PreprintCC BY 4.0arXiv
Shapes of tame abelian number fields
Abstract
The shape of a number field K of degree n≥2 is the similarity class of the lattice obtained by projecting 𝒪K orthogonally away from the rational line in Minkowski space. It defines a point in the space of shapes of rank-(n−1) lattices, 𝒮n−1:=GLn−1(ℤ)backslash GLn−1(ℝ)/GOn−1(ℝ), where GOn−1(ℝ) is the group of orthogonal similitudes. Let G be a fixed nontrivial finite abelian group, and let K/ℚ vary over tamely ramified Galois extensions with Galois group isomorphic to G. We study the variation of the shapes of such number fields in 𝒮n−1, where n=|G|.
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BibTeX
@misc{ray2026shapes,
title = {{Shapes of tame abelian number fields}},
author = {Anwesh Ray},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.19311},
eprint = {2610.19311},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.19311},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Shapes of tame abelian number fields
AU - Anwesh Ray
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.19311
UR - https://arxiv.org/abs/2610.19311
AB - The shape of a number field K of degree n≥2 is the similarity class of the lattice obtained by projecting 𝒪_K orthogonally away from the rational line in Minkowski space. It defines a point in the space of shapes of rank-(n−1) lattices, 𝒮_n−1:=GL_n−1(ℤ)backslash GL_n−1(ℝ)/GO_n−1(ℝ), where GO_n−1(ℝ) is the group of orthogonal similitudes. Let G be a fixed nontrivial finite abelian group, and let K/ℚ vary over tamely ramified Galois extensions with Galois group isomorphic to G. We study the variation of the shapes of such number fields in 𝒮_n−1, where n=|G|.
ER -