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I have to write a paper on elliptic curves over finite fields and I was wondering if anyone had any ideas of some interesting directions to take this? Like what are some subtopics within this general topic?

Not sure if I tagged this correctly for my question, sorry in advance

Edit: I am an undegraduate and this paper is approximately 10 pages

Thanks!

Math Major
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    I guess you’re an undergraduate? When I was around that level, I found amazing the fact that if $E$ is defined over $k=\Bbb F_p$, then the number of points of $E(k)$ determines completely how many points there are of $E$ over the extension of degree $n$, no matter what $n$. – Lubin Mar 30 '15 at 18:57
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    Hi! Why not to resume some facts about the group structure over such an elliptic curve? – MattAllegro Mar 30 '15 at 19:22
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    An interesting result is that two elliptic curves over a finite field $\mathbb F_q$ are $\mathbb F_q$-isogenous if and only if they have the same number of rational points. A fact which is even more interesting, even though it is highly nontrivial, is that two elliptic curves over $\mathbb Q$ which have the same number of points over $\mathbb F_p$ for almost all $p$ are $\mathbb Q$-isogenous. – Ferra Mar 31 '15 at 09:32

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