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On congruent zeta of elliptic curves

Cosider a curve % latex2html id marker 807
$ E:y^2=x(x-1)(x-\lambda) \qquad( \lambda \in \mathbb{F}_q)$ . then:

      % latex2html id marker 808
$\displaystyle \char93 E(\mathbb{F}_{q^r}) = \sum_{x\in \mathbb{F}_{q^r}} ((x(x-1)(x-\lambda))^ {\frac{q-1}{2}} +1) +1$
    $\displaystyle =$ % latex2html id marker 810
$\displaystyle q+1+\sum_{x\in \mathbb{F}_{q^r}} x^{\frac{q-1}{2}} ((x-1)(x-\lambda ^{\frac{q-1}{2}}$

We have on the other hand:

LEMMA 6.6  

\begin{displaymath}
% latex2html id marker 817\sum_{x \in \mathbb{F}_{q^r}} x^...
...)\vert k$ and $k\neq 0$.} \\
0 & \text{otherwise.}
\end{cases}\end{displaymath}

      % latex2html id marker 818
$\displaystyle \char93 E(\mathbb{F}_{q^r}) = q+1- \o...
...orname{coeff}\left( (x-1)(x-\lambda)\right)^{\frac{q-1}{2}}, x^{\frac{q-1}{2}})$

Further computations are found in Clemens: ``A scrapbook of complex curve thery''. Students that are interested in this subject are advised to read the book.



2015-05-28