Selmer groups and the arithmetic of modular symbols

代数数论的一个显著方面在于它发现了性质完全不同的对象之间的联系。 PI研究的首要原则是，某些代数问题可以直接重新解释为更高维度的几何问题，在数学的不同部分之间提供迷人的联系。 通过分析性质的中介对象，已经间接地发现了大量这样的联系。 提案中概述的研究旨在提供一个窗口，通过这个窗口，人们可以以新的、更直接的眼光来看待算术的著名命题和陈述。PI已经阐明了模符号和分圆域算术之间复杂但明确的关系，这可以被视为对岩泽主要猜想的完善。 粗略地说，这个猜想通过Eisenstein理想的作用将分圆域的类群与模曲线的同调群的同调元等同起来。 该奖项的中心项目是将这一猜想扩展到更高的维度和其他全球领域。 期望局部对称空间的几何应该明确地确定伽罗瓦表示中的格的算术，也就是说塞尔默群的结构。

A remarkable aspect of algebraic number theory lies in the connections it finds between objects that appear to be of entirely different natures. The overarching principle of the research of the PI is that certain algebraic problems can be reinterpreted directly as geometric problems in higher dimension, providing fascinating connections between objects from different parts of mathematics. A great wealth of such connections have been found indirectly through intermediate objects of an analytic nature. The research outlined in the proposal aims to provide a window through which well-known conjectures and statements of arithmetic may be seen in a new and more direct light.The PI has conjectured an intricate but explicit relationship between modular symbols and the arithmetic of cyclotomic fields that may be viewed as refining the Iwasawa main conjecture. Roughly speaking, this conjecture identifies class groups of cyclotomic fields with quotients of homology groups of modular curves by actions of Eisenstein ideals. The central project of the award is the extension of this conjecture to higher dimensions and other global fields. The expectation is that the geometry of locally symmetric spaces should explicitly determine the arithmetic of lattices in Galois representations, which is to say the structure of Selmer groups.