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Absolutely simple prymians of trigonal curves

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Abstract

Using Galois theory, we explicitly construct absolutely simple (principally polarized) Prym varieties that are not isomorphic to jacobians of curves even if we ignore the polarizations.

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References

  1. W. Bosma, J. Cannon, and C. Playoust, “The Magma Algebra System. I: The User Language,” J. Symb. Comput. 24, 235–265 (1997); http://magma.maths.usyd.edu.au/magma

    Article  MATH  MathSciNet  Google Scholar 

  2. J. K. Koo, “On Holomorphic Differentials of Some Algebraic Function Field of One Variable over ℂ,” Bull. Aust. Math. Soc. 43, 399–405 (1991).

    Article  MATH  MathSciNet  Google Scholar 

  3. K. Matsumoto and T. Terasoma, “Theta Constants Associated to Cubic Threefolds,” J. Algebr. Geom. 12, 741–775 (2003).

    MATH  MathSciNet  Google Scholar 

  4. D. Mumford, “Prym Varieties. I,” in Contributions to Analysis (Academic Press, New York, 1974), pp. 325–350.

    Google Scholar 

  5. F. Oort, “Endomorphism Algebras of Abelian Varieties,” in Algebraic Geometry and Commutative Algebra (Kinokuniya, Tokyo, 1988), Vol. 2, pp. 469–502.

    Google Scholar 

  6. D. S. Passman, Permutation Groups (W.A. Benjamin, New York, 1968).

    MATH  Google Scholar 

  7. B. Poonen and E. Schaefer, “Explicit Descent for Jacobians of Cyclic Covers of the Projective Line,” J. Reine Angew. Math. 488, 141–188 (1997).

    MATH  MathSciNet  Google Scholar 

  8. K. Ribet, “Galois Action on Division Points of Abelian Varieties with Real Multiplications,” Am. J. Math. 98, 751–804 (1976).

    Article  MATH  MathSciNet  Google Scholar 

  9. E. Schaefer, “Computing a Selmer Group of a Jacobian Using Functions on the Curve,” Math. Ann. 310, 447–471 (1998); “Erratum,” Math. Ann. 339, 1 (2007).

    Article  MATH  MathSciNet  Google Scholar 

  10. J.-P. Serre, Topics in Galois Theory (Jones and Bartlett Publ., Boston, 1992).

    MATH  Google Scholar 

  11. G. Shimura, “On Analytic Families of Polarized Abelian Varieties and Automorphic Functions,” Ann. Math., Ser. 2, 78, 149–192 (1963).

    Article  MathSciNet  Google Scholar 

  12. G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions (Princeton Univ. Press, Princeton, 1971).

    MATH  Google Scholar 

  13. V. V. Shokurov, “Distinguishing Prymians from Jacobians,” Invent. Math. 65(2), 209–219 (1981).

    Article  MATH  MathSciNet  Google Scholar 

  14. V. V. Shokurov, “Prym Varieties: Theory and Applications,” Izv. Akad. Nauk SSSR, Ser. Mat. 47(4), 785–855 (1983) [Math. USSR, Izv. 23 (1), 83–147 (1984)].

    MathSciNet  Google Scholar 

  15. C. Towse, “Weierstrass Points on Cyclic Covers of the Projective Line,” Trans. Am. Math. Soc. 348, 3355–3378 (1996).

    Article  MATH  MathSciNet  Google Scholar 

  16. Yu. G. Zarhin, “Cyclic Covers of the Projective Line, Their Jacobians and Endomorphisms,” J. Reine Angew. Math. 544, 91–110 (2002).

    MATH  MathSciNet  Google Scholar 

  17. Yu. G. Zarhin, “The Endomorphism Rings of Jacobians of Cyclic Covers of the Projective Line,” Math. Proc. Cambridge Philos. Soc. 136, 257–267 (2004).

    Article  MATH  MathSciNet  Google Scholar 

  18. Yu. G. Zarhin, “Endomorphism Algebras of Superelliptic Jacobians,” in Geometric Methods in Algebra and Number Theory, Ed. by F. Bogomolov and Yu. Tschinkel (Birkhäuser, Boston, 2005), Prog. Math. 235, pp. 339–362.

    Chapter  Google Scholar 

  19. Yu. G. Zarhin, “Endomorphisms of Superelliptic Jacobians,” Math. Z. 261, 691–707, 709 (2009).

    Article  MATH  MathSciNet  Google Scholar 

  20. Yu. G. Zarhin, “Cubic Surfaces and Cubic Threefolds, Jacobians and Intermediate Jacobians,” in Algebra, Arithmetic and Geometry: Manin Festschrift (Birkhäuser, Boston, 2009), Vol. 2, Prog. Math. 270 (in press); arXiv:math/0610138v3.

    Google Scholar 

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Correspondence to Yu. G. Zarhin.

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Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2009, Vol. 264, pp. 212–223.

In memory of Vasilii Alekseevich Iskovskikh

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Zarhin, Y.G. Absolutely simple prymians of trigonal curves. Proc. Steklov Inst. Math. 264, 204–215 (2009). https://doi.org/10.1134/S0081543809010210

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  • DOI: https://doi.org/10.1134/S0081543809010210

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