When does a Mordell curve have non-trivial torsion?
$begingroup$
Is there a known simple criteria for when a Mordell curve has non-trivial torsion?
A comment in this question:
Family of elliptic curves with trivial torsion
Suggests that
$$y^2 = x^3 + k$$
has nontrivial torsion when k is 1, a square, or a cube.
If this is true, it would be nice to see it explained or point to a reference I can read more.
number-theory elliptic-curves mordell-curves
asked Jan 9 at 12:12
PineDoorsPineDoors
335
$endgroup$
add a comment |
$begingroup$
Is there a known simple criteria for when a Mordell curve has non-trivial torsion?
A comment in this question:
Family of elliptic curves with trivial torsion
Suggests that
$$y^2 = x^3 + k$$
has nontrivial torsion when k is 1, a square, or a cube.
If this is true, it would be nice to see it explained or point to a reference I can read more.
number-theory elliptic-curves mordell-curves
asked Jan 9 at 12:12
PineDoorsPineDoors
335
$endgroup$
add a comment |
$begingroup$
Is there a known simple criteria for when a Mordell curve has non-trivial torsion?
A comment in this question:
Family of elliptic curves with trivial torsion
Suggests that
$$y^2 = x^3 + k$$
has nontrivial torsion when k is 1, a square, or a cube.
If this is true, it would be nice to see it explained or point to a reference I can read more.
number-theory elliptic-curves mordell-curves
asked Jan 9 at 12:12
PineDoorsPineDoors
335
$endgroup$
Is there a known simple criteria for when a Mordell curve has non-trivial torsion?
A comment in this question:
Family of elliptic curves with trivial torsion
Suggests that
$$y^2 = x^3 + k$$
has nontrivial torsion when k is 1, a square, or a cube.
If this is true, it would be nice to see it explained or point to a reference I can read more.
number-theory elliptic-curves mordell-curves
number-theory elliptic-curves mordell-curves
asked Jan 9 at 12:12
PineDoorsPineDoors
335
asked Jan 9 at 12:12
PineDoorsPineDoors
335
asked Jan 9 at 12:12
PineDoorsPineDoors
335
asked Jan 9 at 12:12
PineDoorsPineDoors
335
asked Jan 9 at 12:12
PineDoorsPineDoors
335
335
add a comment |
add a comment |
1 Answer
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Here is a further reference, extending the result you have linked.
Theorem: If $k$ is square-free and not equal to $1$, the elliptic curve $y^2 = x^3+k$ has no rational torsion points.
Proof: See A. Knapp, Elliptic Curves, Princeton Univ. Press, 1992, Theorem $5.3$.
More generally, there is indeed a classification result as follows.
Theorem: Let $k=m^6 cdot k_0$, where $m, k_0in Bbb Z$ and $k_0$ is free of sixth power prime factors. Then the torsion subgroup of $Ecolon y^2=x^3+k$ over $Bbb Q$ is given as follows:
$$
E_{rm tors}(Bbb Q)=
begin{cases}
Bbb Z/6Bbb Z & text{ if } k_0= 1, \
Bbb Z/3Bbb Z & text{ if $k_0$ is a square different from $1$}, \
Bbb Z/3Bbb Z & text{ if $k_0=−432$}, \
Bbb Z/2Bbb Z & text{ if $k_0$ is a cube different from $1$}, \
{mathcal O} & text{ otherwise.}
end{cases}
$$
answered Jan 9 at 12:21
Dietrich BurdeDietrich Burde
82.1k649107
$endgroup$
$begingroup$
Is there something that explains what makes -432 so very special here? Is it related to the birational mapping to $u^3 + v^3 = 1$?
$endgroup$
– PineDoors
Jan 10 at 16:24
$begingroup$
Yes, $-432=-27cdot 16$ is related to the discriminant of the Mordell curve. For $k=-432m^6$, the torsion points are $mathcal{O},(12m^2,pm 36m^3)$.
$endgroup$
– Dietrich Burde
Jan 10 at 17:27
add a comment |
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1 Answer
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1 Answer
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active
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$begingroup$
Here is a further reference, extending the result you have linked.
Theorem: If $k$ is square-free and not equal to $1$, the elliptic curve $y^2 = x^3+k$ has no rational torsion points.
Proof: See A. Knapp, Elliptic Curves, Princeton Univ. Press, 1992, Theorem $5.3$.
More generally, there is indeed a classification result as follows.
Theorem: Let $k=m^6 cdot k_0$, where $m, k_0in Bbb Z$ and $k_0$ is free of sixth power prime factors. Then the torsion subgroup of $Ecolon y^2=x^3+k$ over $Bbb Q$ is given as follows:
$$
E_{rm tors}(Bbb Q)=
begin{cases}
Bbb Z/6Bbb Z & text{ if } k_0= 1, \
Bbb Z/3Bbb Z & text{ if $k_0$ is a square different from $1$}, \
Bbb Z/3Bbb Z & text{ if $k_0=−432$}, \
Bbb Z/2Bbb Z & text{ if $k_0$ is a cube different from $1$}, \
{mathcal O} & text{ otherwise.}
end{cases}
$$
answered Jan 9 at 12:21
Dietrich BurdeDietrich Burde
82.1k649107
$endgroup$
$begingroup$
Is there something that explains what makes -432 so very special here? Is it related to the birational mapping to $u^3 + v^3 = 1$?
$endgroup$
– PineDoors
Jan 10 at 16:24
$begingroup$
Yes, $-432=-27cdot 16$ is related to the discriminant of the Mordell curve. For $k=-432m^6$, the torsion points are $mathcal{O},(12m^2,pm 36m^3)$.
$endgroup$
– Dietrich Burde
Jan 10 at 17:27
add a comment |
$begingroup$
Here is a further reference, extending the result you have linked.
Theorem: If $k$ is square-free and not equal to $1$, the elliptic curve $y^2 = x^3+k$ has no rational torsion points.
Proof: See A. Knapp, Elliptic Curves, Princeton Univ. Press, 1992, Theorem $5.3$.
More generally, there is indeed a classification result as follows.
Theorem: Let $k=m^6 cdot k_0$, where $m, k_0in Bbb Z$ and $k_0$ is free of sixth power prime factors. Then the torsion subgroup of $Ecolon y^2=x^3+k$ over $Bbb Q$ is given as follows:
$$
E_{rm tors}(Bbb Q)=
begin{cases}
Bbb Z/6Bbb Z & text{ if } k_0= 1, \
Bbb Z/3Bbb Z & text{ if $k_0$ is a square different from $1$}, \
Bbb Z/3Bbb Z & text{ if $k_0=−432$}, \
Bbb Z/2Bbb Z & text{ if $k_0$ is a cube different from $1$}, \
{mathcal O} & text{ otherwise.}
end{cases}
$$
answered Jan 9 at 12:21
Dietrich BurdeDietrich Burde
82.1k649107
$endgroup$
$begingroup$
Is there something that explains what makes -432 so very special here? Is it related to the birational mapping to $u^3 + v^3 = 1$?
$endgroup$
– PineDoors
Jan 10 at 16:24
$begingroup$
Yes, $-432=-27cdot 16$ is related to the discriminant of the Mordell curve. For $k=-432m^6$, the torsion points are $mathcal{O},(12m^2,pm 36m^3)$.
$endgroup$
– Dietrich Burde
Jan 10 at 17:27
add a comment |
$begingroup$
Here is a further reference, extending the result you have linked.
Theorem: If $k$ is square-free and not equal to $1$, the elliptic curve $y^2 = x^3+k$ has no rational torsion points.
Proof: See A. Knapp, Elliptic Curves, Princeton Univ. Press, 1992, Theorem $5.3$.
More generally, there is indeed a classification result as follows.
Theorem: Let $k=m^6 cdot k_0$, where $m, k_0in Bbb Z$ and $k_0$ is free of sixth power prime factors. Then the torsion subgroup of $Ecolon y^2=x^3+k$ over $Bbb Q$ is given as follows:
$$
E_{rm tors}(Bbb Q)=
begin{cases}
Bbb Z/6Bbb Z & text{ if } k_0= 1, \
Bbb Z/3Bbb Z & text{ if $k_0$ is a square different from $1$}, \
Bbb Z/3Bbb Z & text{ if $k_0=−432$}, \
Bbb Z/2Bbb Z & text{ if $k_0$ is a cube different from $1$}, \
{mathcal O} & text{ otherwise.}
end{cases}
$$
answered Jan 9 at 12:21
Dietrich BurdeDietrich Burde
82.1k649107
$endgroup$
Here is a further reference, extending the result you have linked.
Theorem: If $k$ is square-free and not equal to $1$, the elliptic curve $y^2 = x^3+k$ has no rational torsion points.
Proof: See A. Knapp, Elliptic Curves, Princeton Univ. Press, 1992, Theorem $5.3$.
More generally, there is indeed a classification result as follows.
Theorem: Let $k=m^6 cdot k_0$, where $m, k_0in Bbb Z$ and $k_0$ is free of sixth power prime factors. Then the torsion subgroup of $Ecolon y^2=x^3+k$ over $Bbb Q$ is given as follows:
$$
E_{rm tors}(Bbb Q)=
begin{cases}
Bbb Z/6Bbb Z & text{ if } k_0= 1, \
Bbb Z/3Bbb Z & text{ if $k_0$ is a square different from $1$}, \
Bbb Z/3Bbb Z & text{ if $k_0=−432$}, \
Bbb Z/2Bbb Z & text{ if $k_0$ is a cube different from $1$}, \
{mathcal O} & text{ otherwise.}
end{cases}
$$
answered Jan 9 at 12:21
Dietrich BurdeDietrich Burde
82.1k649107
edited Jan 9 at 12:38
answered Jan 9 at 12:21
Dietrich BurdeDietrich Burde
82.1k649107
answered Jan 9 at 12:21
Dietrich BurdeDietrich Burde
82.1k649107
answered Jan 9 at 12:21
Dietrich BurdeDietrich Burde
82.1k649107
82.1k649107
$begingroup$
Is there something that explains what makes -432 so very special here? Is it related to the birational mapping to $u^3 + v^3 = 1$?
$endgroup$
– PineDoors
Jan 10 at 16:24
$begingroup$
Yes, $-432=-27cdot 16$ is related to the discriminant of the Mordell curve. For $k=-432m^6$, the torsion points are $mathcal{O},(12m^2,pm 36m^3)$.
$endgroup$
– Dietrich Burde
Jan 10 at 17:27
add a comment |
$begingroup$
Is there something that explains what makes -432 so very special here? Is it related to the birational mapping to $u^3 + v^3 = 1$?
$endgroup$
– PineDoors
Jan 10 at 16:24
$begingroup$
Yes, $-432=-27cdot 16$ is related to the discriminant of the Mordell curve. For $k=-432m^6$, the torsion points are $mathcal{O},(12m^2,pm 36m^3)$.
$endgroup$
– Dietrich Burde
Jan 10 at 17:27
$begingroup$
Is there something that explains what makes -432 so very special here? Is it related to the birational mapping to $u^3 + v^3 = 1$?
$endgroup$
– PineDoors
Jan 10 at 16:24
$begingroup$
Is there something that explains what makes -432 so very special here? Is it related to the birational mapping to $u^3 + v^3 = 1$?
$endgroup$
– PineDoors
Jan 10 at 16:24
$begingroup$
Yes, $-432=-27cdot 16$ is related to the discriminant of the Mordell curve. For $k=-432m^6$, the torsion points are $mathcal{O},(12m^2,pm 36m^3)$.
$endgroup$
– Dietrich Burde
Jan 10 at 17:27
$begingroup$
Yes, $-432=-27cdot 16$ is related to the discriminant of the Mordell curve. For $k=-432m^6$, the torsion points are $mathcal{O},(12m^2,pm 36m^3)$.
$endgroup$
– Dietrich Burde
Jan 10 at 17:27
add a comment |
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