Level compatibility in Sharifi's conjecture

TL;DR

This paper investigates the relationships between Sharifi's homology-to-$K$-group maps for varying levels, using advanced techniques to understand their compatibility and underlying structure.

Contribution

It analyzes the compatibility of Sharifi's maps across different levels, extending previous work with new methodological insights.

Findings

Established relations between maps for different levels

Extended Sharifi's techniques to new contexts

Provided structural insights into $K$-groups and modular curves

Abstract

Sharifi has constructed a map from the first homology of the modular curve $X_1(M)$ to the $K$-group $K_2(\mathbf{Z}[\zeta_M, \frac{1}{M}])$, where $\zeta_M$ is a primitive $M$th root of unity. We study how these maps relate when $M$ varies. Our method relies on the techniques developed by Sharifi and Venkatesh.