Zeta function of the projective curveaY21 =bX21 +cZ21 over a class of finite fields, for odd primesl
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.
Fore = l, 2l, we determine the class numbers for the function fields associated to each class of curves over Fq. 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).
Volume 131, 2021
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