Topics in Arithmetic Geometry

算术几何专题

• 批准号: 1948356
• 金额: --
• 依托单位: King's College London
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2017
• 资助国家: 英国
• 起止时间: 2017 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-1948356
• 关键词: Topics; Arithmetic; Geometry

The theory of 'Euler systems' was introduced by Kolyvagin in [1] and was later systematically developed by Rubin in [2] and by Mazur and Rubin in [3].It has since played an indispensable role in the proof of many of the most spectacular, and most famous, results in arithmetic geometry concerning relations between the special values of L-series and the structure of the Selmer groups of associated p-adic representations over number fields. With a view towards extending the range of such applications to important new classes of examples and, in particular, to attack key problems that arise in deformation theory, the theory of Euler systems has also recently been expanded by Mazur and Rubin in [4] to a natural 'higher rank' setting. This aspect of the theory is currently undergoing rapid development and is attracting the interest of many leading researchers.However, in order to apply the general theory in any given arithmetic setting, one must first supply an explicit example of an Euler system (of the relevant rank) that is related to the values of L-series. Unfortunately, the search for such examples has so far proven extremely difficult!The most classical example of an Euler system is provided by the so-called 'cyclotomic elements' that arise in the multiplicative group of abelian extensions of the field of rational numbers. In this context, earlier work of Robert Coleman gave a beautiful reinterpretation of the relevant properties of cyclotomic elements in terms of so-called 'circular distributions' and this led him to conjecture that every Euler system that could arise in the aforementioned setting must arise in a straightforward fashion from the Euler system of cyclotomic elements (see [5] and [6]). The validity of this striking conjecture would therefore offer a precise explanation for the mysterious scarcity of Euler systems. It would also seem reasonable to believe that any methods leading to a proof of the conjecture could also shed light on the difficulty of obtaining Euler systems in other significant settings.During the course of my PhD I will be investigating whether or not one can formulate a natural ana- logue of Coleman's conjecture in the very general setting of higher rank Euler systems that arise in the multiplicative group of abelian extensions of arbitrary number fields. In this setting I will first aim to construct a module of so-called 'basic' Euler systems, elements of which can be seen as a natural generalisation of cyclotomic elements. This module of basic Euler systems will be constructed as a 'global' analogue of its namesake that is constructed in the setting of p-adic representations by Burns and Sano in [7]. I will then seek to precisely formulate the aforementioned generalisation of Coleman's conjecture in this setting. Such a conjecture should, modulo minor technical details, in effect state that every higher rank Euler system in this setting is necessarily obtained from a basic Euler system in a straightforward way.I will then aim to provide evidence for this conjecture by building upon the techniques developed in the setting of Coleman's original conjecture by Seo in [9] and [10].I expect a key role to be played by the recent proof of Burns, Sakamato and Sano in [8] of the main conjecture of Mazur and Rubin concerning the theory of higher rank Euler systems.

“欧拉系统”理论是由Kolyvagin在[1]中提出的，后来由Rubin在[1]中系统地发展，由Mazur和Rubin在[3]中系统地发展。从那以后，它在证明算术几何中许多最引人注目、最著名的结果中发挥了不可或缺的作用，这些结果涉及到l级数的特殊值与数域上相关p进表示的Selmer群的结构之间的关系。为了将这种应用范围扩展到重要的新类别的例子中，特别是为了解决变形理论中出现的关键问题，最近，Mazur和Rubin在b[4]中也将欧拉系统理论扩展到一个自然的“更高等级”设置。这方面的理论目前正在迅速发展，并吸引了许多主要研究人员的兴趣。然而，为了将一般理论应用于任何给定的算术设置，必须首先提供与l系列值相关的欧拉系统（相关秩）的显式示例。不幸的是，到目前为止，寻找这样的例子被证明是极其困难的！欧拉系统最经典的例子是所谓的“分环元素”，它出现在有理数域的阿贝尔扩展的乘法群中。在这种背景下，罗伯特·科尔曼（Robert Coleman）的早期工作用所谓的“圆形分布”对分环元素的相关性质进行了漂亮的重新解释，这使他推测，在上述情况下可能出现的每一个欧拉系统都必须以一种直接的方式从分环元素的欧拉系统中产生（见[5]和[6]）。因此，这个惊人猜想的有效性将为欧拉系统的神秘稀缺性提供一个精确的解释。似乎也有理由相信，任何导致证明该猜想的方法，也可以解释在其他重要情况下获得欧拉系统的困难。在我的博士课程中，我将研究是否有人可以在任意数域的阿贝尔扩展的乘法群中出现的高阶欧拉系统的非常一般的设置中形成Coleman猜想的自然模拟。在这种情况下，我将首先致力于构建一个所谓的“基本”欧拉系统模块，其中的元素可以被看作是环切元素的自然推广。基本欧拉系统的这个模块将被构建为其同名的“全局”模拟，该模拟是在Burns和Sano在[7]中的p进表示设置中构建的。然后，我将在这种情况下，试图精确地表述前述对科尔曼猜想的概括。这样的一个猜想应该，模化次要的技术细节，实际上表明，在这种情况下，每一个更高阶的欧拉系统都必须以一种直接的方式从一个基本欧拉系统中得到。然后，我的目标是通过建立在Seo在[9]和[10]中Coleman的原始猜想设置中开发的技术来为这一猜想提供证据。我希望Burns， Sakamato和Sano在Mazur和Rubin关于高阶欧拉系统理论的主要猜想[8]中的最新证明能够发挥关键作用。