Number of Points on the Projective Curves aYl=bXl+cZl and aY2l=bX2l+cZ2l Defined over Finite Fields, l an Odd Prime

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The number of points on the curve aYe=bXe+c (abc≠0) defined over a finite field Fq, q≡1 (mode), is known to be obtainable in terms of Jacobi sums and cyclotomic numbers of order e with respect to this field. In this paper, we obtain explicitly the Jacobi sums and cyclotomic numbers of order e=l and e=2l over finite fields Fq, q=pα≡1 (mode), for odd primes l and any prime p such that the order of p modulo l is even. Contrary to the case p≡1 (mode) considered in the literature, we have obtained these results solely in terms of q and l. We apply these results to evaluate the number of Fqn-rational points on the non-singular projective curves aYl=bXl+cZl and aY2l=bX2l+cZ2l (abc≠0) defined over finite fields Fq, with conditions on q, p, and l as above. Using these evaluations, we obtain explicitly the ζ-function of the former curve aYl=bXl+cZl defined over Fq as a rational function in the variable t. Thereby we corroborate the Weil conjectures (now theorems) for this concrete class of curves.