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.
