English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Let $p \geq 5$ be a prime and for $a, b \in \mathbb{F}_{p}$, let $E_{a,b}$denote the elliptic curve over $\mathbb{F}_{p}$ with equation $y^2=x^3+a\,x +b$. As usual define the trace of Frobenius $a_{p,\,a,\,b}$ by \begin{equation*}  \#E_{a,b}(\mathbb{F}_{p}) = p+1 -a_{p,\,a,\,b}. \end{equation*} We useelementary facts about exponential sums and known results about binaryquadratic forms over finite fields to evaluate the sums$\sum_{t\in\mathbb{F}_{p}} a_{p,\, t,\, b}$, $\sum _{t \in \mathbb{F}_{p}}a_{p,\,a,\, t}$,  $ \sum_{t=0}^{p-1}a_{p,\,t,\,b}^{2}$, $ \sum_{t=0}^{p-1}a_{p,\,a,\,t}^{2}$and $ \sum_{t=0}^{p-1}a_{p,\,t,\,b}^{3}$ for primes $p$ in various congruenceclasses.  As an example of our results, we prove the following: Let $p \equiv 5$ $($mod6$)$ be prime and let $b \in \mathbb{F}_{p}^{*}$. Then \begin{equation*}\sum_{t=0}^{p-1}a_{p,\,t,\,b}^{3}= -p\left((p-2)\left(\frac{-2}{p}\right)+2p\right)\left(\frac{b}{p}\right). \end{equation*}

Some properties of the distribution of the numbers of points on elliptic curves over a finite prime field