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Tate-Shafarevich products in elliptic curves over pseudolocal fields with residue fields of characteristic 3

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Abstract

Let k be a general local field with pseudolocal residue field x, char x=3, and A an elliptic curve defined over k. It is proved that the Tate-Shafarevich product H1(k, A)×Ak→ Q/ℤ of the group H1(k, A) of principal homogeneous spaces of the curve A over k and the group Ak of its k-rational points is left nondegenerate.

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References

  1. Y. R. Shafarevich, “Group of principal homogeneous algebraic manifolds,” Dokl. Akad. Nauk SSSR,124, 42–43 (1959).

    Google Scholar 

  2. J. Tate, “WC-group over p-adic fields,” Sem. Bourbaki, No. 157 (1967).

  3. O. N. Vvedenskii, “On local ‚fields of classes’ of elliptic curves,” Izv. Akad. Nauk SSSR. Ser. Matem.,37, No. 1, 20–88 (1973).

    Google Scholar 

  4. J.-P. Serre, Local Fields [in French], Hermann, Paris (1962).

    Google Scholar 

  5. V. I. Andriychuk, “On elliptic curves over pseudolocal fields,” Mat. Sbornik,110, No. 9, 88–101 (1979).

    Google Scholar 

  6. J. Ax, “The elementary theory of fields,” Ann. Math.,88, No. 2, 239–271 (1968).

    Google Scholar 

  7. O. N. Vvedenskii, “Duality in elliptic curves over a local field. I, II,” Izv. Akad. Nauk SSSR. Ser. Matem.,28, 1091–1112 (1964);30, 891–922 (1966).

    Google Scholar 

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Authors and Affiliations

  1. L'vov University, USSR

    V. I. Andriychuk

Authors
  1. V. I. Andriychuk
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Additional information

Translated from Ukrayins'kyy Matematychnyy Zhurnal, Vol. 44, No. 9, pp. 1157–1165, September, 1992.

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Andriychuk, V.I. Tate-Shafarevich products in elliptic curves over pseudolocal fields with residue fields of characteristic 3. Ukr Math J 44, 1057–1064 (1992). https://doi.org/10.1007/BF01058363

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  • DOI: https://doi.org/10.1007/BF01058363

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