Number fields as curves over F1
Abstract
We study function fields in one variable over the field with one element F1. It is proved that the Galois extensions of Q are isomorphic to the curves over F1 being understood as the Deitmar schemes. Specifically, one gets explicit formulas linking the genus and the number of cusps of an algebraic curve over the extension F1m of order m≥ 1 of the field F1 and the m-th roots of unity of the corresponding number field. It follows that elliptic curves with one cusp over F1 are either the cyclotomic fields or the maximal abelian unramified extensions of the quadratic number fields. Our proof depends on representaion of the Drinfeld modules by the bounded linear operators on a Hilbert space and the crossed product structure of the Cuntz-Krieger algebras.
Related
- Splitting Brauer classes by genus one curves over number fields
- Explicit lower bounds on the conductors of elliptic curves and abelian varieties over number fields
- Prescribed Abscissae on Congruent-Number Curves over Simplest Cubic Fields
- Heavenly Elliptic Curves over Cubic Number Fields
- Fermat curves and explicit class field theory for real quadratic number fields
BibTeX
@misc{nikolaev2026number,
title = {{Number fields as curves over $\mathbfF_1$}},
author = {Igor V. Nikolaev},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.16360},
eprint = {2609.16360},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.16360},
}
RIS
TY - UNPB
TI - Number fields as curves over F_1
AU - Igor V. Nikolaev
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.16360
UR - https://arxiv.org/abs/2609.16360
AB - We study function fields in one variable over the field with one element F_1. It is proved that the Galois extensions of Q are isomorphic to the curves over F_1 being understood as the Deitmar schemes. Specifically, one gets explicit formulas linking the genus and the number of cusps of an algebraic curve over the extension F_1^m of order m≥ 1 of the field F_1 and the m-th roots of unity of the corresponding number field. It follows that elliptic curves with one cusp over F_1 are either the cyclotomic fields or the maximal abelian unramified extensions of the quadratic number fields. Our proof depends on representaion of the Drinfeld modules by the bounded linear operators on a Hilbert space and the crossed product structure of the Cuntz-Krieger algebras.
ER -