# Deuring's result on elliptic curves. Any proof reference

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

• This is Theorem 14.18 in Cox, Primes of the form $x^2+ny^2$. – rogerl May 19 '20 at 14:49
• Sorry I missed this comment (the comment came 2 years after my original post). Exactly which book of Cox (and which Cox) do you refer to? – quantum Feb 8 at 7:28
• Sorry, David Cox, Primes of the form $x^2+ny^2$ – rogerl Feb 8 at 19:34
• Oh sorry, you already mentioned the title (I was a bit surprised because I did not know David Cox worked on this topic). I scanned thru it very fast and see that it relies on Thereom 13.21 which is on the splitting of the prime $p$ in a number field. This Theorem then refers to Deuring (or Lang) for the proof. With reference to Lang I think I can follow the proof without looking at Deuring. You should post this as an answer. Thanks! – quantum Feb 18 at 21:30

## 2 Answers

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, doi:10.1090/S0025-5718-1987-0890272-3.

There is also the paper:

MR0265369, Waterhouse, William C., Abelian varieties over finite fields. Ann. Sci. École Norm. Sup. (4) 2 1969 521–560, doi:10.24033/asens.1183.

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

Theorem 14.18 in Cox, Primes of the form $$𝑥^2+ n y^2$$ (Wiley, 2013, doi:10.1002/9781118400722) contains a proof of this fact. It does rely on a theorem proved both in Deuring's paper and in Lang's Elliptic Functions, Graduate Texts in Mathematics 112 (1987) doi:10.1007/978-1-4612-4752-4.