Modular symbols in Iwasawa theory

TL;DR

This survey explores the deep connections between the geometry of $ ext{GL}_d$ and the arithmetic of $ ext{GL}_{d-1}$ over global fields, highlighting conjectures and results in Iwasawa theory for various dimensions and fields.

Contribution

It presents a conjecture relating modular symbols and Iwasawa theory for general $d$, discusses proven cases for $d=2$, and proposes questions for broader settings.

Findings

Conjecture linking $ ext{GL}_d$ geometry with $ ext{GL}_{d-1}$ arithmetic.

Proven cases of the conjecture for $d=2$ over $ ext{Q}$.

Open questions for higher $d$ and different global fields.

Abstract

This survey paper is focused on a connection between the geometry of $\mathrm{GL}_d$ and the arithmetic of $\mathrm{GL}_{d-1}$ over global fields, for integers $d \ge 2$. 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.