What happens to the group structure of an elliptic curve over a field when the discriminant = 0?
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Working on a question for a number theory class.
So, basically, it asks us what happens to the group structure of an elliptic curve over a field if the discriminant is equal to zero?
So, basically, what I've got is that either is crosses itself, or it ends up having a cusp. In either case, it does not have a well defined derivative at some point. Since lambda depends on a well defined derivative, if an elliptic curve has a singular point at (a, b), then elliptic curve addition would not be well defined for (a, b) + (a, b).
Is this right? Am I missing something else that happens to group structure?
EDIT: I guess, also, when they are this shape, we couldn't guarantee that a tangent line that intersected the line in two places intersected it in a third place. So, then, the operations aren't necessarily well defined anywhere? Is that more right?
group-theory elliptic-curves discriminant
asked Nov 14 at 4:01
Chris N-L
112
add a comment |
up vote
2
down vote
favorite
Working on a question for a number theory class.
So, basically, it asks us what happens to the group structure of an elliptic curve over a field if the discriminant is equal to zero?
So, basically, what I've got is that either is crosses itself, or it ends up having a cusp. In either case, it does not have a well defined derivative at some point. Since lambda depends on a well defined derivative, if an elliptic curve has a singular point at (a, b), then elliptic curve addition would not be well defined for (a, b) + (a, b).
Is this right? Am I missing something else that happens to group structure?
EDIT: I guess, also, when they are this shape, we couldn't guarantee that a tangent line that intersected the line in two places intersected it in a third place. So, then, the operations aren't necessarily well defined anywhere? Is that more right?
group-theory elliptic-curves discriminant
asked Nov 14 at 4:01
Chris N-L
112
Do you mean discriminant?
– Randall
Nov 14 at 4:04
Yes. Sorry. Super tired. Haha. Edited the post.
– Chris N-L
Nov 14 at 4:10
add a comment |
up vote
2
down vote
favorite
up vote
2
down vote
favorite
Working on a question for a number theory class.
So, basically, it asks us what happens to the group structure of an elliptic curve over a field if the discriminant is equal to zero?
So, basically, what I've got is that either is crosses itself, or it ends up having a cusp. In either case, it does not have a well defined derivative at some point. Since lambda depends on a well defined derivative, if an elliptic curve has a singular point at (a, b), then elliptic curve addition would not be well defined for (a, b) + (a, b).
Is this right? Am I missing something else that happens to group structure?
EDIT: I guess, also, when they are this shape, we couldn't guarantee that a tangent line that intersected the line in two places intersected it in a third place. So, then, the operations aren't necessarily well defined anywhere? Is that more right?
group-theory elliptic-curves discriminant
asked Nov 14 at 4:01
Chris N-L
112
Working on a question for a number theory class.
So, basically, it asks us what happens to the group structure of an elliptic curve over a field if the discriminant is equal to zero?
So, basically, what I've got is that either is crosses itself, or it ends up having a cusp. In either case, it does not have a well defined derivative at some point. Since lambda depends on a well defined derivative, if an elliptic curve has a singular point at (a, b), then elliptic curve addition would not be well defined for (a, b) + (a, b).
Is this right? Am I missing something else that happens to group structure?
EDIT: I guess, also, when they are this shape, we couldn't guarantee that a tangent line that intersected the line in two places intersected it in a third place. So, then, the operations aren't necessarily well defined anywhere? Is that more right?
group-theory elliptic-curves discriminant
group-theory elliptic-curves discriminant
asked Nov 14 at 4:01
Chris N-L
112
asked Nov 14 at 4:01
Chris N-L
112
edited Nov 14 at 4:44
asked Nov 14 at 4:01
Chris N-L
112
asked Nov 14 at 4:01
Chris N-L
112
asked Nov 14 at 4:01
Chris N-L
112
112
Do you mean discriminant?
– Randall
Nov 14 at 4:04
Yes. Sorry. Super tired. Haha. Edited the post.
– Chris N-L
Nov 14 at 4:10
add a comment |
Do you mean discriminant?
– Randall
Nov 14 at 4:04
Yes. Sorry. Super tired. Haha. Edited the post.
– Chris N-L
Nov 14 at 4:10
Do you mean discriminant?
– Randall
Nov 14 at 4:04
Do you mean discriminant?
– Randall
Nov 14 at 4:04
Yes. Sorry. Super tired. Haha. Edited the post.
– Chris N-L
Nov 14 at 4:10
Yes. Sorry. Super tired. Haha. Edited the post.
– Chris N-L
Nov 14 at 4:10
add a comment |
1 Answer
1
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If the discriminant is zero, it's not an elliptic curve. Anyway,
consider an singular irreducible plane cubic curve $C$ over an algebraically
closed field.
A singular irreducible cubic has one singular point. The
non-singular points on the curve do have a group structure though. When $C$
has a node, the group of non-singular points is isomorphic to
the multiplicative group
$K^*$ and when $C$ has a cusp, the group is isomorphic to the additive
group of $K$.
You can find details in texts such as Silverman's.
answered Nov 14 at 4:43
Lord Shark the Unknown
97.5k958128
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
If the discriminant is zero, it's not an elliptic curve. Anyway,
consider an singular irreducible plane cubic curve $C$ over an algebraically
closed field.
A singular irreducible cubic has one singular point. The
non-singular points on the curve do have a group structure though. When $C$
has a node, the group of non-singular points is isomorphic to
the multiplicative group
$K^*$ and when $C$ has a cusp, the group is isomorphic to the additive
group of $K$.
You can find details in texts such as Silverman's.
answered Nov 14 at 4:43
Lord Shark the Unknown
97.5k958128
add a comment |
up vote
0
down vote
If the discriminant is zero, it's not an elliptic curve. Anyway,
consider an singular irreducible plane cubic curve $C$ over an algebraically
closed field.
A singular irreducible cubic has one singular point. The
non-singular points on the curve do have a group structure though. When $C$
has a node, the group of non-singular points is isomorphic to
the multiplicative group
$K^*$ and when $C$ has a cusp, the group is isomorphic to the additive
group of $K$.
You can find details in texts such as Silverman's.
answered Nov 14 at 4:43
Lord Shark the Unknown
97.5k958128
add a comment |
up vote
0
down vote
up vote
0
down vote
If the discriminant is zero, it's not an elliptic curve. Anyway,
consider an singular irreducible plane cubic curve $C$ over an algebraically
closed field.
A singular irreducible cubic has one singular point. The
non-singular points on the curve do have a group structure though. When $C$
has a node, the group of non-singular points is isomorphic to
the multiplicative group
$K^*$ and when $C$ has a cusp, the group is isomorphic to the additive
group of $K$.
You can find details in texts such as Silverman's.
answered Nov 14 at 4:43
Lord Shark the Unknown
97.5k958128
If the discriminant is zero, it's not an elliptic curve. Anyway,
consider an singular irreducible plane cubic curve $C$ over an algebraically
closed field.
A singular irreducible cubic has one singular point. The
non-singular points on the curve do have a group structure though. When $C$
has a node, the group of non-singular points is isomorphic to
the multiplicative group
$K^*$ and when $C$ has a cusp, the group is isomorphic to the additive
group of $K$.
You can find details in texts such as Silverman's.
answered Nov 14 at 4:43
Lord Shark the Unknown
97.5k958128
answered Nov 14 at 4:43
Lord Shark the Unknown
97.5k958128
answered Nov 14 at 4:43
Lord Shark the Unknown
97.5k958128
answered Nov 14 at 4:43
Lord Shark the Unknown
97.5k958128
97.5k958128
add a comment |
add a comment |
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Do you mean discriminant?
– Randall
Nov 14 at 4:04
Yes. Sorry. Super tired. Haha. Edited the post.
– Chris N-L
Nov 14 at 4:10