Zeta function of the projective curveaY 21 =bX 21 +cZ 21 over a class of finite fields, for odd primesl

Abstract

Letp andl be rational primes such thatl is odd and the order ofp modulol is even. For such primesp andl, and fore = l, 2l, we consider the non-singular projective curvesaY21 =bX21 +cZ21 defined over finite fields Fq such thatq = pα? l(mode).We see that the Fermat curves correspond precisely to those curves among each class (fore = l, 2l), that are maximal or minimal over Fq. We observe that each Fermat prime gives rise to explicit maximal and minimal curves over finite fields of characteristic 2. Fore = 2l, we explicitly determine the ζ -function(s) for this class of curves, over Fq, as rational functions in the variablet, for distinct cases ofa, b, andc, in Fq*. Theζ-function in each case is seen to satisfy the Weil conjectures (now theorems) for this concrete class of curves.