Number of Points on the Projective Curves aYl=bXl+cZl and aY2l=bX2l+cZ2l Defined over Finite Fields, l an Odd Prime
✍ Scribed by N. Anuradha; S.A. Katre
- Publisher
- Elsevier Science
- Year
- 1999
- Tongue
- English
- Weight
- 201 KB
- Volume
- 77
- Category
- Article
- ISSN
- 0022-314X
No coin nor oath required. For personal study only.
✦ Synopsis
The number of points on the curve aY e =bX e +c (abc{0) defined over a finite field F q , q#1 (mod e), 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 F q , q= p : #1 (mod e), for odd primes l and any prime p such that the order of p modulo l is even. Contrary to the case p#1 (mod e) 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 F q n -rational points on the non-singular projective curves aY l =bX l +cZ l and aY 2l =bX 2l +cZ 2l (abc{0) defined over finite fields F q , with conditions on q, p, and l as above. Using these evaluations, we obtain explicitly the `-function of the former curve aY l =bX l +cZ l defined over F q as a rational function in the variable t. Thereby we corroborate the Weil conjectures (now theorems) for this concrete class of curves.
1999 Academic Press J e (i, j)= : v # F q / i (v) / j (v+1)= :
v # F q "[0, &1]
`i ind # (v)+ j ind # (v+1) ,