approach is known. From another perspective, K3 surfaces are geometrically simply connected 2-dimensional analogues of elliptic curves, so one expects a deep and rich arithmetic theory of K3 surfaces and rational points on them. The only method to prove the existence of rational points on K3 surfaces known today is due to Swinnerton-Dyer. It applies to families of quadratic or cubic twists of abelian varieties, e.g. elliptic curves. The theory of elliptic curves has recently seen massive breakthroughs (due to Bharagava and others), and we hope to be able to use these results to advance our understanding of rational points on K3 surfaces and more general varieties.