Integral trace forms of cyclic number fields without tameness
Abstract
Let K/Q be a cyclic number field and let TK=(𝒪K,(x,y)↦ TrK/Q(xy)) be its integral trace lattice. We prove that two cyclic number fields have isometric integral trace lattices if and only if they have the same degree, signature, and signed discriminant. More generally, an isomorphism between the Dirichlet character groups of two finite abelian extensions that preserves the conductor and parity of every character yields an isometry of their integral trace lattices, equivariant for the dual identification of Galois groups. For cyclic fields, the degree, signature, and discriminant determine the character group together with these two functions. This proves Bolaños's conjecture for wildly ramified cyclic fields and gives the corresponding extension of the tame classification: in general, the signature must be retained as an additional invariant.
Related
- Closed-form evaluation of the cyclic four-electron atomic correlation integral
- Norm-Euclideanity in cyclic fields
- Integral curvature estimates and geodesic limits for stable abelian Higgs fields
- The cyclic-induction Schur cone: Boolean sums, Ramanujan-square positivity, and integral structure
- A number field extension of Ramanujan's identity and its integral representation via the generalized Riemann Ξ-function
BibTeX
@misc{zhang2026integral,
title = {{Integral trace forms of cyclic number fields without tameness}},
author = {Zhi-Lin Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.29876},
eprint = {2610.29876},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.29876},
}
RIS
TY - UNPB
TI - Integral trace forms of cyclic number fields without tameness
AU - Zhi-Lin Zhang
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.29876
UR - https://arxiv.org/abs/2610.29876
AB - Let K/Q be a cyclic number field and let T_K=(𝒪_K,(x,y)↦ Tr_K/Q(xy)) be its integral trace lattice. We prove that two cyclic number fields have isometric integral trace lattices if and only if they have the same degree, signature, and signed discriminant. More generally, an isomorphism between the Dirichlet character groups of two finite abelian extensions that preserves the conductor and parity of every character yields an isometry of their integral trace lattices, equivariant for the dual identification of Galois groups. For cyclic fields, the degree, signature, and discriminant determine the character group together with these two functions. This proves Bolaños's conjecture for wildly ramified cyclic fields and gives the corresponding extension of the tame classification: in general, the signature must be retained as an additional invariant.
ER -