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On trace-zero norm forms of orders of rational quaternion algebras with applications

  • Yifan Luo
  • Tianze Wang
  • Haigang Zhou
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Abstract

In this paper, by normalizing the trace-zero norm forms according to the parity of orders of definite rational quaternion algebras, we give an explicit formulation to the even Clifford correspondence followed by the reciprocity of ternary forms. We obtain an exact identity between ternary representation numbers and the number of elements of an order with prescribed reduced norm and trace, which applies to all orders without squarefreeness constraint on the reduced discriminant. Applications include a criterion for realizing ternary forms as trace-zero norm forms, a formula for traces of Brandt matrices in terms of representation numbers, and explicit theta series relations associated with local reciprocity. We also describe reciprocal orders and order chains, and interpret the ternary forms occurring in the congruent number problem through their associated quaternion orders.

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Cite

BibTeXRIS
BibTeX
@misc{luo2026tracezero,
  title = {{On trace-zero norm forms of orders of rational quaternion algebras with applications}},
  author = {Yifan Luo and Tianze Wang and Haigang Zhou},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.14613},
  eprint = {2610.14613},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.14613},
}
RIS
TY  - UNPB
TI  - On trace-zero norm forms of orders of rational quaternion algebras with applications
AU  - Yifan Luo
AU  - Tianze Wang
AU  - Haigang Zhou
PY  - 2026
DA  - 2026/10/10
PB  - arXiv
DO  - 10.48550/arxiv.2610.14613
UR  - https://arxiv.org/abs/2610.14613
AB  - In this paper, by normalizing the trace-zero norm forms according to the parity of orders of definite rational quaternion algebras, we give an explicit formulation to the even Clifford correspondence followed by the reciprocity of ternary forms. We obtain an exact identity between ternary representation numbers and the number of elements of an order with prescribed reduced norm and trace, which applies to all orders without squarefreeness constraint on the reduced discriminant. Applications include a criterion for realizing ternary forms as trace-zero norm forms, a formula for traces of Brandt matrices in terms of representation numbers, and explicit theta series relations associated with local reciprocity. We also describe reciprocal orders and order chains, and interpret the ternary forms occurring in the congruent number problem through their associated quaternion orders.
ER  -