Thread by @computer_dream, elliptic curves thread Disclaimer: this thread is not meant to be technical [...]

elliptic curves thread

Disclaimer: this thread is not meant to be technical but rather a bunch of facts I find beautiful about elliptic curves. I hope you can find them beautiful as well.
If you want to learn more about this, read Silverman's Arithmetic of Elliptic Curves!

So let's get started: what in life is an elliptic curve? well, if you already did some search, you might have come across some familiar pictures and you can have a mental image already like this:

(wikipedia)

But, allow me to make this a tweet a clickbait for captivating your interest:

an elliptic curve can be any of these as well!

How can this be!? The goal of this thread is to make sense of these!

Let's take it easy and friendly and start from scratch:

all of you have a mental image of what a general curve is. You can construct a curve giving values to polynomials in two variables, as simple as that.

Nevertheless, elliptic curves are more restrictive.

for constructing an elliptic curve, we need to ask our curve some stuff:

- it must be smooth,
- it must be projective,
- its genus must be 1,
- we need to have a given point O in the curve, which will be the origin, and
- it's given by cubic polynomial equations.

every curve which satisfies these conditions is an elliptic curve!

Smoothness is required in order to avoid singularities. We don't want our curve to have nodes or cusps. (An equivalent thing is asking for its discriminant to be non-zero).

What's the genus thing? the genus of a surface is, intuitively, the number of holes of the surface. It's a topological invariant and it helps us to classify them. Now you can guess why I dropped the picture of the torus (one hole)... but it's more complex than that (pun intended)

The projective condition just means that elliptic curves live in the projective plane. This allows us, among other things, to have this O point which will be the point at infinity. There's a good reason for asking this, but first let me talk about the last condition.

I kinda lied at the beginning when I said that a curve is given by a polynomial in two variables.

If our curve lives in the projective plane we'll need to deal with homogeneous polynomials in three variables... but!

we can usually work with affine charts, and it's ok. Just don't forget that our curve is projective!

with all these things in mind we can claim without trouble that an elliptic curve is something of this form:

a x³ + b x²y + c xy² + d y³ + e x² + f xy + g y² + h x + i y + k = 0

this is just the general expression of a polynomial in two variables. But there are amazing news for us!!!

By some algebra magic (change of variables) we can simplify this expression A LOT!!

so if our coefficients don't live in a field of
characteristic 2 or 3, any elliptic curve can be written in the so called short Weierstrass form:

y²=x³+Ax+B (with the discriminant non-zero)

now, with all these (apparently random) restrictions we have created a cool thing:

now, we're able to construct an abelian group structure in our curve!!

We only need to come up with some way of adding points. I don't really want to go into details or this thread will be too long and boring, and pictures work better here. So this how you do it:

(I'm just happy with you knowing that you can verify this has an abelian group structure.)
We have constructed something amazing, not only because abelian groups are cool themselves, but because we can DO CRYPTOGRAPHY involving this addition.
This is related to the first picture.

Finite fields are the standard ambient to do cryptography. I like to think that all you need for doing cryptography is an abelian group and a hard problem.
So indeed, you can set different cryptosystems using elliptic curves over finite fields with this addition!

This is a bit out of topic but I really feel like saying it: classic elliptic curve cryptography is, sadly, broken by Shor's algorithm if quantum computers become a thing eventually.

The good news is that there's a lot of research in a post-quantum cryptosystem...

... which seems to be resistant to it, and it's also based in elliptic curves! It's just, the approach is very different: it's based in maps between elliptic curves (so called isogenies).

If you want to read more, search for Supersingular Isogeny Diffie-Hellman!

Now let's move on to the third picture, which is a... triangle?!

Are you telling me that an elliptic curve is not actually an ellipse but can be a... triangle !?!?

Well, surprisingly, yes!

This is related to an ancient problem involving Diophantine equations.

It's called the "Congruent Number Problem" (CNP). And it's not yet solved!

First, an integer is a congruent number if it's the area of a right triangle with rational sides. CNP is about finding an algorithm which determines if an integer is congruent or not.

Now you can make sense of the picture I posted:

that right triangle has rational (integer, in fact) sides and its area is 3/2*4=6, which means, 6 is a congruent number.

ok Marta so cool but WHERE ARE THE ELLIPTIC CURVES HERE !? well, they are here!! just hidden:

If we analyze this problem a bit, we need to find some rational numbers which satisfy Pythagoras theorem and which area is an integer, i.e.:

a²+b²=c² and ab/2=n,

where a,b,c are the rational sides of the triangle and n is the integer area.

Once again, by some algebra magic (multiplying by certain stuff and changing variables, basically) we can transform these equations in... this!! :

y² = x³-n²x

sounds familiar?

(Note that n is still the original area, and the number we want to know if it's congruent or not)

If you keep studying CNP you will eventually realize that a number is congruent iff the "associated" elliptic curve has positive rank. But isn't this a characterization? Can't we obtain an algorithm from this? Well, turns out the rank of elliptic curves is still a mystery for us.

Let me clarify this a bit: if you consider elliptic curves over Q, you can write them as a direct sum of the torsion part and the free part.

The exponent "r" here is the rank of the elliptic curve E.

I find this really crazy, but we know very good how the torsion part behaves. We have this powerful (Mazur's) theorem which blows my mind and we know
exactly how to parametrize each type of elliptic curves, depending on its torsion. And we even have algorithms for computing stuff

BUT WE DON'T KNOW ABOUT THE RANK OF ELLIPTIC CURVES!

We don't even know if it's bounded or not!!!

It's conjectured that the rank can be arbitrarily large. But the highest rank that it's been found is at least 28. It's very very rare to find curves of rank greater than 3, so!!

Nevertheless, there's this famous Birch and Swinnerton-Dyer conjecture which is a Millennium Problem and it's related to the rank of elliptic curves, and the last picture of the beginning of the thread, as well.

BSD conjecture states that the rank of an elliptic curve over a global field is exactly the order of vanishing of L(E,s) at s=1.

ok this is where things seem to get a bit difficult. What's that L(E,s) thing.

This is probably one of the most surprising things in mathematics: the connection of L-functions with elliptic curves.

An L-function is a meromorphic function
(it has poles), which are "born" from Dirichlet series.

A Dirichlet series is an infinite sum (on n) of the term a_n/n^s where s is a complex number and a_n a complex
sequence.

Most of you know at least one L-function: Riemann zeta function!

This theory is pretty technical and it's the one I know less about so let's just be happy with understanding the statement of BSD and that there's an important relation between elliptic curves and complex analysis.

(Also, proving BSD would imply solving CNP)

I'd like to make a last remark on how powerful elliptic curves are: Fermat's Last Theorem was proved thanks to this connection between elliptic curves (number theory) and modular forms (complex analysis).

The key of this connection is a beautiful theorem: the modularity theorem.

It's also historically known as Taniyama-Shimura conjecture, and states that EVERY elliptic curve over Q has an associated modular form.

This was fully proven in 2001, and Fermat's Last Theorem was proven in 1995. But this is not totally "true", let me explain:

when FLT was proved, what was REALLY proven was a particular case of the modularity theorem, say, for semistable elliptic curves; and FLT's equation just happened to fell into that family of elliptic curves.

So what I find cool here is not actually that FLT was proved, but the fact that elliptic curves over Q and modular forms (the 4th picture I posted at the beginning) are in 1-to-1 correspondence.

I'm going to return to the torus picture for one second and give a cool fact about elliptic curves over C:
they are in 1-to-1 correspondence to lattices! This means, you can either work with lattices and obtain information of the elliptic curve associated to it and viceversa.

As for ending, I'd like to remark that elliptic curves arise in the most unexpected places and I think they do it in a really charming way.

I hope you enjoyed this thread just a bit. Feel free to correct any mistake or adding anything you want!

Thanks for reading

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