On the Sato-Tate conjecture

for QM-curves of genus two

Authors:
Ki-ichiro Hashimoto and Hiroshi Tsunogai

Journal:
Math. Comp. **68** (1999), 1649-1662

MSC (1991):
Primary 11G40; Secondary 11G15, 14H10, 14K15

DOI:
https://doi.org/10.1090/S0025-5718-99-01061-3

Published electronically:
February 19, 1999

MathSciNet review:
1627797

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Abstract | References | Similar Articles | Additional Information

Abstract: An abelian surface is called a QM-abelian surface if its endomorphism ring includes an order of an indefinite quaternion algebra, and a curve 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.

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Additional Information

**Ki-ichiro Hashimoto**

Affiliation:
Department of Mathematics, Waseda University, 3-4-1, Ōkubo, Shinjuku-ku, Tōkyō, 169-8555, Japan

Email:
khasimot@mn.waseda.ac.jp

**Hiroshi Tsunogai**

Affiliation:
Department of Mathematics, Sophia University, 7-1, Kioi-chō, Chiyoda-ku, Tōkyō, 102-8554, Japan

Email:
tsuno@mm.sophia.ac.jp

DOI:
https://doi.org/10.1090/S0025-5718-99-01061-3

Keywords:
Quaternionic multiplication,
$L$-functions

Received by editor(s):
August 22, 1995

Received by editor(s) in revised form:
January 22, 1998

Published electronically:
February 19, 1999

Article copyright:
© Copyright 1999
American Mathematical Society