Higher dimensional local class field theory
Abstract
In this largely expository article we give an introduction to higher dimensional local class field theory, more precisely class field theory for smooth projective schemes over local fields. We begin with a quick summary of some of the main results of classical local class field theory. Then we explain the main definitions and statements in the higher dimensional setting. We give a new and simple proof of the ℓ-part, which is originally due to Jannsen--Saito and Forré. Inspired by this proof, we propose a conjecture which would imply the remaining open cases with p-coefficients and which improves our understanding of the p-part. A particular goal of the exposition is to outline analogies between the classical and the higher dimensional theory and on how the unit filtration and ramification, i.e. the p- and the ℓ-part, can be interpreted in the higher dimensional setting.
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BibTeX
@misc{luders2026higher,
title = {{Higher dimensional local class field theory}},
author = {Morten Lüders},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.34437},
eprint = {2609.34437},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.34437},
}
RIS
TY - UNPB
TI - Higher dimensional local class field theory
AU - Morten Lüders
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.34437
UR - https://arxiv.org/abs/2609.34437
AB - In this largely expository article we give an introduction to higher dimensional local class field theory, more precisely class field theory for smooth projective schemes over local fields. We begin with a quick summary of some of the main results of classical local class field theory. Then we explain the main definitions and statements in the higher dimensional setting. We give a new and simple proof of the ℓ-part, which is originally due to Jannsen--Saito and Forré. Inspired by this proof, we propose a conjecture which would imply the remaining open cases with p-coefficients and which improves our understanding of the p-part. A particular goal of the exposition is to outline analogies between the classical and the higher dimensional theory and on how the unit filtration and ramification, i.e. the p- and the ℓ-part, can be interpreted in the higher dimensional setting.
ER -