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I have heard of this result from Deuring 1941 paper: Given $\mathbb F_p$ ($p$ prime number) and any number $n$ in the Hasse interval $[p+1-2\sqrt p, p+1+2\sqrt p]$ there is an elliptic curve over $\mathbb F_p$ having $n$ points. I have minimal knowledge on more advanced topics on elliptic curves (only know a thing or two on isogenies and the endomorphism ring and enough algebraic geometry that can get me through the proofs). So I wanted to know if there is an easy proof of this result. I looked at the original Deuring's paper (German) and the 78 page paper, at first sight, it does not immediately tell me where I can find this statement (I am sure it is hidden in some context in the paper). So I ask, if someone could kindly point to me:

  1. The location on the paper where the above statement immediately follows (I hope I do not need to read the whole paper for that)
  2. Is there any modern (perhaps english) treatment of Deuring's proof? or at least a simplified one where I do not really need to look at elliptic curves over number fields (aside from $\mathbb Q$) to understand the proof of the statement (at the moment I am only concerned with the simplest case, i.e. elliptic curves over finite fields with prime order).

I tried to look at Lang's book "Elliptic Functions", but I think he does not prove this result. Please correct me if I am wrong.

Edit: It seems I will have no luck here. There is a not so unrelated post asking a similar question: Proof or Translation of Deuring's Theorem

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You might find the following paper useful, although it proves something more general than what you are asking:

MR0890272, Rück, Hans-Georg, A note on elliptic curves over finite fields. Math. Comp. 49 (1987), no. 179, 301–304.

There is also the paper:

MR0265369, Waterhouse, William C., Abelian varieties over finite fields. Ann. Sci. École Norm. Sup. (4) 2 1969 521–560.

The review of this paper says: "In Chapter 4, Deuring's results on elliptic curves are derived, where the classification is very explicit."

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