Modular Symbols in Iwasawa Theory

Conference paper
Part of the Contributions in Mathematical and Computational Sciences book series (CMCS, volume 7)


This survey paper is focused on a connection between the geometry of \(\mathop{\text{GL}}\nolimits _{d}\) and the arithmetic of \(\mathop{\text{GL}}\nolimits _{d-1}\) over global fields, for integers d ≥ 2. For d = 2 over \(\mathbb{Q}\), there is an explicit conjecture of the third author relating the geometry of modular curves and the arithmetic of cyclotomic fields, and it is proven in many instances by the work of the first two authors. The paper is divided into three parts: in the first, we explain the conjecture of the third author and the main result of the first two authors on it. In the second, we explain an analogous conjecture and result for d = 2 over \(\mathbb{F}_{q}(t)\). In the third, we pose questions for general d over the rationals, imaginary quadratic fields, and global function fields.


Tate Kato Betti 



The work of the first two authors (resp., third author) was supported in part by the National Science Foundation under Grant Nos. DMS-1001729 (resp., DMS-0901526).


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© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ChicagoChicagoUSA
  2. 2.Department of MathematicsUniversity of ArizonaTucsonUSA

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