PreprintCC BY 4.0arXiv
Rank functions, ℤp-extensions and class numbers
Abstract
If F is a number field and F = F0 ⊂ F1 ⊂ F2⊂ ⋯ is a tower of Galois extensions with Gal(Fn/F) cyclic of order pn, K. Iwasawa gave a closed formula for the growth of the p-part of the class number of Fn whenever n is large enough. We present a new proof of this formula, obtained by reinterpreting its summands in terms of Sylvester matrix rank functions on the Iwasawa algebra Λ without resorting to the classification of finitely-generated Λ-modules up to pseudo-isomorphism.
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BibTeX
@misc{souza2026rank,
title = {{Rank functions, $\mathbbZ_p$-extensions and class numbers}},
author = {Iq Souza},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.37490},
eprint = {2610.37490},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.37490},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Rank functions, ℤ_p-extensions and class numbers
AU - Iq Souza
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.37490
UR - https://arxiv.org/abs/2610.37490
AB - If F is a number field and F = F_0 ⊂ F_1 ⊂ F_2⊂ ⋯ is a tower of Galois extensions with Gal(F_n/F) cyclic of order p^n, K. Iwasawa gave a closed formula for the growth of the p-part of the class number of F_n whenever n is large enough. We present a new proof of this formula, obtained by reinterpreting its summands in terms of Sylvester matrix rank functions on the Iwasawa algebra Λ without resorting to the classification of finitely-generated Λ-modules up to pseudo-isomorphism.
ER -