Chapter 3: Elliptic Curves

y^2 = x^3 - N^2 x

This equation is an example of an elliptic curve!

Elliptic Curves

Curves of the form y^2 = x^3 + px + q for some p and q

New Points from Old

The Group Law

List of rational points found by adding P_1 to itself

Will this list be infinite?

There's a million-dollar unsolved problem about this!

The Birch and Swinnerton-Dyer (BSD) Conjecture

(Prize for proof: $1,000,000. Unsolved!)

The BSD Conjecture

(Prize for proof: $1,000,000. Unsolved!)

\text{Algebraic Rank} of elliptic curve: Measures "how infinite" the number of rational points is.

The BSD Conjecture

(Prize for proof: $1,000,000. Unsolved!)

\text{Algebraic Rank} of elliptic curve: Measures "how infinite" the number of rational points is.
\text{Analytic Rank} of elliptic curve: Measures # of solutions using different types of numbers

The BSD Conjecture

(Prize for proof: $1,000,000. Unsolved!)

For all elliptic curves:
\text{Algebraic Rank} = \text{Analytic Rank}

Tunnell's Theorem (1983)

Solves the congruent number problem using deep properties of y^2 = x^3 - N^2 x!
N is a congruent number if...

Tunnell's Theorem (1983)

If N is odd:

Does x^2+2y^2+8z^2=N have exactly twice as many integer solutions as x^2+2y^2+32z^2=N?

If N is even:

Does x^2 + 4y^2 + 8z^2 = \frac{N}{2} have exactly twice as many integer solutions as x^2 + 4y^2 + 32z^2 = \frac{N}{2}?

If so, N is a congruent number! Otherwise, N isn't!

But why does that mean N is a congruent number?

If Tunnel's criterion is false, y^2 = x^3 - N^2 x has no rational solutions.

No rational solutions means no triangles with area N.

So the congruent number problem is solved!

Except...
Tunnell's proof assumes the BSD conjecture is true. The million-dollar, unsolved conjecture.

The Case of the Impossible Triangles

August 2021
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The equation y^2 = x^3 - N^2 x, which we got from the three successive squares problem, is a type of equation called an elliptic curve. And elliptic curves, to mathematicians, are very interesting.
(Elliptic curves are different from ellipses. They were first named "elliptic" because similar equations appear when trying to find the perimeter of an ellipse.)

An elliptic curve is the curve drawn by all possible pairs of numbers (x,y) which solve the equation y^2 = x^3 + px + q, defined by some choice of numbers p and q. Depending on the exact values of p and q, the graph of an elliptic curve might look like one smooth curve, or two distinct parts.

In our case, we're interested in the equation y^2 = x^3 - N^2 x, which gives us a different curve for each value of N. When N=6, for example, we get the curve y^2 = x^3 - 36 x, which I've graphed here.

Here's the key idea: every triangle with area 6 corresponds to a point (x,y) on this curve.

Here's how to convert triangles into points on this curve. First, let's start with this rational triangle with area 6.

Comparing this triangle to the one whose sides we found last chapter, with a bit of work we can compute the areas of the three squares we saw in the previous chapter, r^2, s^2 and t^2.

During our algebra last chapter, we decided to take s^2 and rename it to x.

And finally, we can use that x to plot a point on this elliptic curve.
As long as the triangle we started with has area 6, this process is guaranteed to result in a point on this elliptic curve.

We can work backwards, too: if we find an x on the elliptic curve, it's possible to do some algebra to compute r^2, s^2, t^2, which can be turned into a triangle with area N.

But more importantly: this means if we can find a point (x,y) on the elliptic curve where both x and y are rational, we can construct a rational-sided triangle with area 6 - proving 6 is a congruent number!

Therefore, we've turned the problem of "is N a congruent number?" into "Does the equation y^2 = x^3 - N^2 x have any pairs of rational numbers (x,y) which solve the equation?"
That's great for us - mathematicians have years of experience finding solutions to equations! Now we can use their techniques.

Another reason elliptic curves are useful is they have a special property: you can generate new solutions (x,y) from old ones!

Let's say we found two rational points P_1 and P_2 on this curve. Elliptic curves allow you to define a way to "add" two rational points on the curve, and get another, different rational point on the curve!
For us, this means once we find one rational triangle with area N, we can use the elliptic curve to generate more rational triangles with area N!

Here's the recipe for "adding" two points P_1 and P_2 on an elliptic curve:

Step 1: First, draw a line through P_1 and P_2. Because of how elliptic curves are defined, this line will always* intersect the curve in a third point.

Here's the recipe for "adding" two points P_1 and P_2 on an elliptic curve, continued:

Step 2: Reflect the point from Step 1 vertically over the x-axis.
Step 3: You're done! The result of this "addition" is usually written P_1 + P_2, where the symbol + means elliptic curve addition, not your everyday addition.

What if P_1 and P_2 are the same point? No problem! Step 1 is slightly different when adding P_1 + P_1: the line you draw is the tangent line to the curve. But everything still works out - we get a new point on the curve.

Elliptic curve addition obeys many of the same rules as normal addition: Add two numbers points, and you'll get another number point.
Interestingly, just like normal addition, if you add two rational numbers points, you're guaranteed to get another rational number point.
That's good for us - remember, each rational point on this elliptic curve corresponds to a rational-sided triangle with area N!

In other words, by using elliptic curve addition, once we find one rational triangle with area 6, we can generate new rational triangles with area 6!
Here's how: first, find the point P_1 on an elliptic curve which corresponds to your triangle. Then, use elliptic curve addition to compute P_1 + P_1. Then, turn that new point back into a triangle, and we'll get a new rational triangle with area 6!

Let's find P_1 + P_1 by following the same elliptic curve addition recipe. The math guarantees P_1 + P_1 lies on the elliptic curve, but we might need to zoom out a bit to see it.

Those are some big numbers. But what matters is that P_1 + P_1's coordinates are big rational numbers, meaning we can turn this point into a new valid rational triangle with area 6!
It would have taken a long time to find this skinny triangle by brute force. Elliptic curve addition is powerful.

We can keep going! We can compute P_1 + P_1 + P_1 by adding P_1 to (P_1 + P_1). Following the recipe, we draw the line between (P_1 + P_1) and P_1, flip the point over the y-axis, and find a third point. This point gives us yet another rational triangle with area 6!
This means, using elliptic curves, we can turn one triangle with area 6 into as many as we want!

By the way, modern encryption is built around adding points on elliptic curves!
Remember how I said your computer uses elliptic curves to encrypt its connection to the internet? In a nutshell, elliptic curve encryption weaponizes the fact that adding points on elliptic curves is really hard to undo. If someone adds a point P to itself repeatedly and hands you P + P + \ldots + P, it's almost impossible to find P, the point they started with. Then, elliptic curve encryption turns "undoing addition is hard" into "decrypting a message is hard"!

Let's make a list of the points we got by adding P_1 to itself repeatedly. So far, our list has 3 points on it: P_1, (P_1 + P_1), and (P_1 + P_1 + P_1). As we've seen, each point can be turned into a new triangle with area 6.

What would happen if we kept going? If we keep adding P_1 to itself forever, would we visit a new point each time and make an infinitely long list of rational points? Or would we eventually loop back around to the top of the list and start creating points we've already seen?
It's an interesting question. If the answer is "yes" and we generate infinitely many rational points, it'll prove there are infinitely many triangles with area N.

As it turns out, "Are there infinitely many rational points?" is a very hard question. In fact, there's an unsolved problem about elliptic curves, called the BSD Conjecture, which is related to this exact question - and there's a million-dollar prize for the first person to prove it.

The Birch and Swinnerton-Dyer Conjecture, or BSD Conjecture, is a deep unsolved conjecture about elliptic curves. If it's true (which seems very likely), it would create an incredibly deep connection between abstract algebra, number theory, complex analysis, and other branches of math. The details are very technical, but I'll try to give a quick summary.

A quick summary of the BSD conjecture, part 1:
One approach to studying elliptic curves is the one we took: studying rational points on the elliptic curve, and using elliptic curve addition to generate more points. Then, mathematicians can study the structure of elliptic curve addition using a field of math called group theory.

A quick summary of the BSD conjecture, part 1:
Mathematicians answer the question "Are there infinitely many rational points?" using a number called the \text{algebraic rank} of an elliptic curve, which measures whether or not the number of rational points is infinite.

A quick summary of the BSD conjecture, part 2:
But there's also a second popular approach for studying elliptic curves we haven't mentioned. It uses techniques from completely different areas of math, ones which are much harder to turn into nice beautiful pictures. In a nutshell, it involves using mathematical tools from complex analysis and number theory, solving the equation "mod p", and turning elliptic curves into functions with fancy names such as "L-functions".
But what's important is that using these techniques from complex analysis, we can define an important number called the \text{analytic rank} of an elliptic curve.

The BSD conjecture says that these two important numbers associated with an elliptic curve - \text{one} which comes from the elliptic curve addition we've been working with this entire time, and \text{one} which comes from completely different mathematical fields - are equal. These completely different mathematical techniques attack elliptic curves from two different angles, and somehow end up measuring the same thing. And that, I think, is pretty beautiful.

Finally, let's return to the very first question we started with, the congruent number problem. Given an integer N, like 30, can we find a triangle with rational sides and area N? Let me tell you the solution!

In 1983, a mathematician named J. B. Tunnell used deep properties of the elliptic curve y^2 = x^3 - N^2 x to solve the congruent number problem. It's not the neatest answer, so don't be scared, but it still is a complete answer.

Here's Tunnell's answer: To figure out if N is a congruent number, count the number of integer solutions to these two equations and compare them.

What a complicated criterion. What do solutions of these equations have to do with whether or not N is a congruent number?

Here's why it works: Tunnell showed that if N fails his criterion, it means the elliptic curve y^2 = x^3 - N^2 x has no rational solutions. And as we've seen, no rational solutions on the elliptic curve means no rational triangles with area N.
By the way, we started with a geometry question about triangles, and now we're answering it using number theory. Isn't that interesting? Mathematicians love elliptic curves because they connect to so many different areas of math.

Is that where the story of congruent numbers ends? Almost. There's one tiny piece left in the puzzle. One tiny, million-dollar piece.

That's right - it's our friend the BSD conjecture. Tunnell's theorem solves the congruent number problem... using techniques which are only valid if the BSD conjecture is true. And nobody has proved the BSD conjecture yet.
There's lots of evidence suggesting the BSD conjecture is true. But we don't know that for sure. Until someone proves the BSD conjecture and collects that million-dollar prize, there's always the possibility Tunnell's proof will come crashing down.

What a journey. We started with such a simple question: are there any triangles with rational sides and area 30? It took us to the three successive squares problem, and then to elliptic curves. There, we found out how a weird way to "add" points on elliptic curves lets us create new triangles with area 30 out of old ones, and discovered at the heart of our problem lay a million-dollar unsolved conjecture called the BSD conjecture.

Elliptic curves still have many secrets yet to be decoded, and I hope this gave you a taste of why mathematicians care about them so much.

Thanks so much for reading!

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