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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
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 $A$ 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.


References [Enhancements On Off] (What's this?)

  • [D] Deuring, M., Die Typen der Multiplicatorenringe der elliptischer Funktionenkörper, Abh.Math.Sem. Hamburg 14 (1941), 197-272. MR 3:104f
  • [F] Faltings, G., Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math. 73 (1983), 349-366. MR 85g:11026a
  • [HHM] Hasegawa, Y., Hashimoto, K., Momose, F., Modular conjecture for ${\boldsymbol Q}$-curves and QM-curves, preprint, 1997.
  • [HM] Hashimoto, K., Murabayashi, N., Shimura curves as intersection of Humbert surfaces and defining equations of QM-curves of genus two, Tohoku Math. J. 47 (1995), 271-296. MR 96b:14023
  • [K] Kurihara, A., On some examples of equations defining Shimura curves and the Mumford uniformization, F. Fac. Sci. Univ. Tokyo, 25 (1979), 277-301. MR 80e:14010
  • [M] Mumford, D., Abelian varieties, Oxford Univ. Press, London, 1970. MR 44:219
  • [O] Ohta, M., On $l$-adic representations of Galois groups obtained from certain two dimensional abelian varieties, J. Fac. Sci. Univ. Tokyo, 21 (1974), 299-308. MR 54:7389
  • [P] Pyle, E., Abelian varieties over ${\boldsymbol Q}$ with large endmorphism algebras and thier simple components over $\bar {\boldsymbol Q}$, Doctor's thesis, Univ. of California at Berkeley, 1995.
  • [R] Ribet, K., Fields of definition of abelian varieties with real multiplication, Contemp. Math. 174 (1994), 107-118. MR 95i:11057
  • [T] Tate, J., Algebraic Cycles and Poles of Zeta Functions, in ``Arithmetical Algebraic Geometry'', (F.G. Schilling, ed.), Harper and Row, New York, 1965, pp. 93-110. MR 37:1371
  • [Ya] Yamamoto, Y., On Sato Conjecture for two-dimensional abelian varieties (in Japanese), Number Theory Symposium at Kinosaki (1979), 236-244.
  • [Yo1] Yoshida, H., Mumford-Tate groups and its application to abelian varieties (in Japanese), ``Shimura varieties and algebraic geometry'' Symposium at Kinosaki (1983), 106-131.
  • [Yo2] Yoshida, H., On an Analogue of the Sato Conjecture, Invent. Math. 19 (1973), 261-277. MR 49:2746

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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

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