112
P. B. Vinod Kumar
d((x 1 , y 1 ), (x 2 , y 2 )) = |x 1 − x 2 | + |y 1 − y 2 |,
[d((0, 0), (1, 0))]
2
+ [d((0, 0), (0, 1))]
2
= [d((1, 0), (0, 1))]
2
.
Geometric shapes do not remain as it is if we change the topology of R
n .
We will discuss the geometry of elliptic curves in the next section .
2.3 Elliptic Curves
Elliptic curve is simply the locus of points in the x − y plane that satisfies an algebraic
equation of the form Y
2
= (X − x
n
)(Y − y
n
) (with some additional minor technical
conditions). This is deliberately vague as to what sort of values x and y represent.
In the most elementary case, they are real numbers, in which case the elliptic curve
is easily graphed in the usual Cartesian plane. But the theory is much richer when
x and y may be any complex numbers (∈ C). And for arithmetic purposes, x and y
may lie in some other field, such as the rational numbers Q or a finite field F.
So an elliptic curve is an object that is easily definable with simple high school
algebra. Its amazing fruitfulness as an object of investigation may well depend on this
simplicity, which makes possible the study of a number of much more sophisticated
mathematical objects that can be defined in terms of elliptic curves.
It is very natural to work with curves in the complex numbers, since C is the
algebraic closure of the real numbers. That is, it is the smallest algebraically closed
field that contains the roots of all possible polynomials with coefficients in R. Being
algebraically closed means that C contains the roots of all polynomials with coefficients in C itself. It’s natural to work with a curve in an algebraically closed field,
since then the curve is as “full” as possible.
The case of elliptic curves in the complex numbers is especially interesting, not
only because of the algebraic completeness of C, but also because of the rich analytic
theory that exists for complex functions. In particular, the equation of an elliptic curve
defines y as an “algebraic function” of x. For every algebraic function, it is possible
to construct a specific surface such that the function is “single-valued” on the surface
as a domain of definition. It turns out that an elliptic curve, defined as a locus of
points, is also the Riemann surface associated with the algebraic function defined by
the equation.
So an elliptic curve is a Riemann surface. In fact, it is of a special type: a compact
Riemann surface of genus 1. And not only that, but the converse is also true: every
compact Riemann surface of genus 1 is an elliptic curve. In other words, elliptic
curves over the complex numbers represent exactly the “simplest” sorts of compact
Riemann surfaces with non-zero genus. Topologically, the genus counts the number
of “holes” in a surface. A surface with one hole is a torus.
This topological equivalence of an elliptic curve with a torus is actually given
by an explicit mapping involving the Weierstrass ρ-function and its first derivative.
P. B. Vinod Kumar
d((x 1 , y 1 ), (x 2 , y 2 )) = |x 1 − x 2 | + |y 1 − y 2 |,
[d((0, 0), (1, 0))]
2
+ [d((0, 0), (0, 1))]
2
= [d((1, 0), (0, 1))]
2
.
Geometric shapes do not remain as it is if we change the topology of R
n .
We will discuss the geometry of elliptic curves in the next section .
2.3 Elliptic Curves
Elliptic curve is simply the locus of points in the x − y plane that satisfies an algebraic
equation of the form Y
2
= (X − x
n
)(Y − y
n
) (with some additional minor technical
conditions). This is deliberately vague as to what sort of values x and y represent.
In the most elementary case, they are real numbers, in which case the elliptic curve
is easily graphed in the usual Cartesian plane. But the theory is much richer when
x and y may be any complex numbers (∈ C). And for arithmetic purposes, x and y
may lie in some other field, such as the rational numbers Q or a finite field F.
So an elliptic curve is an object that is easily definable with simple high school
algebra. Its amazing fruitfulness as an object of investigation may well depend on this
simplicity, which makes possible the study of a number of much more sophisticated
mathematical objects that can be defined in terms of elliptic curves.
It is very natural to work with curves in the complex numbers, since C is the
algebraic closure of the real numbers. That is, it is the smallest algebraically closed
field that contains the roots of all possible polynomials with coefficients in R. Being
algebraically closed means that C contains the roots of all polynomials with coefficients in C itself. It’s natural to work with a curve in an algebraically closed field,
since then the curve is as “full” as possible.
The case of elliptic curves in the complex numbers is especially interesting, not
only because of the algebraic completeness of C, but also because of the rich analytic
theory that exists for complex functions. In particular, the equation of an elliptic curve
defines y as an “algebraic function” of x. For every algebraic function, it is possible
to construct a specific surface such that the function is “single-valued” on the surface
as a domain of definition. It turns out that an elliptic curve, defined as a locus of
points, is also the Riemann surface associated with the algebraic function defined by
the equation.
So an elliptic curve is a Riemann surface. In fact, it is of a special type: a compact
Riemann surface of genus 1. And not only that, but the converse is also true: every
compact Riemann surface of genus 1 is an elliptic curve. In other words, elliptic
curves over the complex numbers represent exactly the “simplest” sorts of compact
Riemann surfaces with non-zero genus. Topologically, the genus counts the number
of “holes” in a surface. A surface with one hole is a torus.
This topological equivalence of an elliptic curve with a torus is actually given
by an explicit mapping involving the Weierstrass ρ-function and its first derivative.