don’t stop believin’ - university of san diego
TRANSCRIPT
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Don’t Stop Believin’
Don’t Stop Believin’There Is a Group Law on the Cubic
Josh Mollner
MathFest, 2008
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Don’t Stop Believin’
Introduction
Goals
At The End of This Talk, You Should Know:
The set of points on a non-singular, irreducible cubic planecurve can be formed into an abelian group.
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Don’t Stop Believin’
Introduction
Cubic Plane Curves
Cubic Plane Curves.
A cubic plane curve is the set of solutions in R2 to an equationof the form:
ax3 + bx2y + cxy2 + dy3 + ex2 + fxy + gy2 + hx + iy + j = 0
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Don’t Stop Believin’
Introduction
Cubic Plane Curves
Examples of Cubics.
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Don’t Stop Believin’
Introduction
Cubic Plane Curves
Special Types of Cubics.
Irreducible Cubic: A cubic whose equation cannot befactored.Non-Singular Cubic:
A cubic is singular at a point (a, b) if:
∂P∂x
(a, b) = 0 and∂P∂y
(a, b) = 0.
A cubic is non-singular if it has no points of singularity.
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Don’t Stop Believin’
Introduction
Cubic Plane Curves
Special Types of Cubics.
Irreducible Cubic: A cubic whose equation cannot befactored.Non-Singular Cubic:
A cubic is singular at a point (a, b) if:
∂P∂x
(a, b) = 0 and∂P∂y
(a, b) = 0.
A cubic is non-singular if it has no points of singularity.
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Don’t Stop Believin’
Introduction
Abelian Groups
Definition of an Abelian Group.
An abelian group is a set of elements, G, together with anoperation, +, such that the following properties hold:
1 There is an identity element.2 Each element has an inverse.3 Addition is associative.4 Addition is commutative.
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Don’t Stop Believin’
Introduction
Abelian Groups
Definition of an Abelian Group.
An abelian group is a set of elements, G, together with anoperation, +, such that the following properties hold:
1 There is an identity element.2 Each element has an inverse.3 Addition is associative.4 Addition is commutative.
![Page 9: Don’t Stop Believin’ - University of San Diego](https://reader035.vdocuments.site/reader035/viewer/2022081505/62a2bbba0de4137d9a2b6304/html5/thumbnails/9.jpg)
Don’t Stop Believin’
Introduction
Abelian Groups
Definition of an Abelian Group.
An abelian group is a set of elements, G, together with anoperation, +, such that the following properties hold:
1 There is an identity element.2 Each element has an inverse.3 Addition is associative.4 Addition is commutative.
![Page 10: Don’t Stop Believin’ - University of San Diego](https://reader035.vdocuments.site/reader035/viewer/2022081505/62a2bbba0de4137d9a2b6304/html5/thumbnails/10.jpg)
Don’t Stop Believin’
Introduction
Abelian Groups
Definition of an Abelian Group.
An abelian group is a set of elements, G, together with anoperation, +, such that the following properties hold:
1 There is an identity element.2 Each element has an inverse.3 Addition is associative.4 Addition is commutative.
![Page 11: Don’t Stop Believin’ - University of San Diego](https://reader035.vdocuments.site/reader035/viewer/2022081505/62a2bbba0de4137d9a2b6304/html5/thumbnails/11.jpg)
Don’t Stop Believin’
Introduction
Abelian Groups
Definition of an Abelian Group.
An abelian group is a set of elements, G, together with anoperation, +, such that the following properties hold:
1 There is an identity element.2 Each element has an inverse.3 Addition is associative.4 Addition is commutative.
![Page 12: Don’t Stop Believin’ - University of San Diego](https://reader035.vdocuments.site/reader035/viewer/2022081505/62a2bbba0de4137d9a2b6304/html5/thumbnails/12.jpg)
Don’t Stop Believin’
The Group Law.
Addition of Points
At The End of This Talk, You Should Know:
We will show that the set of points of C, where C is anon-singular, irreducible cubic, is an abelian group.
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Don’t Stop Believin’
The Group Law.
Addition of Points
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Don’t Stop Believin’
The Group Law.
Addition of Points
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Don’t Stop Believin’
The Group Law.
Addition of Points
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Don’t Stop Believin’
The Group Law.
Addition of Points
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Don’t Stop Believin’
The Group Law.
Addition of Points
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Don’t Stop Believin’
The Group Law.
Addition of Points
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Don’t Stop Believin’
The Group Law.
Addition of Points
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Don’t Stop Believin’
The Group Law.
Well-Definiedness of Addition of Points.
Is Addition of Points is Well-Defined?
We might worry that addition of points is not well-defined.What if the line between P and Q does not intersect C at athird point?What if the line between P and Q intersects C at twoadditional points?
Don’t Stop Believin’
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Don’t Stop Believin’
The Group Law.
Well-Definiedness of Addition of Points.
Is Addition of Points is Well-Defined?
We might worry that addition of points is not well-defined.What if the line between P and Q does not intersect C at athird point?What if the line between P and Q intersects C at twoadditional points?
Don’t Stop Believin’
![Page 22: Don’t Stop Believin’ - University of San Diego](https://reader035.vdocuments.site/reader035/viewer/2022081505/62a2bbba0de4137d9a2b6304/html5/thumbnails/22.jpg)
Don’t Stop Believin’
The Group Law.
Well-Definiedness of Addition of Points.
Is Addition of Points is Well-Defined?
We might worry that addition of points is not well-defined.What if the line between P and Q does not intersect C at athird point?What if the line between P and Q intersects C at twoadditional points?
Don’t Stop Believin’
![Page 23: Don’t Stop Believin’ - University of San Diego](https://reader035.vdocuments.site/reader035/viewer/2022081505/62a2bbba0de4137d9a2b6304/html5/thumbnails/23.jpg)
Don’t Stop Believin’
The Group Law.
Well-Definiedness of Addition of Points.
Is Addition of Points is Well-Defined?
We might worry that addition of points is not well-defined.What if the line between P and Q does not intersect C at athird point?What if the line between P and Q intersects C at twoadditional points?
Don’t Stop Believin’
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Don’t Stop Believin’
The Group Law.
Well-Definiedness of Addition of Points.
Yes! Addition of Points is Well-Defined.
TheoremLet l be a line that intersects an irreducible cubic C at leasttwice, counting multiplicities. Then l intersects C exactly threetimes, counting multiplicities.
A line tangent to C at a point P intersects C twice at P.A line tangent to C at an inflection point P intersects Cthree times at P.
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Don’t Stop Believin’
The Group Law.
Well-Definiedness of Addition of Points.
Yes! Addition of Points is Well-Defined.
TheoremLet l be a line that intersects an irreducible cubic C at leasttwice, counting multiplicities. Then l intersects C exactly threetimes, counting multiplicities.
A line tangent to C at a point P intersects C twice at P.A line tangent to C at an inflection point P intersects Cthree times at P.
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Don’t Stop Believin’
The Group Law.
The Properties of an Abelian Group Are Satisfied.
There Is An Identity.There exists an O on C such that P + O = P for all P on C.
We will show that O is the identity. We must show thatP + O = P for all P.
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Don’t Stop Believin’
The Group Law.
The Properties of an Abelian Group Are Satisfied.
There Is An Identity.There exists an O on C such that P + O = P for all P on C.
We will show that O is the identity. We must show thatP + O = P for all P.
![Page 28: Don’t Stop Believin’ - University of San Diego](https://reader035.vdocuments.site/reader035/viewer/2022081505/62a2bbba0de4137d9a2b6304/html5/thumbnails/28.jpg)
Don’t Stop Believin’
The Group Law.
The Properties of an Abelian Group Are Satisfied.
There Is An Identity.There exists an O on C such that P + O = P for all P on C.
We will show that O is the identity. We must show thatP + O = P for all P.
![Page 29: Don’t Stop Believin’ - University of San Diego](https://reader035.vdocuments.site/reader035/viewer/2022081505/62a2bbba0de4137d9a2b6304/html5/thumbnails/29.jpg)
Don’t Stop Believin’
The Group Law.
The Properties of an Abelian Group Are Satisfied.
There Is An Identity.There exists an O on C such that P + O = P for all P on C.
We will show that O is the identity. We must show thatP + O = P for all P.
![Page 30: Don’t Stop Believin’ - University of San Diego](https://reader035.vdocuments.site/reader035/viewer/2022081505/62a2bbba0de4137d9a2b6304/html5/thumbnails/30.jpg)
Don’t Stop Believin’
The Group Law.
The Properties of an Abelian Group Are Satisfied.
There Is An Identity.There exists an O on C such that P + O = P for all P on C.
We will show that O is the identity. We must show thatP + O = P for all P.
![Page 31: Don’t Stop Believin’ - University of San Diego](https://reader035.vdocuments.site/reader035/viewer/2022081505/62a2bbba0de4137d9a2b6304/html5/thumbnails/31.jpg)
Don’t Stop Believin’
The Group Law.
The Properties of an Abelian Group Are Satisfied.
There Is An Identity.There exists an O on C such that P + O = P for all P on C.
We will show that O is the identity. We must show thatP + O = P for all P.
![Page 32: Don’t Stop Believin’ - University of San Diego](https://reader035.vdocuments.site/reader035/viewer/2022081505/62a2bbba0de4137d9a2b6304/html5/thumbnails/32.jpg)
Don’t Stop Believin’
The Group Law.
The Properties of an Abelian Group Are Satisfied.
Inverses Exist.For every P on C, there is a −P on C such that P + (−P) = O.
Given a point P on C, we construct −P in the following manner:
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Don’t Stop Believin’
The Group Law.
The Properties of an Abelian Group Are Satisfied.
Inverses Exist.For every P on C, there is a −P on C such that P + (−P) = O.
Given a point P on C, we construct −P in the following manner:
![Page 34: Don’t Stop Believin’ - University of San Diego](https://reader035.vdocuments.site/reader035/viewer/2022081505/62a2bbba0de4137d9a2b6304/html5/thumbnails/34.jpg)
Don’t Stop Believin’
The Group Law.
The Properties of an Abelian Group Are Satisfied.
Inverses Exist.For every P on C, there is a −P on C such that P + (−P) = O.
Given a point P on C, we construct −P in the following manner:
![Page 35: Don’t Stop Believin’ - University of San Diego](https://reader035.vdocuments.site/reader035/viewer/2022081505/62a2bbba0de4137d9a2b6304/html5/thumbnails/35.jpg)
Don’t Stop Believin’
The Group Law.
The Properties of an Abelian Group Are Satisfied.
Inverses Exist.For every P on C, there is a −P on C such that P + (−P) = O.
Given a point P on C, we construct −P in the following manner:
![Page 36: Don’t Stop Believin’ - University of San Diego](https://reader035.vdocuments.site/reader035/viewer/2022081505/62a2bbba0de4137d9a2b6304/html5/thumbnails/36.jpg)
Don’t Stop Believin’
The Group Law.
The Properties of an Abelian Group Are Satisfied.
Inverses Exist.For every P on C, there is a −P on C such that P + (−P) = O.
Given a point P on C, we construct −P in the following manner:
![Page 37: Don’t Stop Believin’ - University of San Diego](https://reader035.vdocuments.site/reader035/viewer/2022081505/62a2bbba0de4137d9a2b6304/html5/thumbnails/37.jpg)
Don’t Stop Believin’
The Group Law.
The Properties of an Abelian Group Are Satisfied.
Addition is Commutative.For every P and Q on C, P + Q = Q + P.
We must show that P + Q = Q + P. But this is clear, since theline between P and Q is the same as the line between Q and P.
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Don’t Stop Believin’
The Group Law.
The Properties of an Abelian Group Are Satisfied.
Addition is Commutative.For every P and Q on C, P + Q = Q + P.
We must show that P + Q = Q + P. But this is clear, since theline between P and Q is the same as the line between Q and P.
![Page 39: Don’t Stop Believin’ - University of San Diego](https://reader035.vdocuments.site/reader035/viewer/2022081505/62a2bbba0de4137d9a2b6304/html5/thumbnails/39.jpg)
Don’t Stop Believin’
The Group Law.
The Properties of an Abelian Group Are Satisfied.
Associativity.The Cayley-Bacharach Theorem.
TheoremLet C1 and C2 be two cubic curves which intersect in exactlynine points. Suppose C is a third cubic curve which passesthrough eight of these nine points. Then C also passes throughthe ninth point.
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Don’t Stop Believin’
The Group Law.
The Properties of an Abelian Group Are Satisfied.
Associativity.For all P, Q, and R on C, (P + Q) + R = P + (Q + R).
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Don’t Stop Believin’
Conclusion
Mordell’s Theorem
The Set of Rational Points Is a Group Too.
We have shown that the set of points on an irreducible,non-singular cubic is an abelian group.It is also true that the set of rational points on a rational,irreducible, non-singular cubic with at least one rationalpoint is an abelian group.
A rational cubic is one whose equation can be written in theform
ax3 + bx2y + cxy2 + dy3 + ex2 + fxy + gy2 + hx + iy + j = 0
Where a, b, c, d , e, f , g, h, i , and j are rational numbers.
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Don’t Stop Believin’
Conclusion
Mordell’s Theorem
The Set of Rational Points Is a Group Too.
We have shown that the set of points on an irreducible,non-singular cubic is an abelian group.It is also true that the set of rational points on a rational,irreducible, non-singular cubic with at least one rationalpoint is an abelian group.
A rational cubic is one whose equation can be written in theform
ax3 + bx2y + cxy2 + dy3 + ex2 + fxy + gy2 + hx + iy + j = 0
Where a, b, c, d , e, f , g, h, i , and j are rational numbers.
![Page 43: Don’t Stop Believin’ - University of San Diego](https://reader035.vdocuments.site/reader035/viewer/2022081505/62a2bbba0de4137d9a2b6304/html5/thumbnails/43.jpg)
Don’t Stop Believin’
Conclusion
Mordell’s Theorem
The Set of Rational Points Is a Group Too.
We have shown that the set of points on an irreducible,non-singular cubic is an abelian group.It is also true that the set of rational points on a rational,irreducible, non-singular cubic with at least one rationalpoint is an abelian group.
A rational cubic is one whose equation can be written in theform
ax3 + bx2y + cxy2 + dy3 + ex2 + fxy + gy2 + hx + iy + j = 0
Where a, b, c, d , e, f , g, h, i , and j are rational numbers.
![Page 44: Don’t Stop Believin’ - University of San Diego](https://reader035.vdocuments.site/reader035/viewer/2022081505/62a2bbba0de4137d9a2b6304/html5/thumbnails/44.jpg)
Don’t Stop Believin’
Conclusion
Mordell’s Theorem
Mordell’s Theorem.
Mordell’s Theorem: The set of rational points on a rational,irreducible, non-singular cubic with at least one rationalpoint is a finitely-generated abelian group.There is a set of rational points, P1, ..., Pn, such that if Q isa rational point on C, we can write:
Q =n∑
i=1
niPi .
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Don’t Stop Believin’
Conclusion
Mordell’s Theorem
Mordell’s Theorem.
Mordell’s Theorem: The set of rational points on a rational,irreducible, non-singular cubic with at least one rationalpoint is a finitely-generated abelian group.There is a set of rational points, P1, ..., Pn, such that if Q isa rational point on C, we can write:
Q =n∑
i=1
niPi .
![Page 46: Don’t Stop Believin’ - University of San Diego](https://reader035.vdocuments.site/reader035/viewer/2022081505/62a2bbba0de4137d9a2b6304/html5/thumbnails/46.jpg)
Don’t Stop Believin’
Conclusion
An Example
Mordell’s Theorem Is Useful In The Real World.
Here is a real-world example of how Mordell’s Theorem isuseful:Let’s say you are LOST on an island ...
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Don’t Stop Believin’
Conclusion
An Example
Mordell’s Theorem Is Useful In The Real World.
Here is a real-world example of how Mordell’s Theorem isuseful:Let’s say you are LOST on an island ...
![Page 48: Don’t Stop Believin’ - University of San Diego](https://reader035.vdocuments.site/reader035/viewer/2022081505/62a2bbba0de4137d9a2b6304/html5/thumbnails/48.jpg)
Don’t Stop Believin’
Conclusion
An Example
Mordell’s Theorem Is Useful In The Real World.
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Don’t Stop Believin’
Conclusion
An Example
LOST
You have a dream:
![Page 50: Don’t Stop Believin’ - University of San Diego](https://reader035.vdocuments.site/reader035/viewer/2022081505/62a2bbba0de4137d9a2b6304/html5/thumbnails/50.jpg)
Don’t Stop Believin’
Conclusion
An Example
LOST
You have a dream:
![Page 51: Don’t Stop Believin’ - University of San Diego](https://reader035.vdocuments.site/reader035/viewer/2022081505/62a2bbba0de4137d9a2b6304/html5/thumbnails/51.jpg)
Don’t Stop Believin’
Conclusion
An Example
LOST
You have a dream:
![Page 52: Don’t Stop Believin’ - University of San Diego](https://reader035.vdocuments.site/reader035/viewer/2022081505/62a2bbba0de4137d9a2b6304/html5/thumbnails/52.jpg)
Don’t Stop Believin’
Conclusion
An Example
LOST
You have a dream:
![Page 53: Don’t Stop Believin’ - University of San Diego](https://reader035.vdocuments.site/reader035/viewer/2022081505/62a2bbba0de4137d9a2b6304/html5/thumbnails/53.jpg)
Don’t Stop Believin’
Conclusion
An Example
LOST
You have a dream:
![Page 54: Don’t Stop Believin’ - University of San Diego](https://reader035.vdocuments.site/reader035/viewer/2022081505/62a2bbba0de4137d9a2b6304/html5/thumbnails/54.jpg)
Don’t Stop Believin’
Conclusion
An Example
LOST
You wake up:
![Page 55: Don’t Stop Believin’ - University of San Diego](https://reader035.vdocuments.site/reader035/viewer/2022081505/62a2bbba0de4137d9a2b6304/html5/thumbnails/55.jpg)
Don’t Stop Believin’
Conclusion
An Example
LOST
You try P2 + P2:
![Page 56: Don’t Stop Believin’ - University of San Diego](https://reader035.vdocuments.site/reader035/viewer/2022081505/62a2bbba0de4137d9a2b6304/html5/thumbnails/56.jpg)
Don’t Stop Believin’
Conclusion
An Example
LOST
You try P2 + P2:
![Page 57: Don’t Stop Believin’ - University of San Diego](https://reader035.vdocuments.site/reader035/viewer/2022081505/62a2bbba0de4137d9a2b6304/html5/thumbnails/57.jpg)
Don’t Stop Believin’
Conclusion
An Example
LOST
And you find the hatch!
![Page 58: Don’t Stop Believin’ - University of San Diego](https://reader035.vdocuments.site/reader035/viewer/2022081505/62a2bbba0de4137d9a2b6304/html5/thumbnails/58.jpg)
Don’t Stop Believin’
Conclusion
Open Questions
Open Questions.
Given a cubic, which are the rational points whichcompose this finite generating set?Given a cubic, what is the minimum number of pointsneeded in a finite generating set?It is not yet known how to determine in a finite number ofsteps whether a given cubic has a rational point at all.
![Page 59: Don’t Stop Believin’ - University of San Diego](https://reader035.vdocuments.site/reader035/viewer/2022081505/62a2bbba0de4137d9a2b6304/html5/thumbnails/59.jpg)
Don’t Stop Believin’
Conclusion
Open Questions
Open Questions.
Given a cubic, which are the rational points whichcompose this finite generating set?Given a cubic, what is the minimum number of pointsneeded in a finite generating set?It is not yet known how to determine in a finite number ofsteps whether a given cubic has a rational point at all.
![Page 60: Don’t Stop Believin’ - University of San Diego](https://reader035.vdocuments.site/reader035/viewer/2022081505/62a2bbba0de4137d9a2b6304/html5/thumbnails/60.jpg)
Don’t Stop Believin’
Conclusion
Open Questions
Open Questions.
Given a cubic, which are the rational points whichcompose this finite generating set?Given a cubic, what is the minimum number of pointsneeded in a finite generating set?It is not yet known how to determine in a finite number ofsteps whether a given cubic has a rational point at all.
![Page 61: Don’t Stop Believin’ - University of San Diego](https://reader035.vdocuments.site/reader035/viewer/2022081505/62a2bbba0de4137d9a2b6304/html5/thumbnails/61.jpg)
Don’t Stop Believin’
Conclusion
Open Questions
Open Questions.
Given a cubic, which are the rational points whichcompose this finite generating set?Given a cubic, what is the minimum number of pointsneeded in a finite generating set?It is not yet known how to determine in a finite number ofsteps whether a given cubic has a rational point at all.
![Page 62: Don’t Stop Believin’ - University of San Diego](https://reader035.vdocuments.site/reader035/viewer/2022081505/62a2bbba0de4137d9a2b6304/html5/thumbnails/62.jpg)
Don’t Stop Believin’
Conclusion
Open Questions
Open Questions.
Given a cubic, which are the rational points whichcompose this finite generating set?Given a cubic, what is the minimum number of pointsneeded in a finite generating set?It is not yet known how to determine in a finite number ofsteps whether a given cubic has a rational point at all.
![Page 63: Don’t Stop Believin’ - University of San Diego](https://reader035.vdocuments.site/reader035/viewer/2022081505/62a2bbba0de4137d9a2b6304/html5/thumbnails/63.jpg)
Don’t Stop Believin’
Conclusion
Open Questions
THANK YOU!
AND
DON’T STOP BELIEVIN’
![Page 64: Don’t Stop Believin’ - University of San Diego](https://reader035.vdocuments.site/reader035/viewer/2022081505/62a2bbba0de4137d9a2b6304/html5/thumbnails/64.jpg)
Don’t Stop Believin’
Conclusion
Open Questions
THANK YOU!
AND
DON’T STOP BELIEVIN’