Selmer groups and the arithmetic of modular symbols

代数论的一个显著特点在于，它发现了看起来性质完全不同的物体之间的联系。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，在来自数学不同部分的对象之间提供了迷人的联系。通过分析性质的中间对象间接地发现了大量这样的联系。该提案中概述的研究旨在提供一个窗口，通过它可以更直接地新地看到众所周知的猜想和算术语句。PI猜想了模符号和割圆域的算法之间复杂而明确的关系，这可以被认为是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，用模曲线的同调群的商来确定分圆域的类群。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决定Galois表示中的格的算术，也就是Selmer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.