Deuring's result on elliptic curves. Any proof reference
$begingroup$
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 $[p+1-2sqrt p, p+1+2sqrt p]$ 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
reference-request alternative-proof elliptic-curves
asked Dec 2 '18 at 13:07
quantumquantum
1563
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migrated from math.stackexchange.com Dec 12 '18 at 18:04
This question came from our site for people studying math at any level and professionals in related fields.
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$begingroup$
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 $[p+1-2sqrt p, p+1+2sqrt p]$ 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
reference-request alternative-proof elliptic-curves
asked Dec 2 '18 at 13:07
quantumquantum
1563
$endgroup$
migrated from math.stackexchange.com Dec 12 '18 at 18:04
This question came from our site for people studying math at any level and professionals in related fields.
add a comment |
$begingroup$
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 $[p+1-2sqrt p, p+1+2sqrt p]$ 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
reference-request alternative-proof elliptic-curves
asked Dec 2 '18 at 13:07
quantumquantum
1563
$endgroup$
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 $[p+1-2sqrt p, p+1+2sqrt p]$ 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
reference-request alternative-proof elliptic-curves
reference-request alternative-proof elliptic-curves
asked Dec 2 '18 at 13:07
quantumquantum
1563
asked Dec 2 '18 at 13:07
quantumquantum
1563
asked Dec 2 '18 at 13:07
quantumquantum
1563
asked Dec 2 '18 at 13:07
quantumquantum
1563
asked Dec 2 '18 at 13:07
quantumquantum
1563
1563
migrated from math.stackexchange.com Dec 12 '18 at 18:04
This question came from our site for people studying math at any level and professionals in related fields.
migrated from math.stackexchange.com Dec 12 '18 at 18:04
This question came from our site for people studying math at any level and professionals in related fields.
add a comment |
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1 Answer
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$begingroup$
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."
answered Dec 12 '18 at 18:59
Joe SilvermanJoe Silverman
30.6k182158
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1 Answer
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$begingroup$
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."
answered Dec 12 '18 at 18:59
Joe SilvermanJoe Silverman
30.6k182158
$endgroup$
add a comment |
$begingroup$
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."
answered Dec 12 '18 at 18:59
Joe SilvermanJoe Silverman
30.6k182158
$endgroup$
add a comment |
$begingroup$
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."
answered Dec 12 '18 at 18:59
Joe SilvermanJoe Silverman
30.6k182158
$endgroup$
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."
answered Dec 12 '18 at 18:59
Joe SilvermanJoe Silverman
30.6k182158
edited Dec 12 '18 at 19:48
answered Dec 12 '18 at 18:59
Joe SilvermanJoe Silverman
30.6k182158
answered Dec 12 '18 at 18:59
Joe SilvermanJoe Silverman
30.6k182158
answered Dec 12 '18 at 18:59
Joe SilvermanJoe Silverman
30.6k182158
30.6k182158
add a comment |
add a comment |
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