ISSN 0253-2778

CN 34-1054/N

open
Free to Publish. Free to Read.
Open AccessOpen Access JUSTC Research Articles

The generating fields of two twisted Kloosterman sums

  • CAS Wu Wen-Tsun Key Laboratory of Mathematics, School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, China

Cite this:
https://doi.org/10.52396/JUST-2021-0077
  • Received Date: 17 March 2021
  • Rev Recd Date: 18 May 2021
  • Publish Date: 31 December 2021
    Abstract

  • Abstract

    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.

    Abstract

    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.
    • FullText(HTML)
  • loading
    • References(11)
  • [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.
    • Supplements(0)
    • Related Articles
    • Track Citations
  • 加载中

Catalog

    |
    Get Citation
    PDF
    XML
    [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.
    Recommended articles
    TrendMD
    Volume 51 Issue 12 page: 879-888
    Cover

    Article Metrics

    Article views (273) PDF downloads(2743)
    Proportional views

    Copyright © Editorial Office of JUSTC, All Rights Reserved. 皖ICP备05002528号

    Supported by: Beijing Renhe Information Technology Co. Ltd    

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return