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。此后，它在许多证明中发挥了不可或缺的作用。最引人注目、最著名的是关于 L 级数的特殊值与数域上相关 p 进数表示的 Selmer 群结构之间关系的算术几何结果。为了将此类应用的范围扩展到重要的新类别示例，特别是为了解决变形理论中出现的关键问题，Euler 系统的理论最近也被 Mazur 和 Rubin 在 [4] 中扩展为自然的“更高级别”设置。该理论的这一方面目前正在快速发展，并吸引了许多领先研究人员的兴趣。然而，为了将一般理论应用于任何给定的算术设置，必须首先提供欧拉系统的明确示例（相关的等级）与 L 系列的值相关。不幸的是，到目前为止，寻找这样的例子极其困难！欧拉系统最经典的例子是由所谓的“分圆元素”提供的，这些元素出现在有理数域的阿贝尔扩张的乘法群中。在这种背景下，罗伯特·科尔曼的早期工作用所谓的“循环分布”对分圆元素的相关属性进行了漂亮的重新解释，这使他推测在上述环境中可能出现的每个欧拉系统都必须出现在来自分圆元素欧拉系统的一种简单方式（参见[5]和[6]）。因此，这一引人注目的猜想的有效性将为欧拉系统的神秘稀缺性提供精确的解释。似乎也有理由相信，任何导致猜想证明的方法也可以揭示在其他重要环境中获得欧拉系统的困难。在我的博士学位课程中，我将研究是否可以制定一个科尔曼猜想在高阶欧拉系统的非常一般的设置中的自然类似物，该系统出现在任意数域的阿贝尔扩展的乘法群中。在这种情况下，我首先致力于构建一个所谓“基本”欧拉系统的模块，其中的元素可以被视为分圆元素的自然概括。基本欧拉系统的这个模块将被构造为其同名的“全局”类似物，该类似物是在 Burns 和 Sano 在 [7] 中的 p 进表示设置中构造的。然后，我将在这种情况下精确地阐述科尔曼猜想的上述概括。这样的猜想应该以次要技术细节为模，实际上表明，在这种情况下，每个更高阶的欧拉系统都必然以简单的方式从基本欧拉系统获得。然后，我将致力于通过建立在这些技术的基础上，为这个猜想提供证据Seo 在 [9] 和 [10] 中在 Coleman 的原始猜想的基础上发展起来。我预计 Burns、Sakamato 和 Sano 在 [8] 中的主要猜想的最新证明将发挥关键作用Mazur 和 Rubin 关于高阶欧拉系统的理论。