Topics in Arithmetic Geometry

“欧拉系统”理论由Kolyvagin在[1]中提出，后来由Rubin在[2]中以及Mazur和Rubin在[3]中系统地发展。结果在算术几何中涉及L-级数的特殊值和相关p-的塞尔默群的结构之间的关系。数字字段上的进表示。为了将此类应用的范围扩展到重要的新类别示例，特别是解决变形理论中出现的关键问题，Mazur和Rubin最近也在[4]中将欧拉系统理论扩展到自然的“更高等级”设置。这方面的理论目前正在迅速发展，并吸引了许多领先的研究人员的兴趣。然而，为了将一般理论应用于任何给定的算术设置，必须首先提供一个显式的例子欧拉系统（相关的秩），这是相关的L-级数的值。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有理数域的阿贝尔扩张的乘法群中出现的所谓的“分圆元”。在这种情况下，罗伯特科尔曼早期的工作给出了一个美丽的重新解释的相关性质的分圆元素在所谓的“圆形分布”，这使他猜想，每一个欧拉系统，可以出现在上述设置必须产生在一个简单的方式从欧拉系统的分圆元素（见[5]和[6]）。因此，这一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。这似乎也是合理的，相信任何方法导致的一个证明的猜想，也可以阐明的困难，获得欧拉系统在其他重要的settings.During我的博士学位的过程中，我将调查是否可以制定一个自然的分析，在高阶欧拉系统的非常一般的设置中的科尔曼猜想的logue，该高阶欧拉系统出现在任意数字字段。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，其中的元素可以被看作是一个自然的概括分圆元素的模块。基本欧拉系统的这个模块将被构造为在[7]中由Burns和佐野在p-adic表示的设置中构造的同名模块的“全局”模拟。然后，我将寻求精确地制定上述推广的科尔曼猜想在这种设置。这样的推测，模小的技术细节，实际上，在这种情况下，每一个高阶欧拉系统都必须以一种直接的方式从基本欧拉系统中得到。然后，我的目标是通过建立在科尔曼的原始猜想的基础上的技术来为这个猜想提供证据。我希望Burns、Sakamato和佐野在[8]中对Mazur和Rubin关于高阶Euler系统理论的主要猜想的证明能起到关键作用。

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.