
\titleofpresentation{On the Singular Elliptic Curves over Finite Fields}

\presenter{Betul Gezer}

This is joint work with Osman Bizim and Ahmet Tekcan.
We consider the
rational points on singular elliptic curves over finite fields
${\mathbb{F}}_{p}$. We give results concerning the number of the
points on the singular elliptic curves, $y^{2}=x^{3}$ and
$y^{2}=x^{3}+ax^{2}$ over $\mathbb{F}_{p}$ where $p$ is a prime.
We give the number of solutions to $y^{2}=x^{3}$ and
$y^{2}=x^{3}+ax^{2}$ over $\mathbb{F}_{p}$ by means of quadratic
residue character, and also for the curve $y^{2}=x^{3}$ we use the
cubic residue character. Also some results are given on the sum of
$x$-and $y$-coordinates of the points $(x,y)$ on these curves. We
determine the structure of the group of the rational points and
torsion points on these curves and finally we generalize the
results concerning the number of points in $\mathbb{F}_{p}$ to
$\mathbb{F}_{p^{n}}$.
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\bibitem{gezer4} R. A. Mollin, \textit{Fundamental Number Theory with Applications}%
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\bibitem{gezer5} S. Schmitt and H. Zimmer, \textit{Elliptic Curves A Computational
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Springer-Verlag, 1986.

\bibitem{gezer7} J.H.Silverman, and J.Tate, \textit{Rational Points on Elliptic
Curves, } Undergraduate Texts in Mathematics, Springer, 1992.

\bibitem{gezer8} L.C.Washington, \textit{Elliptic Curves, Number Theory and
Cryptography. }Chapman\&Hall/CRC, 2003.
\end{ourbiblio}

\presenteraddress{Uludag University, Faculty of Science,
Department of Mathematics, G\"{o}r\"{u}kle 16059. Bursa-TURKEY.}

\email{betulgezer@uludag.edu.tr}%, obizim@uludag.edu.tr, tekcan@uludag.edu.tr}

