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丢番图方程的局部到全局的方法，最终形成了类场论和数域上二次型的闵可夫斯基-哈斯定理。从本质上讲，这是关于从多项式同余模自然数到多项式方程的整数解的通道的问题。在1970年马宁发现了一种方法，适用于类领域理论的问题存在合理的点任意代数品种的全球领域。由此产生的Brauer-Manin阻塞理论有很多应用。它后来与追溯到费马、莫德尔、塞尔默、卡塞尔的下降法合并，并与追溯到哈塞的纤维化法合并。这些方法可以用来证明Brauer-Manin阻塞控制某些几何有理簇上有理点的存在和分布。正如数论中的传统，代数技巧的成功取决于解析数论的结果。最近绿色，陶和齐格勒通过加法组合学的方法得到了很强的分析结果。作为应用，解决了二次曲线和二次曲线有理族长期以来的重要特例。另一方面，对于由亏格至少为1的曲线参数化的二次曲线和二次曲线族，已经发现了反例。K3曲面是代数簇的下一个关键类，在某种意义上占据了有理簇之间的中间地带，其中人们期望有理点的行为由Brauer-Manin障碍控制，以及更一般的变种，其中没有已知的有效的局部到全局方法。从另一个角度看，K3曲面是椭圆曲线的几何单连通二维类似物，因此人们期待着K3曲面及其上的有理点的深入而丰富的算术理论。唯一的方法来证明存在的合理点的K3曲面今天已知的是由于Swinnerton-Dyer。它适用于阿贝尔簇的二次或三次扭曲的族，例如椭圆曲线。椭圆曲线理论最近有了巨大的突破（由于Bharagava等人），我们希望能够利用这些结果来推进我们对K3曲面和更一般的曲面上的有理点的理解。

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.