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Let $E$ be an elliptic curve defined over a finite field ${\bf F}_p,$ where $p$ is prime. From Hasse theorem we get $p+1-2\sqrt{p} \leq |E({\bf F}_p)|\leq p+1+2\sqrt{p}.$ Now say that we choose in random the coefficients of $E$ from the interval $[0,p)\cap {\bf Z}$ in such way $E$ is elliptic.

Then, (i) can we say anything about $Pr(|E({\bf F}_p)|=k)?$

(ii) The number $|E({\bf F}_p)|$ takes uniformly, all the values in the interval $[p+1-2\sqrt{p},p+1+2\sqrt{p}]\cap {\bf Z}\ ?$

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    $\begingroup$ @Jyrki Are you sure Sato-Tate is relevant? Sato-Tate is concerned with a single elliptic curve, not averages over all elliptic curves... $\endgroup$ – Bruno Joyal Dec 7 '14 at 18:52
  • $\begingroup$ Oops. You're right, @Bruno. I had this recollection that the distribution was known (and involves the sine function), and was googling for the result. Sato-Tate appeared to fit the bill, but I didn't read it carefully enough. Sorry about creating confusion. $\endgroup$ – Jyrki Lahtonen Dec 7 '14 at 21:19
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This is remarkably similar to, but not quite, a classical result of Deuring which classifies the number of elliptic curves over a finite field with a given number of points up to isomorphism. I suggest you read this paper of Schoof. This is not enough to give the full probability, but it does do a lot of the work. It also shows that part (ii) of your question has a negative answer.

[1] Schoof, R, Nonsingular planar cubics over finite fields, J. Combin. Theory Series A 46. (1987), no. 2, 183-211.

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