Hints of Tori

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Slicing Bagels: Plane Sections of

Real and Complex Tori

Asilomar - December 2004

Bruce Cohen

Lowell High School, SFUSD

[email protected]://www.cgl.ucsf.edu/home/bic

David Sklar

San Francisco State University

[email protected]



Part I - Slicing a Real Circular Torus

Equations for the torus in R 3

The Spiric Sections of Perseus

Ovals of Cassini and The Lemniscate of Bernoulli

Other Slices

The Villarceau Circles

A Characterization of the torus



Bibliography

Part II - Slicing a Complex Torus

and 2 2 2 2 2 2 2( 1 )( 2 ) ( ) y x x x x g

Elliptic curves and number theory

2 2( 1) y x x c Some graphs of

Hints of toric sections

Two closures: Algebraic and Geometric

2 2

( 1) y x x Algebraic closure, C2, R 4, and the graph of

Geometric closure, Projective spaces

P1(R ), P2(R ), P1(C), and P2(C)

2 2 2 2 2( 1), ( ), y x x y x x n The graphs of




2( )( )

n n y x x a x b

Roughly, an elliptic curve over a field F is the graph of an equation of the

form where p( x) is a cubic polynomial with three distinct rootsand coefficients in F . The fields of most interest are the rational numbers,

finite fields, the real numbers, and the complex numbers.

2

( ) y p x

Within a year it was shown that Fermat’s last theorem would follow from awidely believed conjecture in the arithmetic theory of elliptic curves.

In 1985, after mathematicians had been working on Fermat’s Last Theorem

for about 350 years, Gerhard Frey suggested that if we assumed Fermat’s

Last Theorem was false, the existence of an elliptic curve

where a, b and c are distinct integers such that with integer

exponent n > 2, might lead to a contradiction.

n n na b c

Less than 10 years later Andrew Wiles proved a form of the Taniyama

conjecture sufficient to prove Fermat’s Last Theorem.




The strategy of placing a centuries old number theory problem in the context

of the arithmetic theory of elliptic curves has led to the complete or partialsolution of at least three major problems in the last thirty years.

The Congruent Number Problem – Tunnell 1983

The Gauss Class Number Problem – Goldfeld 1976, Gross & Zagier 1986

Fermat’s Last Theorem – Frey 1985, Ribet 1986, Wiles 1995, Taylor 1995

Although a significant discussion of the theory of elliptic curves and why

they are so nice is beyond the scope of this talk I would like to try to show

you that, when looked at in the right way, the graph of an elliptic curve is a

beautiful and familiar geometric object. We’ll do this by studying the graphof the equation 2 2( 1). y x x



2 2( 1) y x x 2 2

( 1) 0.3 y x x

2 2( 1) 1 y x x 2 2

( 1) 0.385 y x x

Graphs of 2 2( 1) y x x c : Hints of Toric Sections

x

y

x

y

x

y

x

y

2 2( 1) y x x

If we close up the algebra to include the complex numbers and the geometry to

include points at infinity, we can argue that the graph of is a torus.



Geometric Closure: an Introduction to Projective GeometryPart I – Real Projective Geometry

One-Dimension - the Real Projective Line P1(R )

The real (affine) line R is the

ordinary real number line

The real projective line P1(R ) is

the set R

0

It is topologically equivalent to the open

interval (-1, 1) by the map (1 ) x x x

01 1

and topologically equivalent to a puncturedcircle by stereographic projection

0

It is topologically equivalent to a closed

interval with the endpoints identified

0 P P

0

and topologically equivalent to a circle by stereographic projection

0

P



Geometric Closure: an Introduction to Projective GeometryPart I – Real Projective Geometry

Two-Dimensions - the Real Projective Plane P2(R )

The real (affine) plane R 2 isthe ordinary x, y -plane

It is topologically equivalent to a closed

disk with antipodal points on the

boundary circle identified.

x

y

2 2 2 2 ( , ) ,

1 1

x y x y

x y x y

It is topologically equivalent to

the open unit disk by the map

( )

x

y

x

y

The real projective plane P2(R ) is theset . It is R 2 together with a

“line at infinity”, . Every line in R 2

intersects , parallel lines meet at the

same point on , and nonparallel

lines intersect at distinct points.Every line in P2(R ) is a P1(R ).

2 LR

L

L

L

L

Two distinct lines intersect

at one and only one point.



A Projective View of the Conics

x

y

Ellipse

x

y

x

y

Parabola

x

y

x

y

Hyperbola



A Projective View of the Conics

Ellipse Parabola

Hyperbola



2 2( 1) y x x 2 2

( 1) 0.3 y x x

2 2( 1) 1 y x x 2 2

( 1) 0.385 y x x

Graphs of 2 2( 1) y x x c : Hints of Toric Sections

including pointtopological view

at infinity

If we close up the algebra, by extending to the , and the geometry,complex numbers2 2

by including points at infinity we can argue that the graph of ( 1) is a torus. y x x

x

y

x

y

x

y

x

y





Graph of with x and y complex

Algebraic closure

2 2( 1) y x x

2 2 3 2

1 2 1 1 1 23 y y x x x x

3 2

1 2 2 2 1 22 3 y y x x x x

1 2 1 2Letting and , then solving for and in terms of and , x s x t y y s t

1 2 1 1 2 2we would essentially have , , ( , ) and ( , ) x s x t y y s t y y s t

1 2 1 2These are parametric equations for a surface in , , , space x x y y

1 2for ( , ) and ( , ) which can be pieced together to get the whole graph. y s t y s t

The situation is a little more complicated in that the algebra leads to several solutions

Some comments on why the graph of the system

is a surface.

1 2 1 2(a nice mapping of a 2-D , plane into 4-D , , , space.) s t x x y y



Graph of with x and y complex

Algebraic closure

2 2( 1) y x x

1 x

1 y

1 x

2 y

2 21 1 1( 1) y x x

2 1for 0, 0 x y

2 22 1 1( 1) y x x

2 2 3 2

1 2 1 1 1 2

3 y y x x x x 3 2

1 2 2 2 1 22 3 y y x x x x

2 2for 0, 0 x y

1 1(the , - plane) x y2 2

( 1) becomes y x x 1 2(the , - plane) x y

2 2( 1) becomes y x x

21 1( )[( ) 1] x x



Graph of with x and y complex2 2( 1) y x x

2 2 3

1 2 1 1

1 22 0

y y x x y y

The system of equations becomes

2 2 2Recall, the graph of ( 1) in is equivalent to the graph of the system y x x C

2 2 3 2

1 2 1 1 1 2

3 2

1 2 2 2 1 2

32 3

y y x x x x y y x x x x

Now lets look at the intersection of 4in .R

2the graph with the 3-space 0. x 1

x

2 y

1 y

2 1so 0 or 0 y y

1 1 1 2and the intersection (a curve) lies in only the , - plane or the , - plane. x y x y

1 x

1 y

1 x

2 y

2for 0, y 2 2

1 1 1( 1) y x x

1for 0, y 2 2

2 1 1( 1) y x x




1 x

1 y

2 y

1 x

1 y

1 x

2 y

2

1 1 1 2

The intersection of the graph with the 3-space 0 is a curve whose branches

lie only the , - plane or the , - plane so we can put together this picture.

x

x y x y

2 2 4

2 ( 1) in intersecting the 3-space =0 y x x x R

P

1 0 1

2 Topological view in projective C

2 (roughly with points at infinity)C



Geometric Closure: an Introduction to Projective GeometryPart II – Complex Projective Geometry

One-Dimension - the Complex Projective Line or Riemann Sphere P1(C)

The complex (affine) line C is the

ordinary complex plane where ( x, y)

corresponds to the number z = x + iy.

x

y

It is topologically a punctured sphere

by stereographic projection

The complex projective line P1(C) is

the set the complex plane

with one number adjoined. C

It is topologically a sphere by

stereographic projection with the

north pole corresponding to . It is

often called the Riemann Sphere.

(Note: 1-D over the complex numbers, but, 2-D over the real numbers)



Geometric Closure: an Introduction to Projective GeometryPart II – Complex Projective Geometry

Two-Dimensions - the Complex Projective Plane P2(C)

The complex (affine) “plane” C2 or

better complex 2-space is a lot like

R 4. A line in C2 is the graph of an

equation of the form ,

where a, b and c are complexconstants and x and y are complex

variables. (Note: not every plane in

R 4 corresponds to a complex line)

ax by c

(Note: 2-D over the complex numbers, but, 4-D over the real numbers)

Complex projective 2-space P2(C) is

the set . It is C2 together with

a complex “line at infinity”, . Every

line in R 2 intersects , parallel lines

meet at the same point on , andnonparallel lines intersect at

distinct points. Every line in P2(C) is a

P1(C), a Riemann sphere, including the

“line at infinity”. Basically P2(C) is C2

closed up nicely by a Riemann Sphereat infinity.

2 LC

L

L

L

L

Two distinct lines intersect at one and

only one point.




1 x

1 y

2 y

1 x

1 y

1 x

2 y

2

1 1 1 2

The intersection of the graph with the 3-space 0 is a curve whose branches

lie only the , - plane or the , - plane so we can put together this picture.

x

x y x y

2 2 4


P

1 0 1

2 Topological view in projective C

2 (roughly with points at infinity)C



P

1 0 1

P

1 0 1


1 x

1 y

2 y

2 2 4


2intersecting the 3-space = > 0 x



2 2 2 2 2 2 2 2 2 2 2The graph of ( 1 )( 2 )( 3 )( 4 )( 5 ) y x x x x x x

2 2 2 2 2 2 2( 1 )( 2 ) ( ) y x x x x g

A Generalization: the Graph of

2intersected with the 3-space 0 x

1 x

1 y

2 y



2 2 2 2 2 2 2( 1 )( 2 ) ( ) y x x x x g

2 2 2 2 2 2 2 2 2 2 2( 1 )( 2 )( 3 )( 4 )( 5 ) y x x x x x x

A Generalization: the Graph of



A depiction of the toric graphs

of the elliptic curves

2 2 2 ( ) y x x n

by A. T. Fomenko

This drawing is the frontispiece

of Neal Koblitz's book

Introduction to Elliptic Curves

and Modular Forms



Bibliography

8. T. Needham, Visual Complex Analysis, Oxford University Press, Oxford 1997

1. E. Brieskorn & H. Knorrer, Plane Algebraic Curves, Birkhauser Verlag,

Basel, 1986

5. K. Kendig, Elementary Algebraic Geometry, Springer-Verlag, New York 1977

7. Z. A. Melzak, Companion to Concrete Mathematics, John Wiley & Sons,

New York, 1973

9. J. Stillwell, Mathematics and Its History, Springer-Verlag, New York 1989

6. Z. A. Melzak, Invitation to Geometry, John Wiley & Sons, New York, 1983

3. D. Hilbert & H. Cohn-Vossen, Geometry and the Imagination, Chelsea

Publishing Company, New York, 1952

4. N. Koblitz, Introduction to Elliptic Curves and Modular Forms,

Springer-Verlag, New York 1984

10. M. Villarceau, "Théorème sur le tore." Nouv. Ann. Math. 7, 345-347, 1848.

2. M. Berger, Geometry I and Geometry II, Springer-Verlag, New York 1987

Documents

Hints of Tori