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](https://math.stackexchange.com/questions/2517425/proof-or-translation-of-deurings-theorem)



Deuring's result on elliptic curves. Any proof reference
You might find the following paper useful, although it proves something more general than what you are asking:


[MR0890272](https://mathscinet.ams.org/mathscinet-getitem?mr=890272),
Rück, Hans-Georg,
A note on elliptic curves over finite fields.
*Math. Comp*. **49** (1987), no. 179, 301–304,
doi:[10.1090/S0025-5718-1987-0890272-3](https://doi.org/10.1090/S0025-5718-1987-0890272-3).


There is also the paper:


[MR0265369](https://mathscinet.ams.org/mathscinet-getitem?mr=265369),
Waterhouse, William C.,
Abelian varieties over finite fields.
*Ann. Sci. École Norm. Sup*. (4) **2** 1969 521–560,
doi:[10.24033/asens.1183](https://doi.org/10.24033/asens.1183).


The [review](https://mathscinet.ams.org/mathscinet-getitem?mr=265369) of this second paper says: "In Chapter 4, Deuring's results on elliptic curves are derived, where the classification is very explicit."


