### Abstract

An abelian surface A is called a QM-abelian surface if its endomorphism ring includes an order of an indefinite quaternion algebra, and a curve C of genus two is called a QM-curve if its jacobian variety is a QM-abelian surface. We give a computational result about the distribution of the arguments of the eigenvalues of the Frobenius endomorphisms of QM-abelian surfaces modulo good primes, which supports an analogue of the Sato-Tate Conjecture for such abelian surfaces. We also make some remarks on the field of definition of QM-curves and their endomorphisms.

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

Pages (from-to) | 1649-1662 |

Number of pages | 14 |

Journal | Mathematics of Computation |

Volume | 68 |

Issue number | 228 |

Publication status | Published - 1999 Oct |

### Fingerprint

### Keywords

- L-functions
- Quaternionic multiplication

### ASJC Scopus subject areas

- Algebra and Number Theory
- Applied Mathematics
- Computational Mathematics

### Cite this

*Mathematics of Computation*,

*68*(228), 1649-1662.

**On the Sato-Tate conjecture for QM-curves of genus two.** / Hashimoto, Kiichiro; Tsunogai, Hiroshi.

Research output: Contribution to journal › Article

*Mathematics of Computation*, vol. 68, no. 228, pp. 1649-1662.

}

TY - JOUR

T1 - On the Sato-Tate conjecture for QM-curves of genus two

AU - Hashimoto, Kiichiro

AU - Tsunogai, Hiroshi

PY - 1999/10

Y1 - 1999/10

N2 - An abelian surface A is called a QM-abelian surface if its endomorphism ring includes an order of an indefinite quaternion algebra, and a curve C of genus two is called a QM-curve if its jacobian variety is a QM-abelian surface. We give a computational result about the distribution of the arguments of the eigenvalues of the Frobenius endomorphisms of QM-abelian surfaces modulo good primes, which supports an analogue of the Sato-Tate Conjecture for such abelian surfaces. We also make some remarks on the field of definition of QM-curves and their endomorphisms.

AB - An abelian surface A is called a QM-abelian surface if its endomorphism ring includes an order of an indefinite quaternion algebra, and a curve C of genus two is called a QM-curve if its jacobian variety is a QM-abelian surface. We give a computational result about the distribution of the arguments of the eigenvalues of the Frobenius endomorphisms of QM-abelian surfaces modulo good primes, which supports an analogue of the Sato-Tate Conjecture for such abelian surfaces. We also make some remarks on the field of definition of QM-curves and their endomorphisms.

KW - L-functions

KW - Quaternionic multiplication

UR - http://www.scopus.com/inward/record.url?scp=0033465032&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0033465032&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0033465032

VL - 68

SP - 1649

EP - 1662

JO - Mathematics of Computation

JF - Mathematics of Computation

SN - 0025-5718

IS - 228

ER -