Cyclotomic problem, Gauss sums and Legendre curve

Abstract

In this paper, explicit determination of the cyclotomic numbers of order l and 2 l , for odd prime l ≡ 3 (mod 4), over finite field 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.