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:
- The location on the paper where the above statement immediately follows (I hope I do not need to read the whole paper for that)
- 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