Zeta function of the projective curveaY^{21} =bX^{21} +cZ^{21} over a class of finite fields, for odd primesl
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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 curvesaY^{21} =bX^{21} +cZ^{21} defined over finite fields F_{q} such thatq = p^{α}? 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 F_{q}. 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 F_{q}, as rational functions in the variablet, for distinct cases ofa, b, andc, in F_{q}^{*}. Theζ-function in each case is seen to satisfy the Weil conjectures (now theorems) for this concrete class of curves.
Fore = l, 2l, we determine the class numbers for the function fields associated to each class of curves over F_{q}. As a consequence, when the field of definition of the curve(s) is fixed, this provides concrete information on the growth of class numbers for constant field extensions of the function field(s) of the curve(s).
N Anuradha^{1} ^{}
Volume 130, 2020
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