Non-commutative class field theory and Shimura varieties

从包括几何观点在内的多个方面研究了非阿贝尔类场理论。在刚性几何的基础方面，与F.Kato和O Gabber(IHES)在国际合作的基础上进行了联合研究，得出了关于交换环的Hausdorff完备化的结果。作为这项研究的结果，刚性几何的基础现在被建立在更一般的框架中，从而在应用中提供了更多的灵活性。作为非阿贝尔类场论的一部分，我们提出了伽罗瓦表示的形变理论(伽罗瓦形变理论)可直接应用于数论问题的新观点。以素数p为例，研究了二次扩张的相对类数的不可分性。这一点在总体上是确定的。我们的合作者Y.Takai给出了当场是有理数上的伽罗瓦且p足够大时，这种二次扩张的数目的一个下界估计。

Non-abelian class field theory is studied from various aspects, including a geometric viewpoint. As for foundations of rigid geometry, a joint research with F. Kato and O Gabber (IHES) went on based on international collaboration, yielding results on the Hausdorff completions of commutative rings. As a result of this research, the foundation of rigid geometry is now established in a more general framework,giving more flexibility in applications. We have also obtained a clear explanation of the relationships between the notion of R. Huber’s adic spaces and V. Berkovich’s Berkovich spaces.As part of non-abelian class field theory, we provide a new viewpointthat the deformation theory of Galois representations (Galois deformation theory) can be applied directly to number-theoretical problems. The author has studied the indivisibility of relative class numbers of quadratic extensions by a prime number p as a first example. This is established in general. Our collaborator Y. Takai gave a lower bound estimate for the number of such quadratic extensions, when the field is Galois over the rationals and p is sufficiently large.