The Summer 2017 Mathematics A To Z: L-function


I’m brought back to elliptic curves today thanks to another request from Gaurish, of the For The Love Of Mathematics blog. Interested in how that’s going to work out? Me too.

Summer 2017 Mathematics A to Z, featuring a coati (it's kind of the Latin American raccoon) looking over alphabet blocks, with a lot of equations in the background.
Art courtesy of Thomas K Dye, creator of the web comic Newshounds. He has a Patreon for those able to support his work. He’s also open for commissions, starting from US$10.

So stop me if you’ve heard this one before. We’re going to make something interesting. You bring to it a complex-valued number. Anything you like. Let me call it ‘s’ for the sake of convenience. I know, it’s weird not to call it ‘z’, but that’s how this field of mathematics developed. I’m going to make a series built on this. A series is the sum of all the terms in a sequence. I know, it seems weird for a ‘series’ to be a single number, but that’s how that field of mathematics developed. The underlying sequence? I’ll make it in three steps. First, I start with all the counting numbers: 1, 2, 3, 4, 5, and so on. Second, I take each one of those terms and raise them to the power of your ‘s’. Third, I take the reciprocal of each of them. That’s the sequence. And when we add —

Yes, that’s right, it’s the Riemann-Zeta Function. The one behind the Riemann Hypothesis. That’s the mathematical conjecture that everybody loves to cite as the biggest unsolved problem in mathematics now that we know someone did something about Fermat’s Last Theorem. The conjecture is about what the zeroes of this function are. What values of ‘s’ make this sum equal to zero? Some boring ones. Zero, negative two, negative four, negative six, and so on. It has a lot of non-boring zeroes. All the ones we know of have an ‘s’ with a real part of ½. So far we know of at least 36 billion values of ‘s’ that make this add up to zero. They’re all ½ plus some imaginary number. We conjecture that this isn’t coincidence and all the non-boring zeroes are like that. We might be wrong. But it’s the way I would bet.

Anyone who’d be reading this far into a pop mathematics blog knows something of why the Riemann Hypothesis is interesting. It carries implications about prime numbers. It tells us things about a host of other theorems that are nice to have. Also they know it’s hard to prove. Really, really hard.

Ancient mathematical lore tells us there are a couple ways to solve a really, really hard problem. One is to narrow its focus. Try to find as simple a case of it as you can solve. Maybe a second simple case you can solve. Maybe a third. This could show you how, roughly, to solve the general problem. Not always. Individual cases of Fermat’s Last Theorem are easy enough to solve. You can show that a^3 + b^3 = c^3 doesn’t have any non-boring answers where a, b, and c are all positive whole numbers. Same with a^5 + b^5 = c^5 , though it takes longer. That doesn’t help you with the general a^n + b^n = c^n .

There’s another approach. It sounds like the sort of crazy thing Captain Kirk would get away with. It’s to generalize, to make a bigger, even more abstract problem. Sometimes that makes it easier.

For the Riemann-Zeta Function there’s one compelling generalization. It fits into that sequence I described making. After taking the reciprocals of integers-raised-to-the-s-power, multiply each by some number. Which number? Well, that depends on what you like. It could be the same number every time, if you like. That’s boring, though. That’s just the Riemann-Zeta Function times your number. It’s more interesting if what number you multiply by depends on which integer you started with. (Do not let it depend on ‘s’; that’s more complicated than you want.) When you do that? Then you’ve created an L-Function.

Specifically, you’ve created a Dirichlet L-Function. Dirichlet here is Peter Gustav Lejeune Dirichlet, a 19th century German mathematician who got his name on like everything. He did major work on partial differential equations, on Fourier series, on topology, in algebra, and on number theory, which is what we’d call these L-functions. There are other L-Functions, with identifying names such as Artin and Hecke and Euler, which get more directly into group theory. They look much like the Dirichlet L-Function. In building the sequence I described in the top paragraph, they do something else for the second step.

The L-Function is going to look like this:

L(s) = \sum_{n \ge 1}^{\infty} a_n \cdot \frac{1}{n^s}

The sigma there means to evaluate the thing that comes after it for each value of ‘n’ starting at 1 and increasing, by 1, up to … well, something infinitely large. The a_n are the numbers you’ve picked. They’re some value that depend on the index ‘n’, but don’t depend on the power ‘s’. This may look funny but it’s a standard way of writing the terms in a sequence.

An L-Function has to meet some particular criteria that I’m not going to worry about here. Look them up before you get too far into your research. These criteria give us ways to classify different L-Functions, though. We can describe them by degree, much as we describe polynomials. We can describe them by signature, part of those criteria I’m not getting into. We can describe them by properties of the extra numbers, the ones in that fourth step that you multiply the reciprocals by. And so on. LMFDB, an encyclopedia of L-Functions, lists eight or nine properties usable for a taxonomy of these things. (The ambiguity is in what things you consider to depend on what other things.)

What makes this interesting? For one, everything that makes the Riemann Hypothesis interesting. The Riemann-Zeta Function is a slice of the L-Functions. But there’s more. They merge into elliptic curves. Every elliptic curve corresponds to some L-Function. We can use the elliptic curve or the L-Function to prove what we wish to show. Elliptic curves are subject to group theory; so, we can bring group theory into these series.

And then it gets deeper. It always does. Go back to that formula for the L-Function like I put in mathematical symbols. I’m going to define a new function. It’s going to look a lot like a polynomial. Well, that L(s) already looked a lot like a polynomial, but this is going to look even more like one.

Pick a number τ. It’s complex-valued. Any number. All that I care is that its imaginary part be positive. In the trade we say that’s “in the upper half-plane”, because we often draw complex-valued numbers as points on a plane. The real part serves as the horizontal and the imaginary part serves as the vertical axis.

Now go back to your L-Function. Remember those a_n numbers you picked? Good. I’m going to define a new function based on them. It looks like this:

f(\tau) = \sum_{n \ge 1}^{\infty} a_n \left(  e^{2 \pi \imath \tau}\right)^n

You see what I mean about looking like a polynomial? If τ is a complex-valued number, then e^{2 \pi \imath \tau} is just another complex-valued number. If we gave that a new name like ‘z’, this function would look like the sum of constants times z raised to positive powers. We’d never know it was any kind of weird polynomial.

Anyway. This new function ‘f(τ)’ has some properties. It might be something called a weight-2 Hecke eigenform, a thing I am not going to explain without charging someone by the hour. But see the logic here: every elliptic curve matches with some kind of L-Function. Each L-Function matches with some ‘f(τ)’ kind of function. Those functions might or might not be these weight-2 Hecke eigenforms.

So here’s the thing. There was a big hypothesis formed in the 1950s that every rational elliptic curve matches to one of these ‘f(τ)’ functions that’s one of these eigenforms. It’s true. It took decades to prove. You may have heard of it, as the Taniyama-Shimura Conjecture. In the 1990s Wiles and Taylor proved this was true for a lot of elliptic curves, which is what proved Fermat’s Last Theorem after all that time. The rest of it was proved around 2000.

As I said, sometimes you have to make your problem bigger and harder to get something interesting out of it.

I mentioned this above. LMFDB is a fascinating site worth looking at. It’s got a lot of L-Function and Riemann-Zeta function-related materials.

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The Summer 2017 Mathematics A To Z: Height Function (elliptic curves)


I am one letter closer to the end of Gaurish’s main block of requests. They’re all good ones, mind you. This gets me back into elliptic curves and Diophantine equations. I might be writing about the wrong thing.

Summer 2017 Mathematics A to Z, featuring a coati (it's kind of the Latin American raccoon) looking over alphabet blocks, with a lot of equations in the background.
Art courtesy of Thomas K Dye, creator of the web comic Newshounds. He has a Patreon for those able to support his work. He’s also open for commissions, starting from US$10.

Height Function.

My love’s father has a habit of asking us to rate our hobbies. This turned into a new running joke over a family vacation this summer. It’s a simple joke: I shuffled the comparables. “Which is better, Bon Jovi or a roller coaster?” It’s still a good question.

But as genial yet nasty as the spoof is, my love’s father asks natural questions. We always want to compare things. When we form a mathematical construct we look for ways to measure it. There’s typically something. We’ll put one together. We call this a height function.

We start with an elliptic curve. The coordinates of the points on this curve satisfy some equation. Well, there are many equations they satisfy. We pick one representation for convenience. The convenient thing is to have an easy-to-calculate height. We’ll write the equation for the curve as

y^2 = x^3 + Ax + B

Here both ‘A’ and ‘B’ are some integers. This form might be unique, depending on whether a slightly fussy condition on prime numbers hold. (Specifically, if ‘p’ is a prime number and ‘p4‘ divides into ‘A’, then ‘p6‘ must not divide into ‘B’. Yes, I know you realized that right away. But I write to a general audience, some of whom are learning how to see these things.) Then the height of this curve is whichever is the larger number, four times the cube of the absolute value of ‘A’, or 27 times the square of ‘B’. I ask you to just run with it. I don’t know the implications of the height function well enough to say why, oh, 25 times the square of ‘B’ wouldn’t do as well. The usual reason for something like that is that some obvious manipulation makes the 27 appear right away, or disappear right away.

This idea of height feeds in to a measure called rank. “Rank” is a term the young mathematician encounters first while learning matrices. It’s the number of rows in a matrix that aren’t equal to some sum or multiple of other rows. That is, it’s how many different things there are among a set. You can see why we might find that interesting. So many topics have something called “rank” and it measures how many different things there are in a set of things. In elliptic curves, the rank is a measure of how complicated the curve is. We can imagine the rational points on the elliptic curve as things generated by some small set of starter points. The starter points have to be of infinite order. Starter points that don’t, don’t count for the rank. Please don’t worry about what “infinite order” means here. I only mention this infinite-order business because if I don’t then something I have to say about two paragraphs from here will sound daft. So, the rank is how many of these starter points you need to generate the elliptic curve. (WARNING: Call them “generating points” or “generators” during your thesis defense.)

There’s no known way of guessing what the rank is if you just know ‘A’ and ‘B’. There are algorithms that can calculate the rank given a particular ‘A’ and ‘B’. But it’s not something like the quadratic formula where you can just do a quick calculation and know what you’re looking for. We don’t even know if the algorithms we have will work for every elliptic curve.

We think that there’s no limit to the height of elliptic curves. We don’t know this. We know there exist curves with ranks as high as 28. They seem to be rare [*]. I don’t know if that’s proven. But we do know there are elliptic curves with rank zero. A lot of them, in fact. (See what I meant two paragraphs back?) These are the elliptic curves that have only finitely many rational points on them.

And there’s a lot of those. There’s a well-respected that the average rank, of all the elliptic curves there are, is ½. It might be. What we have been able to prove is that the average rank is less than or equal to 1.17. Also that it should be larger than zero. So we’re maybe closing in on the ½ conjecture? At least we know something. I admit this essay I’ve started wondering what we do know of elliptic curves.

What do the height, and through it the rank, get us? I worry I’m repeating myself. By themselves they give us families of elliptic curves. Shapes that are similar in a particular and not-always-obvious way. And they feed into the Birch and Swinnerton-Dyer conjecture, which is the hipster’s Riemann Hypothesis. That is, it’s this big, unanswered, important problem that would, if answered, tell us things about a lot of questions that I’m not sure can be concisely explained. At least not why they’re interesting. We know some special cases, at least. Wikipedia tells me nothing’s proved for curves with rank greater than 1. Humanity’s ignorance on this point makes me feel slightly better pondering what I don’t know about elliptic curves.

(There are some other things within the field of elliptic curves called height functions. There’s particularly a height of individual points. I was unsure which height Gaurish found interesting so chose one. The other starts by measuring something different; it views, for example, \frac{1}{2} as having a lower height than does \frac{51}{101} , even though the numbers are quite close in value. It develops along similar lines, trying to find classes of curves with similar behavior. And it gets into different unsolved conjectures. We have our ideas about how to think of fields.).


[*] Wikipedia seems to suggest we only know of one, provided by Professor Noam Elkies in 2006, and let me quote it in full. I apologize that it isn’t in the format I suggested at top was standard. Elkies way outranks me academically so we have to do things his way:

y^2 + xy + y = x^3 - x^2 -  20,067,762,415,575,526,585,033,208,209,338,542,750,930,230,312,178,956,502 x + 34,481,611,795,030,556,467,032,985,690,390,720,374,855,944,359,319,180,361,266,008,296,291,939,448,732,243,429

I can’t figure how to get WordPress to present that larger. I sympathize. I’m tired just looking at an equation like that. This page lists records of known elliptic curve ranks. I don’t know if the lack of any records more recent than 2006 reflects the page not having been updated or nobody having found a rank-29 curve. I fully accept the field might be more difficult than even doing maintenance on a web page’s content is.

The Summer 2017 Mathematics A To Z: Elliptic Curves


Gaurish, of the For The Love Of Mathematics gives me another subject today. It’s one that isn’t about ellipses. Sad to say it’s also not about elliptic integrals. This is sad to me because I have a cute little anecdote about a time I accidentally gave my class an impossible problem. I did apologize. No, nobody solved it anyway.

Summer 2017 Mathematics A to Z, featuring a coati (it's kind of the Latin American raccoon) looking over alphabet blocks, with a lot of equations in the background.
Art courtesy of Thomas K Dye, creator of the web comic Newshounds. He has a Patreon for those able to support his work. He’s also open for commissions, starting from US$10.

Elliptic Curves.

Elliptic Curves start, of course, with polynomials. Particularly, they’re polynomials with two variables. We call the ‘x’ and ‘y’ because we have no reason to be difficult. They’re of at most third degree. That is, we can have terms like ‘x’ and ‘y2‘ and ‘x2y’ and ‘y3‘. Something with higher powers, like, ‘x4‘ or ‘x2y2‘ — a fourth power, all together — is right out. Doesn’t matter. Start from this and we can do some slick changes of variables so that we can rewrite it to look like this:

y^2 = x^3 + Ax + B

Here, ‘A’ and ‘B’ are some numbers that don’t change for this particular curve. Also, we need it to be true that 4A^3 + 27B^2 doesn’t equal zero. It avoids problems. What we’ll be looking at are coordinates, values of ‘x’ and ‘y’ together which make this equation true. That is, it’s points on the curve. If you pick some real numbers ‘A’ and ‘B’ and draw all the values of ‘x’ and ‘y’ that make the equation true you get … well, there’s different shapes. They all look like those microscope photos of a water drop emerging and falling from a tap, only rotated clockwise ninety degrees.

So. Pick any of these curves that you like. Pick a point. I’m going to name your point ‘P’. Now pick a point once more. I’m going to name that point ‘Q’. Now draw a line from P through Q. Keep drawing it. It’ll cross the original elliptic curve again. And that point is … not actually special. What is special is the reflection of that point. That is, the same x-coordinate, but flip the plus or minus sign for the y-coordinate. (WARNING! Do not call it “the reflection” at your thesis defense! Call it the “conjugate” point. It means “reflection”.) Your elliptic curve will be symmetric around the x-axis. If, say, the point with x-coordinate 4 and y-coordinate 3 is on the curve, so is the point with x-coordinate 4 and y-coordinate -3. So that reflected point is … something special.

Kind of a curved-out less-than-sign shape.
y^2 = x^3 - 1 . The water drop bulges out from the surface.

This lets us do something wonderful. We can think of this reflected point as the sum of your ‘P’ and ‘Q’. You can ‘add’ any two points on the curve and get a third point. This means we can do something that looks like addition for points on the elliptic curve. And this means the points on this curve are a group, and we can bring all our group-theory knowledge to studying them. It’s a commutative group, too; ‘P’ added to ‘Q’ leads to the same point as ‘Q’ added to ‘P’.

Let me head off some clever thoughts that make fair objections. What if ‘P’ and ‘Q’ are already reflections, so the line between them is vertical? That never touches the original elliptic curve again, right? Yeah, fair complaint. We patch this by saying that there’s one more point, ‘O’, that’s off “at infinity”. Where is infinity? It’s wherever your vertical lines end. Shut up, this can too be made rigorous. In any case it’s a common hack for this sort of problem. When we add that, everything’s nice. The ‘O’ serves the role in this group that zero serves in arithmetic: the sum of point ‘O’ and any point ‘P’ is going to be ‘P’ again.

Second clever thought to head off: what if ‘P’ and ‘Q’ are the same point? There’s infinitely many lines that go through a single point so how do we pick one to find an intersection with the elliptic curve? Huh? If you did that, then we pick the tangent line to the elliptic curve that touches ‘P’, and carry on as before.

The curved-out less-than-sign shape has a noticeable c-shaped bulge on the end.
y^2 = x^3 + 1 . The water drop is close to breaking off, but surface tension has not yet pinched off the falling form.

There’s more. What kind of number is ‘x’? Or ‘y’? I’ll bet that you figured they were real numbers. You know, ordinary stuff. I didn’t say what they were, so left it to our instinct, and that usually runs toward real numbers. Those are what I meant, yes. But we didn’t have to. ‘x’ and ‘y’ could be in other sets of numbers too. They could be complex-valued numbers. They could be just the rational numbers. They could even be part of a finite collection of possible numbers. As the equation y^2 = x^3 + Ax + B is something meaningful (and some technical points are met) we can carry on. The elliptical curves, and the points we “add” on them, might not look like the curves we started with anymore. They might not look like anything recognizable anymore. But the logic continues to hold. We still create these groups out of the points on these lines intersecting a curve.

By now you probably admit this is neat stuff. You may also think: so what? We can take this thing you never thought about, draw points and lines on it, and make it look very loosely kind of like just adding numbers together. Why is this interesting? No appreciation just for the beauty of the structure involved? Well, we live in a fallen world.

It comes back to number theory. The modern study of Diophantine equations grows out of studying elliptic curves on the rational numbers. It turns out the group of points you get for that looks like a finite collection of points with some collection of integers hanging on. How long that collection of numbers is is called the ‘rank’, and there are deep mysteries at work. We know there are elliptic equations that have a rank as big as 28. Nobody knows if the rank can be arbitrary high, though. And I believe we don’t even know if there are any curves with rank of, like, 27, or 25.

Yeah, I’m still sensing skepticism out there. Fine. We’ll go back to the only part of number theory everybody agrees is useful. Encryption. We have roughly the same goals for every encryption scheme. We want it to be easy to encode a message. We want it to be easy to decode the message if you have the key. We want it to be hard to decode the message if you don’t have the key.

The curved-out sign has a bulge with convex loops to it, so that it resembles the cut of a jigsaw puzzle piece.
y^2 = 3x^2 - 3x + 3 . The water drop is almost large enough that its weight overcomes the surface tension holding it to the main body of water.

Take something inside one of these elliptic curve groups. Especially one that’s got a finite field. Let me call your thing ‘g’. It’s really easy for you, knowing what ‘g’ is and what your field is, to raise it to a power. You can pretty well impress me by sharing the value of ‘g’ raised to some whole number ‘m’. Call that ‘h’.

Why am I impressed? Because if all I know is ‘h’, I have a heck of a time figuring out what ‘g’ is. Especially on these finite field groups there’s no obvious connection between how big ‘h’ is and how big ‘g’ is and how big ‘m’ is. Start with a big enough finite field and you can encode messages in ways that are crazy hard to crack.

We trust. At least, if there are any ways to break the code quickly, nobody’s shared them. And there’s one of those enormous-money-prize awards waiting for someone who does know how to break such a code quickly. (I don’t know which. I’m going by what I expect from people.)

And then there’s fame. These were used to prove Fermat’s Last Theorem. Suppose there are some non-boring numbers ‘a’, ‘b’, and ‘c’, so that for some prime number ‘p’ that’s five or larger, it’s true that a^p + b^p = c^p . (We can separately prove Fermat’s Last Theorem for a power that isn’t a prime number, or a power that’s 3 or 4.) Then this implies properties about the elliptic curve:

y^2 = x(x - a^p)(x + b^p)

This is a convenient way of writing things since it showcases the ap and bp. It’s equal to:

y^2 = x^3 + \left(b^p - a^p\right)x^2 + a^p b^p x

(I was so tempted to leave an arithmetic error in there so I could make sure someone commented.)

A little ball off to the side of a curved-out less-than-sign shape.
y^2 = 3x^3 - 4x . The water drop has broken off, and the remaining surface rebounds to its normal meniscus.

If there’s a solution to Fermat’s Last Theorem, then this elliptic equation can’t be modular. I don’t have enough words to explain what ‘modular’ means here. Andrew Wiles and Richard Taylor showed that the equation was modular. So there is no solution to Fermat’s Last Theorem except the boring ones. (Like, where ‘b’ is zero and ‘a’ and ‘c’ equal each other.) And it all comes from looking close at these neat curves, none of which looks like an ellipse.

They’re named elliptic curves because we first noticed them when Carl Jacobi — yes, that Carl Jacobi — while studying the length of arcs of an ellipse. That’s interesting enough on its own. But it is hard. Maybe I could have fit in that anecdote about giving my class an impossible problem after all.