Year: 2026
Type: preprint
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.
Source: arXiv (Cornell University)
Author: Sharifi, Romyar
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7220739555