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Cyclotomic problem, Gauss sums and Legendre curve

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Abstract

In this paper, explicit determination of the cyclotomic numbers of order l and 2l, for odd prime l ≡ 3 (mod 4), over finite field \(\mathbb{F}_q\) in the index 2 case are obtained, utilizing the explicit formulas on the corresponding Gauss sums. The main results in this paper are related with the number of rational points of certain elliptic curve, called “Legendre curve”, and the properties and value distribution of such number are also presented.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, China

    LingLi Xia & Jing Yang

  2. Department of Basic Courses, Beijing Union University, Beijing, 100101, China

    LingLi Xia

Authors
  1. LingLi Xia
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  2. Jing Yang
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Correspondence to Jing Yang.

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Xia, L., Yang, J. Cyclotomic problem, Gauss sums and Legendre curve. Sci. China Math. 56, 1485–1508 (2013). https://doi.org/10.1007/s11425-013-4653-6

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  • DOI: https://doi.org/10.1007/s11425-013-4653-6

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