\titleofpresentation{The Connection Between Principal Indefinite
  Quadratic Forms and Elliptic Curves}
\presenter{Ahmet Tekcan}

This is joint work with Bet\"{u}l Gezer and Osman Bizim.
Let $p$ be a prime number such
that $p \equiv 1(\text{mod}\,4)$, let $\mathbb{F}_{p}$ be a finite
filed. Let $F(x,y)=x^{2}+xy+\frac{1-p}{4}y^{2}$ be the principal
indefinite binary quadratic form of discriminant $\Delta=p$ and
let $E:y^{2}=x^{3}+x^{2}+\frac{1-p}{4}x$ be the corresponding
elliptic curve over $\mathbb{F}_{p}$. In this paper we prove that
the order of $E$ over $\mathbb{F}_{p}$ is $p$ if $p \equiv
1(\text{mod}\,8)$ or $p+2$ if $p \equiv 5(\text{mod}\,8)$. Further
we obtain some formulas on the sum of $x-$and $y-$coordinates of
the points $(x,y)$ on $E$. Later, we consider the reduction of
$F$. We prove that the reduction of $F$ is
$R^{2}(F)=(1,1+2j,j^{2}+j-2k)$ if $p\equiv 1(\text{mod}\,8)$ or
$R^{2}(F)=(1,1+2j,j^{2}+j-1-2k)$ if $p\equiv 5(\text{mod}\,8),$
where $k=\frac{p-1}{8}$ or $\frac{p-5}{8}$, respectively and $j$
is any positive integer such that $k\in A_{j}.$

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

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

