Distribution of torsion subgroups of elliptic curve
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By Mazur's theorem, we know the list of possible torsion subgroups of elliptic curves over $mathbb{Q}$. Now, if we order them by height, can we compute the distribution of each possible groups? According to wikipedia, it is known that each group occurs infinitely many times, so maybe all of the possible groups have positive density.
elliptic-curves
asked Dec 5 '18 at 23:38
Seewoo LeeSeewoo Lee
6,676927
$endgroup$
add a comment |
$begingroup$
By Mazur's theorem, we know the list of possible torsion subgroups of elliptic curves over $mathbb{Q}$. Now, if we order them by height, can we compute the distribution of each possible groups? According to wikipedia, it is known that each group occurs infinitely many times, so maybe all of the possible groups have positive density.
elliptic-curves
asked Dec 5 '18 at 23:38
Seewoo LeeSeewoo Lee
6,676927
$endgroup$
add a comment |
$begingroup$
By Mazur's theorem, we know the list of possible torsion subgroups of elliptic curves over $mathbb{Q}$. Now, if we order them by height, can we compute the distribution of each possible groups? According to wikipedia, it is known that each group occurs infinitely many times, so maybe all of the possible groups have positive density.
elliptic-curves
asked Dec 5 '18 at 23:38
Seewoo LeeSeewoo Lee
6,676927
$endgroup$
By Mazur's theorem, we know the list of possible torsion subgroups of elliptic curves over $mathbb{Q}$. Now, if we order them by height, can we compute the distribution of each possible groups? According to wikipedia, it is known that each group occurs infinitely many times, so maybe all of the possible groups have positive density.
elliptic-curves
elliptic-curves
asked Dec 5 '18 at 23:38
Seewoo LeeSeewoo Lee
6,676927
asked Dec 5 '18 at 23:38
Seewoo LeeSeewoo Lee
6,676927
asked Dec 5 '18 at 23:38
Seewoo LeeSeewoo Lee
6,676927
asked Dec 5 '18 at 23:38
Seewoo LeeSeewoo Lee
6,676927
asked Dec 5 '18 at 23:38
Seewoo LeeSeewoo Lee
6,676927
6,676927
add a comment |
add a comment |
1 Answer
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Curves with no nontrivial torsion are density 1 in the set of all elliptic curves over $mathbf{Q}$. This is a result of Duke from 1997 (paper is titled Courbes elliptiques sur Q sans nombres premiers exceptionnels).
More specifically, by work of Harron and Snowden, we actually know the leading term in the asymptotic growth of the number of elliptic curves with a given torsion subgroup ordered by height. See Theorems 1.1 and 1.5 in their article.
answered Dec 16 '18 at 4:39
Brandon CarterBrandon Carter
7,28822538
$endgroup$
$begingroup$
This is exactly I wanted!
$endgroup$
– Seewoo Lee
Dec 17 '18 at 4:04
add a comment |
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1 Answer
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1 Answer
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active
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votes
$begingroup$
Curves with no nontrivial torsion are density 1 in the set of all elliptic curves over $mathbf{Q}$. This is a result of Duke from 1997 (paper is titled Courbes elliptiques sur Q sans nombres premiers exceptionnels).
More specifically, by work of Harron and Snowden, we actually know the leading term in the asymptotic growth of the number of elliptic curves with a given torsion subgroup ordered by height. See Theorems 1.1 and 1.5 in their article.
answered Dec 16 '18 at 4:39
Brandon CarterBrandon Carter
7,28822538
$endgroup$
$begingroup$
This is exactly I wanted!
$endgroup$
– Seewoo Lee
Dec 17 '18 at 4:04
add a comment |
$begingroup$
Curves with no nontrivial torsion are density 1 in the set of all elliptic curves over $mathbf{Q}$. This is a result of Duke from 1997 (paper is titled Courbes elliptiques sur Q sans nombres premiers exceptionnels).
More specifically, by work of Harron and Snowden, we actually know the leading term in the asymptotic growth of the number of elliptic curves with a given torsion subgroup ordered by height. See Theorems 1.1 and 1.5 in their article.
answered Dec 16 '18 at 4:39
Brandon CarterBrandon Carter
7,28822538
$endgroup$
$begingroup$
This is exactly I wanted!
$endgroup$
– Seewoo Lee
Dec 17 '18 at 4:04
add a comment |
$begingroup$
Curves with no nontrivial torsion are density 1 in the set of all elliptic curves over $mathbf{Q}$. This is a result of Duke from 1997 (paper is titled Courbes elliptiques sur Q sans nombres premiers exceptionnels).
More specifically, by work of Harron and Snowden, we actually know the leading term in the asymptotic growth of the number of elliptic curves with a given torsion subgroup ordered by height. See Theorems 1.1 and 1.5 in their article.
answered Dec 16 '18 at 4:39
Brandon CarterBrandon Carter
7,28822538
$endgroup$
Curves with no nontrivial torsion are density 1 in the set of all elliptic curves over $mathbf{Q}$. This is a result of Duke from 1997 (paper is titled Courbes elliptiques sur Q sans nombres premiers exceptionnels).
More specifically, by work of Harron and Snowden, we actually know the leading term in the asymptotic growth of the number of elliptic curves with a given torsion subgroup ordered by height. See Theorems 1.1 and 1.5 in their article.
answered Dec 16 '18 at 4:39
Brandon CarterBrandon Carter
7,28822538
answered Dec 16 '18 at 4:39
Brandon CarterBrandon Carter
7,28822538
answered Dec 16 '18 at 4:39
Brandon CarterBrandon Carter
7,28822538
answered Dec 16 '18 at 4:39
Brandon CarterBrandon Carter
7,28822538
7,28822538
$begingroup$
This is exactly I wanted!
$endgroup$
– Seewoo Lee
Dec 17 '18 at 4:04
add a comment |
$begingroup$
This is exactly I wanted!
$endgroup$
– Seewoo Lee
Dec 17 '18 at 4:04
$begingroup$
This is exactly I wanted!
$endgroup$
– Seewoo Lee
Dec 17 '18 at 4:04
$begingroup$
This is exactly I wanted!
$endgroup$
– Seewoo Lee
Dec 17 '18 at 4:04
add a comment |
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