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

Hashimoto, Ki-ichiro; Tsunogai, Hiroshi

doi:10.1090/S0025-5718-99-01061-3

On the Sato-Tate conjecture for QM-curves of genus two
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by Ki-ichiro Hashimoto and Hiroshi Tsunogai PDF

Math. Comp. 68 (1999), 1649-1662 Request permission

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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