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Genus of quaternion algebras over formally real fields
Abstract
We prove the finiteness of the genus of quaternion division algebras over some formally real fields. In particular, if F is a formally real field and the second power of the fundamental ideal I2(F) is torsion-free, then the genus of any quaternion division F-algebra is trivial. For instance, this immediately implies that the genus vanishes for any quaternion division algebra over a formally real pythagorean field. The proofs are based on methods from the theory of quadratic forms.
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BibTeX
@misc{tikhonov2026genus,
title = {{Genus of quaternion algebras over formally real fields}},
author = {Sergey V. Tikhonov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.05392},
eprint = {2610.05392},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.05392},
}
RIS
TY - UNPB
TI - Genus of quaternion algebras over formally real fields
AU - Sergey V. Tikhonov
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2610.05392
UR - https://arxiv.org/abs/2610.05392
AB - We prove the finiteness of the genus of quaternion division algebras over some formally real fields. In particular, if F is a formally real field and the second power of the fundamental ideal I^2(F) is torsion-free, then the genus of any quaternion division F-algebra is trivial. For instance, this immediately implies that the genus vanishes for any quaternion division algebra over a formally real pythagorean field. The proofs are based on methods from the theory of quadratic forms.
ER -