Modular Symbols in Arithmetic

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The focus of this research project is in number theory, an ancient subject that grew out of the study of arithmetic properties of the integers. Today, connections between arithmetic objects of different mathematical natures (algebraic, geometric, and analytic) form a hallmark of this field. The research in this award fits into this theme while exploring deep connections that were perhaps unexpected until recently. The general principle of the research is that certain algebraic problems can be reinterpreted as geometric problems in a higher dimension: for example, paths on a certain curve determine certain “products” of arithmetically interesting numbers. The research of the PI aims to uncover intricate but explicit relationships of this sort with the broad aim of providing a window through which well-known conjectures and statements of arithmetic may be seen in a new light. Graduate students supported by the award will receive training to contribute towards the research.The research supported by this award concerns analogues of a partly conjectural relationship between modular symbols and Steinberg symbols of cyclotomic units. This says roughly that class groups of cyclotomic fields are explicitly determined by and determine homology groups of modular curves reduced modulo Eisenstein ideals: an explicit map on symbols is expected to be inverse to a map constructed from the Galois action on the cohomology of a modular curve. The primary focus of the award is the extension of the framework and known results to higher-dimensional algebraic groups and other global fields. In particular, the PI plans to significantly generalize a motivic construction of the explicit map in the original conjecture and to analyze Galois representations that should give the inverse maps in cases of arithmetic interest. The overall 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.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.