Zeta function of the projective curve aY 2 l = bX 2 l + cZ 2 l over a class of finite fields , for odd primes

Zeta function of the projective curve aY 2 l = bX 2 l + cZ 2 l aY 2 l = bX 2 l + cZ 2 l aY 2 l = bX 2 l + cZ 2 l over a class of finite fields, for odd primes l l l Abstract. Let p and l be rational primes such that l is odd and the order of p mod-ulo l is even. For such primes p and l, and for e = l, 2l, we consider the non-singular projective curves aY e = bX e + cZ e (abc = 0) defined over finite fields F q such that q = p α ≡ 1(mod e). We see that the Fermat curves correspond precisely to those curves among each class (for e = 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. For e = 2l, we explicitly determine the ζ-function(s) for this class of curves, over F q , as rational functions in the variable t, for distinct cases of a, b, and c, in F * q. The ζ-function in each case is seen to satisfy the Weil conjectures (now theorems) for this concrete class of curves. For e = 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).