### Abstract

Let A be a quaternion algebra over a number field k and assume that A satisfies the Eichler condition so that some infinite place of k is unramified in A. Let L be a quadratic extension of k which embeds in A. Let Rk denote the ring of integers of k and let B be an Rk-order in L. Suppose that E is an Eichler order of A of square-free level S. In this paper, we determine when there exists an embedding s : L¿A over k which gives an optimal embedding of B into E in the sense that s(L)nE = s(B). This generalises previous work of Eichler [M. Eichler, Zur Zahlentheorie der Quaternionenalgebren, J. Reine Angew. Math. 195 (1955) 127–155] and Chinburg and Friedman [T. Chinburg, E. Friedman, An embedding theorem for quaternion

algebras, J. London Math. Soc. 60 (1999) 33–44].

algebras, J. London Math. Soc. 60 (1999) 33–44].

Original language | English |
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Pages (from-to) | 2852-2860 |

Number of pages | 9 |

Journal | Journal of Number Theory |

Volume | 128 |

Issue number | 10 |

DOIs | |

Publication status | Published - Oct 2008 |

### Keywords

- quaternion algebra
- order
- maximal order
- Eichler order

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## Cite this

MacLachlan, C. (2008). Optimal embeddings in quaternion algebras.

*Journal of Number Theory*,*128*(10), 2852-2860. https://doi.org/10.1016/j.jnt.2007.12.007