Through the lens of the Eisenstein ideal

Abstract

This paper exposits relationships between the geometry of modular curves and the arithmetic of cyclotomic fields through the Eisenstein ideal. For an integer $N \ge 5$ and an odd prime $p$, we define two conjecturally inverse maps between the real part $P$ of the Eisenstein reduction of the homology $P$ of $X_1(N)$ and the real part of the $p$-part $Y$ of the second $K$-group of the $N$th cyclotomic integer ring. We recall a motivic construction of the map from $P$ to $Y$ and describe the recent proof of the Eisenstein property of the underlying map on modular symbols. We provide a fully equivariant construction of the second map from $Y$ to $P$ using the Eisenstein reduction of the first étale cohomology of $X_1(N)$. We then explain an equivariant construction of the Euler system of cyclotomic units using extension classes in relative cohomology groups of modular curves and describe its place in our program.