Taiwanese Journal of Mathematics

Numerical Invariants of Totally Imaginary Quadratic $\mathbb{Z}[\sqrt{p}]$-orders

Jiangwei Xue, Tse-Chung Yang, and Chia-Fu Yu

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Abstract

Let $A$ be a real quadratic order of discriminant $p$ or $4p$ with a prime $p$. In this paper we classify all proper totally imaginary quadratic $A$-orders $B$ with index $w(B) = [B^\times : A^\times] \gt 1$. We also calculate numerical invariants of these orders including the class number, the index $w(B)$ and the numbers of local optimal embeddings of these orders into quaternion orders. These numerical invariants are useful for computing the class numbers of totally definite quaternion algebras.

Article information

Source
Taiwanese J. Math., Volume 20, Number 4 (2016), 723-741.

Dates
First available in Project Euclid: 1 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1498874487

Digital Object Identifier
doi:10.11650/tjm.20.2016.6464

Mathematical Reviews number (MathSciNet)
MR3535670

Zentralblatt MATH identifier
1366.11112

Subjects
Primary: 11R52: Quaternion and other division algebras: arithmetic, zeta functions 11G10: Abelian varieties of dimension > 1 [See also 14Kxx]

Keywords
class number formula arithmetic of quaternion algebras

Citation

Xue, Jiangwei; Yang, Tse-Chung; Yu, Chia-Fu. Numerical Invariants of Totally Imaginary Quadratic $\mathbb{Z}[\sqrt{p}]$-orders. Taiwanese J. Math. 20 (2016), no. 4, 723--741. doi:10.11650/tjm.20.2016.6464. https://projecteuclid.org/euclid.twjm/1498874487


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