Zeta function of the projective curveaY 21 =bX 21 +cZ 21 over a class of finite fields, for odd primesl

Anuradha, N.

doi:10.1007/bf02829836

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“…The computations for the ζ -functions are similar to those obtained in [2,3], and are derived by considering the numbers a l (n) and a 2l (n) for distinct cases of in (mod l) and in, j n (mod 2), and taking into account the values that arise when l | n, l n, 2 | n, and 2 n, for distinct cases of i (mod l) and i, j (mod 2). Theorem 6.…”

Section: Determination Of Zeta Functionsmentioning

confidence: 90%

Number of points on certain hyperelliptic curves defined over finite fields

Anuradha

2008

Finite Fields and Their Applications

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For odd primes p and l such that the order of p modulo l is even, we determine explicitly the Jacobsthal sums φ l (v), ψ l (v), and ψ 2l (v), and the Jacobsthal-Whiteman sums φ n l (v) and φ n 2l (v), over finite fields F q such that q = p α ≡ 1 (mod 2l). These results are obtained only in terms of q and l. We apply these results pertaining to the Jacobsthal sums, to determine, for each integer n 1, the exact number of F q nrational points on the projective hyperelliptic curves aY 2 Z e−2 = bX e + cZ e (abc = 0) (for e = l, 2l), and aY 2 Z l−1 = X(bX l + cZ l ) (abc = 0), defined over such finite fields F q . As a consequence, we obtain the exact form of the ζ -functions for these three classes of curves defined over F q , as rational functions in the variable t, for all distinct cases that arise for the coefficients a, b, c. Further, we determine the exact cases for the coefficients a, b, c, for each class of curves, for which the corresponding non-singular models are maximal (or minimal) over F q .

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Section: Determination Of Zeta Functionsmentioning

confidence: 90%