Arithmetic equivalence of modular curves and Shimura varieties

这个项目是数论和代数几何，以及这些主题相遇的界面。该项目的核心是Shimura品种。Shimura变体是编码大量算术和几何信息的几何对象。它们是数学中中心对象的参数空间，即阿贝尔簇，也是数论的许多重要方面的设定，尤其是朗兰兹规划。我们将感兴趣的几个自然问题与下村簇的几何和算术有关。第一个问题涉及Shimura品种的Zeta功能。Zeta函数是分析对象，可以附加到许多不同的数学结构上，并编码各种属性。众所周知，非同构的Shimura簇可以具有相同的Zeta函数，并且已经找到了明确的例子。我们打算探索这些例子，并构造具有相同Zeta函数的非同构的Shimura变种的新族。其中第二个问题涉及Shimura变种中的所谓的“不可能的交集”。现在有一个成熟的研究领域，围绕着所谓的特殊亚种在下村品种中的分布，换句话说，较小的下村品种如何位于较大的环境下的下村品种中。模型理论的新结果催生了研究这些问题的新方法，我们打算将这些方法应用于不太可能相交的未解决问题，这些问题现在看起来很容易处理。其中第三个问题涉及Shimura簇的几何和算术性质，其中许多与上面第二个问题有关。我们打算调查与所谓的Hecke对应的程度、特殊亚种的定义领域以及特殊亚种的复杂性有关的各种问题，所有这些都是不太可能的交叉中的技术成分，但它们本身也是有趣的数学片段。研究上述问题的方法论集中在从模型论、数论和代数几何的新工具的开发和应用，以及寻求不同数学分支之间的新联系。该项目将要求学生学习大量的新数学，以及这些特定研究领域的最新方法，以使他能够获得新的结果。

This project is in number theory and algebraic geometry, and the interface at which these topics meet. At the centre of the project are Shimura varieties. Shimura varieties are geometric objects that encode a great deal of arithmetic and geometric information. They are parameter spaces for central objects in mathematics, namely, abelian varieties, and the setting for many important aspects of number theory, not least the Langlands Programme.We will be interested in several natural questions pertaining the geometry and arithmetic of Shimura varieties. The first of these questions relates to zeta functions of Shimura varieties. Zeta functions are analytic objects that can be attached to many different mathematical structures and encode a variety of properties. It is known that non-isomorphic Shimura varieties may possess the same zeta function, and explicit examples have been found. We intend to explore these examples and construct new families of non-isomorphic Shimura varieties sharing the same zeta function.The second of these questions relates to so-called "unlikely intersections" in Shimura varieties. There is now a well-established area of research centred around the distribution of so-called special subvarieties in a Shimura variety, in other words, how smaller Shimura varieties sit inside a larger ambient Shimura variety. New results from model theory have given rise to new methods through which to investigate these questions, and we intend to apply these methods to unresolved problems of unlikely intersections that now appear tractable.The third of these questions relates to geometric and arithmetic properties of Shimura varieties, many of which have relevance for the second question above. We intend to investigate various problems relating to the degrees of so-called Hecke correspondences, the fields of definition of special subvarieties, and the complexities of special subvarities, all of which are technical ingredients in unlikely intersections, but also interesting pieces of mathematics in their own right.The methodology for studying the above questions is centred in the development and application of new tools, from model theory, number theory, and algebraic geometry, as well as pursuing new connections between different branches of mathematics. The project will require the student to learn a substantial amount of new mathematics, as well as state of the art methodologies in these specific areas of research, in order to enable him to obtain new results.