### Abstract

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

Original language | English |
---|---|

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 |

### Fingerprint

### Keywords

- quaternion algebra
- order
- maximal order
- Eichler order

### Cite this

*Journal of Number Theory*,

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

**Optimal embeddings in quaternion algebras.** / MacLachlan, C.

Research output: Contribution to journal › Article

*Journal of Number Theory*, vol. 128, no. 10, pp. 2852-2860. https://doi.org/10.1016/j.jnt.2007.12.007

}

TY - JOUR

T1 - Optimal embeddings in quaternion algebras

AU - MacLachlan, C.

PY - 2008/10

Y1 - 2008/10

N2 - 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].

AB - 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].

KW - quaternion algebra

KW - order

KW - maximal order

KW - Eichler order

U2 - 10.1016/j.jnt.2007.12.007

DO - 10.1016/j.jnt.2007.12.007

M3 - Article

VL - 128

SP - 2852

EP - 2860

JO - Journal of Number Theory

JF - Journal of Number Theory

SN - 0022-314X

IS - 10

ER -