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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
Published electronically: February 19, 1999
MathSciNet review: 1627797
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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.

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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

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

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

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