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A construction of rational elliptic surfaces with the non-surjective boundary map on K 2

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Abstract

We present some rational elliptic surfaces over a certain field such that certain open subschemes do not satisfy the surjectivity of the boundary map on K 2 arising from the localization sequence. We consider two cases that the base field is transcendental over its prime subfield and that the base field is characteristic zero.

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References

  1. Asakura, M. Sato, K.: Beilinson’s Tate Conjecture for K 2 and finiteness of torsion zero-cycles on elliptic Surface (preprint). arXiv:0904.3672v3 mathAG (2009)

  2. Bass, H., Tate, J.: The Milnor ring of a global field. Algebraic K-theory, II: “Classical” algebraic K-theory and connections with arithmetic. In: Proceedings of the Conference, Seattle, Wash., Battelle Memorial Inst., 1972. Lecture Notes in Mathematics, vol. 342, pp. 349–446. Springer, Berlin (1973)

  3. Beĭlinson, A. A.: Higher regulators of modular curves, theory, part I, II. In: Boulder, Colo., 1983, Contemporary Mathematics, vol. 55, pp. 1–34. American Mathematical Society, Providence (1986)

  4. Colliot-Thélène J.-L.: Hilbert’s theorem 90 for K 2, with application to the Chow groups of rational surfaces. Invent. Math. 71(1), 1–20 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  5. Fulton W.: Intersection Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge. A Series of Modern Surveys in Mathematics, vol. 2. Springer, Berlin (1998)

    Google Scholar 

  6. Kato K.: A generalization of local class field theory by using K-groups. II. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27(3), 603–683 (1980)

    MATH  MathSciNet  Google Scholar 

  7. Kondo S., Yasuda S.: On the second rational K-group of an elliptic curve over global fields of positive characteristic. Proc. Lond. Math. Soc. (3) 102(6), 1053–1098 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  8. Shioda T.: On the Modell-Weil lattices. Commentarii mathmatici universitatis sancti pauli 39, 211–240 (1990)

    MATH  MathSciNet  Google Scholar 

  9. Soulé C.: Opérations en K-théorie algébrique. Canad. J. Math 37(3), 488–550 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  10. Tate, J.: Algorithm for determining the type of a singular fiber in an elliptic pencil, Modular functions of one variable, IV. In: Proceedings of the International Summer School, University Antwerp, Antwerp, 1972. Lecture Notes in Mathematics, vol. 476, pp. 33–52. Springer, Berlin (1975)

  11. Quillen, D.: Higher algebraic K-theory. I Algebraic K-theory, I: higher K-theories. In: Proceedings of the Conference, Battelle Memorial Institute, Seattle, Washingdon, 1972. Lecture Notes in Mathematics, vol. 341, pp. 85–147. Springer, Berlin (1973)

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  1. Mathematical Institute, Tohoku University, Aoba, Sendai, 980-8578, Japan

    Mariko Ohara

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  1. Mariko Ohara
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Correspondence to Mariko Ohara.

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Ohara, M. A construction of rational elliptic surfaces with the non-surjective boundary map on K 2 . manuscripta math. 143, 379–388 (2014). https://doi.org/10.1007/s00229-013-0623-0

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  • DOI: https://doi.org/10.1007/s00229-013-0623-0

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