What is so special about elliptic curves?

[pierce15]

The definition of an elliptic curve is an equation in the form:

$$y^2 = x^3 + ax + b $$

Moreover, the curve must be non-singular, i.e. its graph has no cusps or self-intersections. This seems like an awfully specific definition for a family of functions. Can someone shed some light on why they are some important or interesting?

 

[HallsofIvy]

There are a specific formulas for solving any first or second degree equations. Fourth degree equations, on the other hand, can be completely intractible. Third degree equations are hard enough to be "interesting" while still solvable.

 

[pierce15]

But what is so "interesting" about them?

 

[HallsofIvy]

They are hard!

 

[jackmell]

The definition of an elliptic curve is an equation in the form:

$$y^2 = x^3 + ax + b $$

Moreover, the curve must be non-singular, i.e. its graph has no cusps or self-intersections. This seems like an awfully specific definition for a family of functions. Can someone shed some light on why they are some important or interesting?

Some important or so important? You can check out Wikipedia:

http://en.wikipedia.org/wiki/Elliptic_curves

Elliptic curves are especially important in number theory, and constitute a major area of current research; for example, they were used in the proof, by Andrew Wiles (assisted by Richard Taylor), of Fermat's Last Theorem. They also find applications in elliptic curve cryptography (ECC) and integer factorization.

 

[pierce15]

Good catch on that typo- I meant so. Does anyone know specifically how they relate to number theory and the proof of Fermat's Last Theorem?

 

[jackmell]

Good catch on that typo- I meant so. Does anyone know specifically how they relate to number theory and the proof of Fermat's Last Theorem?

Yeah really pierce. Me too. The problem I fear is that's it's not going to be easy to understand if we're not already familiar with the subject. How about if we ask, "specifically how do they relate to the simplest application to Number Theory in a manner that's not too hard to comprehend to the novice?"

For example, is it even possible for someone to explain how elliptic curves are used in cryptography in such a way that the novice would understand it? Really in my opinion that is a defining characteristic of a gifted teacher. Not that I'm trying to seduce anyone into trying. Just saying that's all.

 

[AlephZero]

For example, is it even possible for someone to explain how elliptic curves are used in cryptography in such a way that the novice would understand it?

Hmm ... to quote one google reference, "Elliptic curves are interesting because they are the simplest algebraic structure that is not yet completely understood".

The interesting stuff starts when you consider only the points on the curve where x and y are rational numbers (i.e. fractions). The set of those points has some interesting properties and forms a mathematical structure called a group (and groups are one of the first things you would learn about in a course on abstract algebra). For example, if you know two points on the curve with rational coordinates, you can always construct another point from them. But if a and b are huge numbers, finding two points with rational coordinates to get you started might be hard (which is where cryptography comes in...)

There are other (similar) ways to get a mathematical group of points on an elliptic curve - and the "not completely understood" part starts if you want to choose an arbitrary curve and predict exactly what structure the group will have. Nobody knows, yet...

 

[Therodre]

Hi,
I'm not sure the best way to get why Elliptic Curves are important, is to understand the relationship between then and Fermat's Theroem. To be honnest, the relationship seems rather fortuitous, and to this day, we do not have a very deep understanding of the nature of this link, if it is something systematic or rather coincidental.
Essentially it is a very clever trick that enabled to link Fermat's last theorem (FLT), to the general theory of elliptic curves.

In my opinion, it's more enlightenning to understand the general framework in which elliptic curves are important. I don't know exactly what your mathematical knowledge is, so i'll try to remain as elementary as possible.

One of the most interresting questions of arithmetic, is the study of so called diophantine equations. That is the search of integral (or rationnal) solutions of equations, which are given by polynomials.
For instance what are the integral solutions of x²-dy²=1, when d is a square free integer (this is called Pell-Fermat's equation and it is completely understood). This is certainly interresting in itself, and in fact many questions in number theory ca be interpreted as diophantine equations e.g the fact that √2 is irrationnal is equivalent to the non existence of non trivial integral solutions to x²-2y²=0.

When you look at a diophatine equation, such as f(x,y)=0 where f is a certain polynomial with integral coefficient. You can certainly look at all the solutions for x and y complex numbers. This defines in the plane what is called a complex algebraic curve. This is like a curve in the plane, but you have to stretch your vision a little bit because the axis are complex instead of real.
It is called a complex curve, but really it is a real surface. This does make sense, locally you need one complex number to describe the curve, but you need two real numbers (namely the imaginary and real parts of the complex number) to describe the same object.

We might wonder what is the shape of the surface defined in this way. And it turns out that we have a nice classification of such surfaces, it can be either a sphere, or a torus (a donut), or two tori glued together (like a pretzel), or 3 tori etc... The number of tori you need to glue together is called the genus of the surface, and you can characterise it nicely in function of the degree of the polynomial f(x,y) you took.

The general problem of arithmetic geometry, is the following "Is the general geometry of this surface related to the artihmetic of that surface, that is the integral solutions of f(x,y)=0?"
And the answer is amazingly, yes!

Now, to investigate the situation, we may start to look at the simpler surfaces. The sphere, first, i.e the genus 0. Well, everything is understood about its geometry and its arithmetic, in fact, it was understood even before than we thought about the problem in those terms. It corresponds to the study of conics.

The next simplest case, is that case of the torus, genus=1. We call such curves, elliptic curves. And already the situation is very rich, and it turns out that we can always write down the equation that defines an elliptic curve (at least in charracteristic different than 2 and 3), it will be the equation you have written down in your first post.

This is just a small part of the picture. By no means, it is the sole reason why mathematicians are interrested in elliptic curves. They're a very versatile object, that appears naturally in all algebraic geometry.
They are the simplest example of what is called an abelian variety (this is the fact that on an elliptic curve you can "add up" points, as strange as it may sound), and they're curves, which are the simplest objets in algebraic geometry (in fact this is the only kind of abelian variety that is a curve). Abelian varieties and curves are the nicest objects in algebraic geometry, and the ones that we understand the most. So they occupy a central place in algebraic geometry too.

 

[pierce15]

Thank you very much for the elaborate explanation, Therodre. I'm just wondering now why algebraic geometry is studied in the first place- could someone explain?

 

[Therodre]

Well, that certainly is a pickle.

Algebraic geometry is one of the broadest and most important field in mathematics. It has deep connections with almost every area of mathematics, and even theoretical physics (through complex geometry). Number theory, complex geometry, algebraic topology... are all fields that interract very positively with algebraic geometry. So that is already a good reason.
Personally I've been drawn to algebraic geometry through number theory.

And, perhaps an even better reason, is that it is indeed a very beautiful area of mathematics.

But i guess that's not the kind of answers you're looking for.

There are of course very important open questions in the field, but they will be a little difficult to grasp by a non expert. So let's try a more historical approach.

People wanted to study algebraic curves and surfaces in the plane or space. One the first questions to be tackled was a generalisation of d'Alembert-Gauss theorem, that every non constant polynomial will have a root over the complex numbers. Stated in geometric terms it states that a curve defined by the equation y=f(x) where f is a polynomial will generically intersect a line deg(f) times.
So people wanted to generalise this result to arbitrary curves and surfaces.
This led to what is now called intersection theory. How can we properly define the number of intersection of two curves? Can we do it in such a way that two curves will intersect in deg(f)deg(g) points (where the curves are defined by f and g)?
For instance, the intersection of a line and circle tangent to it should count as 2 even though they intersect at a single point, since that if you slightly move the line, you will get 2 distinct points.
This is a very rich and active subfield, and is strongly connected to things like the topology of algebraic varieties, and such things coming from algebraic topology.

There was also enumerative geometry (in fact it is now a part of intersection theory), people wanted to count different type of objects. For instance, on a (non trivial) cubic surface there are exactly 27 lines. Those results have (for long) seem of no particular interrest to me, until i actually looked at the proof, and it is mesmerizing! The amazing point is not that there is 27 lines on a cubic surface, but that we can actually compute such a number (and pretty easlity too).

In the same spirit, people wanted to classify objects. How many genuinely different curves could we build up etc?

You should read this post on MSE.
http://math.stackexchange.com/questions/255063/why-study-algebraic-geometry
Especially the answer from Javier Alavarez (the first answer) which is very complete and i guess accessible to a non expert.