Arithmetic equivalence of modular curves and Shimura varieties

这个项目是在数论和代数几何，以及接口，这些主题满足。该项目的核心是志村品种。志村变种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，以及数论许多重要方面的背景，尤其是朗兰兹纲领。我们将对有关志村簇的几何和算术的几个自然问题感兴趣。这些问题中的第一个涉及志村品种的zeta函数。Zeta函数是可以附加到许多不同的数学结构并编码各种属性的分析对象。已知非同构的Shimura簇可能具有相同的zeta函数，并且已经发现了明确的例子，我们打算探索这些例子并构造新的具有相同zeta函数的非同构的Shimura簇族。第二个问题涉及Shimura簇中所谓的“不可能交叉”。现在有一个完善的研究领域，集中在志村品种中所谓的特殊子品种的分布上，换句话说，较小的志村品种如何位于较大的环境志村品种中。模型论的新结果产生了新的方法来研究这些问题，我们打算把这些方法应用到那些不太可能相交的未解决的问题上，这些问题现在看来是易于处理的。第三个问题涉及志村变种的几何和算术性质，其中许多与上面的第二个问题有关。我们打算研究与所谓的Hecke对应的程度有关的各种问题，特殊子变种的定义领域，以及特殊子变种的复杂性，所有这些都是不太可能的交叉点的技术成分，但它们本身也是有趣的数学作品。研究上述问题的方法论集中在新工具的开发和应用上，从模型论，数论，代数几何，以及追求数学不同分支之间的新联系。该项目将要求学生学习大量的新数学，以及在这些特定研究领域的最先进的方法，以使他能够获得新的成果。

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.