Generating sets for maximal orders in rational quaternion algebras

Abstract

Given a maximal order O in a rational definite quaternion algebra, and a prime ℓ coprime to the discriminant of O, this paper considers subsets of O consisting of elements with ℓ-power norms that together generate O as a ℤ-algebra. We prove two theorems about the existence of such sets: the first states that O is generated by elements of norm ℓ^k for any k larger than an explicit bound, and the second states that there is a generating set for O consisting of at most three elements, each with norm a power of ℓ. We discuss implications for the study of supersingular isogeny graphs. As steps towards these theorems, for quaternion orders 𝒪 that are not necessarily maximal, we also prove structural results about the algebra of Brandt matrices for 𝒪 and explicit bounds on the coefficients of the theta function of 𝒪. Computational experiments are also discussed.

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BibTeX RIS

BibTeX

@misc{eisentrager2026generating,
  title = {{Generating sets for maximal orders in rational quaternion algebras}},
  author = {Kirsten Eisenträger and Eyal Z. Goren and Annamaria Iezzi and Harun Kir and Eda Kırımlı and Jonathan R. Love and William E. Mahaney and Jennifer Park and Maria Sabitova},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.10139},
  eprint = {2610.10139},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.10139},
}

RIS

TY  - UNPB
TI  - Generating sets for maximal orders in rational quaternion algebras
AU  - Kirsten Eisenträger
AU  - Eyal Z. Goren
AU  - Annamaria Iezzi
AU  - Harun Kir
AU  - Eda Kırımlı
AU  - Jonathan R. Love
AU  - William E. Mahaney
AU  - Jennifer Park
AU  - Maria Sabitova
PY  - 2026
DA  - 2026/10/07
PB  - arXiv
DO  - 10.48550/arxiv.2610.10139
UR  - https://arxiv.org/abs/2610.10139
AB  - Given a maximal order O in a rational definite quaternion algebra, and a prime ℓ coprime to the discriminant of O, this paper considers subsets of O consisting of elements with ℓ-power norms that together generate O as a ℤ-algebra. We prove two theorems about the existence of such sets: the first states that O is generated by elements of norm ℓ^k for any k larger than an explicit bound, and the second states that there is a generating set for O consisting of at most three elements, each with norm a power of ℓ. We discuss implications for the study of supersingular isogeny graphs. As steps towards these theorems, for quaternion orders 𝒪 that are not necessarily maximal, we also prove structural results about the algebra of Brandt matrices for 𝒪 and explicit bounds on the coefficients of the theta function of 𝒪. Computational experiments are also discussed.
ER  -

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