Number of points on certain hyperelliptic curves defined over finite fields

Anuradha, N (2008) Number of points on certain hyperelliptic curves defined over finite fields. In: Finite Fields and Their Applications, 14 (2). pp. 314-328.

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Abstract

For odd primes p and l such that the order of p modulo l is even, we determine explicitly the Jacobsthal sums $\phi_l(\upsilon)$, $\psi_l(\upsilon)$, and $\psi_{2l}(\upsilon)$, and the Jacobsthal–Whiteman sums $\phi^n_l(\upsilon)$ and $\phi^n_{2l}(\upsilon)$, over finite fields $F_q$ such that $q = p^\alpha \equiv 1$ (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 $n \geq 1$, the exact number of $F_{q^{n-}}$ rational points on the projective hyperelliptic curves $aY^2Z^{e-2} = bX^e + cZ^e$ $(abc \neq 0)$ (for e = l, 2l), and $aY^2Z^{l-1} = X(bX^l + cZ^l )$ $(abc \neq 0)$, defined over such finite fields $F_q$. As a consequence, we obtain the exact form of the \zeta -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 $F_q$ .

Item Type:	Journal Article
Publication:	Finite Fields and Their Applications
Publisher:	Elsevier
Additional Information:	Copyright of this article belongs to Elsevier.
Keywords:	Jacobsthal sums;Jacobsthal–Whiteman sums;Curves;Finite fields;Zeta functions.
Department/Centre:	Division of Physical & Mathematical Sciences > Mathematics
Date Deposited:	09 Jul 2008
Last Modified:	19 Sep 2010 04:47
URI:	http://eprints.iisc.ac.in/id/eprint/14879

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