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Slicing Bagels: Plane Sections of Real and Complex Tori

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Slicing Bagels: Plane Sections of Real and Complex Tori

David Sklar

San Francisco State University

Bruce Cohen

Lowell High School, SFUSD

http://www.cgl.ucsf.edu/home/bic

Asilomar - December 2004

Part I - Slicing a Real Circular Torus

Equations for the torus in R3

The Spiric Sections of Perseus

Ovals of Cassini and The Lemniscate of Bernoulli

Other Slices

The Villarceau Circles

A Characterization of the torus

Some graphs of

Algebraic closure, C2, R4, and the graph of

The graphs of

Part II - Slicing a Complex Torus

Elliptic curves and number theory

Hints of toric sections

Two closures: Algebraic and Geometric

Geometric closure, Projective spaces

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

Bibliography

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 roots and coefficients in F. The fields of most interest are the rational numbers, finite fields, the real numbers, and the complex numbers.

where a, b and c are distinct integers such that with integer exponent n > 2, might lead to a contradiction.

Elliptic curves and number theory

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

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

Less than 10 years later Andrew Wiles proved a form of the Taniyama conjecture sufficient to prove Fermat’s Last Theorem.

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 graph of the equation

Elliptic curves and number theory

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 partial solution 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

Graphs of

: Hints of Toric Sections

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.

The real projective line P1(R) is

the set

It is topologically equivalent to the open interval (-1, 1) by the map

Geometric Closure: an Introduction to Projective Geometry

Part I – Real Projective Geometry

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

The real (affine) line R is the

ordinary real number line

It is topologically equivalent to a closed interval with the endpoints identified

and topologically equivalent to a punctured circle by stereographic projection

and topologically equivalent to a circle by stereographic projection

The real projective plane P2(R) is the set . It is R2 together with a “line at infinity”, . Every line in R2 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).

It is topologically equivalent to the open unit disk by the map

( )

Geometric Closure: an Introduction to Projective Geometry

Part I – Real Projective Geometry

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

The real (affine) plane R2is

the ordinary x, y -plane

It is topologically equivalent to a closed disk with antipodal points on the boundary circle identified.

Two distinct lines intersect at one and only one point.

Parabola

Ellipse

Hyperbola

A Projective View of the Conics

Ellipse

Parabola

Hyperbola

A Projective View of the Conics

Graphs of

: Hints of Toric Sections

Graph of with x and y complex

Algebraic closure

Graph of with x and y complex

Algebraic closure

Some comments on why the graph of the system

is a surface.

Graph of with x and y complex

Algebraic closure

Graph of with x and y complex

Graph of with x and y complex

The complex projective line P1(C)is the set the complex plane with one number adjoined.

It is topologically a sphere by stereographic projection with the north pole corresponding to . It is often called the Riemann Sphere.

Geometric Closure: an Introduction to Projective Geometry

Part II – Complex Projective Geometry

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

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

The complex (affine) line C is the ordinary complex plane where (x, y) corresponds to the number z = x + iy.

It is topologically a punctured sphere by stereographic projection

Complex projective 2-space P2(C) is the set . It is C2 together with a complex “line at infinity”, . Every line in R2 intersects , parallel lines meet at the same point on , and nonparallel 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 Sphere at infinity.

The complex (affine) “plane” C2or better complex 2-space is a lot like R4. A line in C2 is the graph of an equation of the form , where a, b and c are complex constants and x and y are complex variables. (Note: not every plane in R4 corresponds to a complex line)

Geometric Closure: an Introduction to Projective Geometry

Part II – Complex Projective Geometry

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

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

Two distinct lines intersect at one and only one point.

Graph of with x and y complex

Graph of with x and y complex

A Generalization: the Graph of

A Generalization: the Graph of

Bibliography

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

Basel, 1986

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

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

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

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

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

New York, 1973

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

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

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