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 ≥ 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 X1(N) and the real part of the p-part Y of the second K-group of the Nth 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 X1(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.
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BibTeX
@misc{sharifi2026through,
title = {{Through the lens of the Eisenstein ideal}},
author = {Romyar Sharifi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.04943},
eprint = {2610.04943},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.04943},
}
RIS
TY - UNPB
TI - Through the lens of the Eisenstein ideal
AU - Romyar Sharifi
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2610.04943
UR - https://arxiv.org/abs/2610.04943
AB - This paper exposits relationships between the geometry of modular curves and the arithmetic of cyclotomic fields through the Eisenstein ideal. For an integer N ≥ 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 Nth 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.
ER -