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CN 34-1054/N

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The generating fields of two twisted Kloosterman sums

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https://doi.org/10.52396/JUST-2021-0077
  • Received Date: 17 March 2021
  • Rev Recd Date: 18 May 2021
  • Publish Date: 31 December 2021
  • The generating fields of the twisted Kloosterman sums Kl (q, a, χ) and the partial Gauss sums g(q, a, χ) are studied.We require that the characteristic p is large with respect to the order d of the character χ and the trace of the coefficient a is nonzero.When p≡±1 mod d,we can characterize the generating fields of these character sums.For general p,when a lies in the prime field,we propose a combinatorial condition on (p, d) to ensure one can determine the generating fields.
    The generating fields of the twisted Kloosterman sums Kl (q, a, χ) and the partial Gauss sums g(q, a, χ) are studied.We require that the characteristic p is large with respect to the order d of the character χ and the trace of the coefficient a is nonzero.When p≡±1 mod d,we can characterize the generating fields of these character sums.For general p,when a lies in the prime field,we propose a combinatorial condition on (p, d) to ensure one can determine the generating fields.
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  • [1]
    Davenport H, Hasse H. Die Nullstellen der Kongruenzzetafunktionen in gewissen zyklischen Fällen. J. Reine Angew. Math., 1935, 172: 151-182.
    [2]
    Wan D. Algebraic theory of exponential sums over finite fields. https://www.math.uci.edu/~dwan/Wan_HIT_2019.pdf.
    [3]
    Zhang S. The degrees of exponential sums of binomials. https://arxiv.org/abs/2010.08342.
    [4]
    Wan D, Yin H. Algebraic degree periodicity in recurrence sequences. https://arxiv.org/abs/2009.14382.
    [5]
    Bombieri E. On exponential sums in finite fields. II. Invent. Math., 1978, 47: 29-39.
    [6]
    Wan D. Minimal polynomials and distinctness of Kloosterman sums. Finite Fields and Their Applications, 1995, 1: 189-203.
    [7]
    Fisher B. Distinctness of Kloosterman sums. In: p-Adic Methods in Number Theory and Algebraic Geometry. Providence, RI: Amer. Math. Soc., 1992.
    [8]
    Kononen K, Rinta-aho M, Väänänen K. On the degree of a Kloosterman sum as an algebraic integer. https://arxiv.org/abs/1107.0188.
    [9]
    Katz N M. Gauss Sums, Kloosterman Sums, and Monodromy Groups. Princeton, NJ: Princeton University Press, 1988.
    [10]
    Stickelberger L. Ueber eine Verallgemeinerung der Kreistheilung. Math. Ann., 1890, 37(3): 321-367.
    [11]
    Washington L C. Introduction to Cyclotomic Fields. New York: Springer-Verlag, 1982.
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    [1]
    Davenport H, Hasse H. Die Nullstellen der Kongruenzzetafunktionen in gewissen zyklischen Fällen. J. Reine Angew. Math., 1935, 172: 151-182.
    [2]
    Wan D. Algebraic theory of exponential sums over finite fields. https://www.math.uci.edu/~dwan/Wan_HIT_2019.pdf.
    [3]
    Zhang S. The degrees of exponential sums of binomials. https://arxiv.org/abs/2010.08342.
    [4]
    Wan D, Yin H. Algebraic degree periodicity in recurrence sequences. https://arxiv.org/abs/2009.14382.
    [5]
    Bombieri E. On exponential sums in finite fields. II. Invent. Math., 1978, 47: 29-39.
    [6]
    Wan D. Minimal polynomials and distinctness of Kloosterman sums. Finite Fields and Their Applications, 1995, 1: 189-203.
    [7]
    Fisher B. Distinctness of Kloosterman sums. In: p-Adic Methods in Number Theory and Algebraic Geometry. Providence, RI: Amer. Math. Soc., 1992.
    [8]
    Kononen K, Rinta-aho M, Väänänen K. On the degree of a Kloosterman sum as an algebraic integer. https://arxiv.org/abs/1107.0188.
    [9]
    Katz N M. Gauss Sums, Kloosterman Sums, and Monodromy Groups. Princeton, NJ: Princeton University Press, 1988.
    [10]
    Stickelberger L. Ueber eine Verallgemeinerung der Kreistheilung. Math. Ann., 1890, 37(3): 321-367.
    [11]
    Washington L C. Introduction to Cyclotomic Fields. New York: Springer-Verlag, 1982.

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