Kevin Knudson: Welcome to My Favorite Theorem, a podcast about theorems and math and all kinds of things. I'm one of your host,s Kevin Knudson, professor of mathematics at the University of Florida. Here is your other host.

Evelyn Lamb: Hi, I'm Evelyn Lamb. I'm a freelance math and science writer in Salt Lake City. How are you today?

KK: I have a sunburn.

EL: Yeah. Can’t sympathize.

KK: No, no, I was, you know, Ellen and I went out birdwatching on Saturday, and it didn't seem like it was sunny at all, and I didn't wear a hat. So I got my head a little sunburned. And then yesterday, she was doing a print festival down in St. Pete. And even though I thought we were in the shade—look my, arms. They're like totally red. I don’t know. This is what happens.

EL: You know, March in Florida, you really can’t get away without SPF.

KK: No, you really can't. You would think I would have learned this lesson after 10 years of living here, but it just doesn't work. So anyway. Yeah. How are you?

EL: Oh, I'm all right. Yeah. Not sunburned.

KK: Okay. Good for you. Yeah. I'm on spring break. So, you know, I'm feeling pretty good. I got some time to breathe at least. So anyway, enough about us. This is actually a podcast where we invite guests on instead of boring the world with our chit chat. Today, we're pleased to welcome Matilde Lalín, you want to introduce yourself?

Matilde Lalín: Hi. Okay. Thank you for having me here. So I'm originally from Argentina. I grew up in Buenos Aires, and I did my undergraduate there. And then I moved to the US to do my Ph.D., mostly at the University of Texas at Austin. And then I moved to Canada for postdocs, and I stayed in Canada. So right now, I'm a professor at the University of Montreal, and I work in number theory.

EL: And I'm guessing you do not have a sunburn, being in Montreal in March.

ML: So maybe I should say we are celebrating that we are very close to zero Celsius.

KK: Oh, okay.

EL: Yeah, so exciting times.

ML: Yeah. So some of the snow actually is melting.

KK: Oh, okay. I haven’t seen snow in quite a while. I kind of miss it sometimes. But anyway.

EL: Oh, It is very pretty.

KK: Yeah, it is. It’s lovely. Until you have to shovel it every week for six months. But yeah, so Matilde, what is your favorite theorem?

ML: Okay, so I wanted to talk about a problem more than theorem. Well, it will lead to some theorems eventually, and a conjecture. So my favorite problem, let's say, is the congruent number problem.

KK: Okay.

ML: So okay, so basically, a positive integer number is called congruent if it is the area of a right triangle with rational sides.

EL: All three sides, right?

ML: Exactly, exactly. So the question will be, you know, how can you tell that a particular number is congruent? But more generally, can you give a list of all congruent numbers? So for example, six is congruent, because it is the area of the right triangle with sides three, four, and five. So that's easy, but then seven is congruent because it’s the area of the triangle with sides 24/5, 35/12, and 337/60.

KK: Ah, okay.

EL: So that’s not quite as obvious.

ML: Not quite as obvious, exactly. And in fact, there is an example, due to Zagier: 157 is congruent, and so the size of the triangle, they are fractions that have—okay, so the hypotenuse has 46 and 47 digits, the numerator and denominator. And so it can be very big. Okay, let me clarify for a congruent number, there are actually infinitely many triangles that satisfy this. But the example I'm giving you is the smallest, in a sense.

EL: Okay.

ML: So actually it can be very complicated, a priori, to decide whether a number is congruent or not.

KK: Sure.

ML: So this problem appears for the first time in an Arab manuscript in the 10th century, and then it was—

EL: Oh, wow, that's shocking!

ML: Yes. Well, because triangles—I mean, it's a very natural question. But then it was picked up by Fibonacci, who was actually looking at this question from a different point of view. So he was studying arithmetic sequences. So he posed the question of whether you can have a three-term arithmetic sequence whose terms are all squares. So basically, let me give you an example. So 1-25-49. Okay? So those are three squares, and 25−1 is 24. And 49−25 is 24. So that makes it an arithmetic sequence. And each of the three members are squares.

EL: Yeah.

ML: And he said that the difference—so in this case it would be 24, okay? 25−1 is 24, 49−25 is 24—so the difference is called a congruum, if you can build a sequence with this difference, basically. So it turns out that this this problem is essentially equivalent to the congruent number problem, so that's where the name, the word congruent, comes from. Fibonacci was calling this congruum. So congruent has to do with things that sort of congregate.

EL: Okay.

ML: And so kind of this difference of the arithmetic sequence. And you can prove that from such a sequence you can build your triangle. So in the example I gave you, this is a sequence that shows that six is congruent. Well, technically it shows that 24 is congruent, but 24 is a square times six. And so if you have a triangle, you can always multiply the size by the constant, and that would be equivalent to multiplying the area by some square.

KK: Sure, yeah,

EL: Right. Right, and so if it has a square in it, then there's a rational relationship that will still be preserved.

ML: Exactly. So Fibonacci actually managed to prove that seven is congruent. And then he posed as a question, as a conjecture, that one wasn't congruent. So when you say that one is not congruent, you are also saying that the squares are not congruent. The square of any rational number.

EL: Oh.

KK: Okay.

ML: It’s actually kind of a nicer statement, in a sense. It's like a very special case. And then, like 400 years after, Fermat came, and so he actually managed to solve Fibonacci’s question. So he actually proved, using his famous descent, he proved that one is not congruent. And also that two and three are not congruent. So basically, he settled the question for those. And five is known to be congruent, also six and seven. So well, that takes care of the first few numbers. Because four is one in this case.

KK: Four is one, that’s right.

ML: Yeah, exactly. And well, one thing that happens with this problem is that actually, if you go in the direction that Fibonacci was looking, okay, so this sequence of three squares, actually, if you can think of them as—say you call the middle square x, and then one is x−n and the other is x+n. So when you multiply these three together, it gives you a square. And what this is telling you, is that actually giving you a solution to an equation that you could write as, say, y2=x(x−n)(x+n). And that's what is called an elliptic curve.

EL: Oh, okay.

ML: Yes. So basically, an elliptic curve in this context is more general. You could think of it as y2 equals a cubic polynomial in x. And so basically, the congruent numbers problem is asking whether, for such an equation, you have a solution such that y is different from 0. So so then you can study the problem from that point of view. There is a lot, there is a big theory about elliptic curves.

EL: Right. And so I've been wondering, like, is this where people got the idea to bring elliptic curves into number theory? That's always seemed mysterious to me—like, when you first learn about Fermat’s Last Theorem, and you learn there's all this elliptic curve stuff involved in proving that, like, how do people think to bring elliptic curves in this way?

ML: As a matter of fact, okay, so elliptic curves in general, it’s actually a very natural object to study. So I don't know if it came exactly via the congruent number problem, because essentially—okay, so essentially, a natural problem more generally is Diophantine equations. So basically, I give you a polynomial with integer coefficients, and I am asking you about solutions that are either integers or rational. And we understand very well what happens when the degree is one, say an equation of a form ax+by=c. Okay? So those we understand completely. We actually understand very well what happens when the degree is two and actually, degree three is elliptic curves. So it's a very natural progression.

EL: Okay.

ML: So it doesn't necessarily have to come with congruent numbers. However, it is true that many people chose choose to introduce elliptic curves via numbers, because it's such a natural question, such a natural problem. But of course, it leads you to a very specific family of elliptic curves. I mean, not just the whole story. So what is known about elliptic curves that can help understand this question of the congruent number problem: So in 1922 Mordell actually proved that the solutions of an elliptic curve—Actually, I should have said this before. So the solutions of an elliptic curve, say, over the rationals, so if you look at all the rational numbers that are solutions to an equation like that, y2 equals some cubic polynomial in x, they form a group. And actually an abelian group.

And as I was saying, Mordell proved that this group actually is finally-generated. So you can actually give a finite list of elements in the group, and then every element in the group is a combination of those. Okay? So basically, it's very tempting to say, “Well, I mean, if you give me an elliptic curve, I want to find what the group is. So I just give the generators. So this should be very easy, okay? Yeah. [laughter] But actually, it's not easy. So there's no systematic way to find all the generators to determine what the group is. And even—so, you will always have, you may have, points of finite order. So, elements that if you, take some multiple, you get back to 0. So those are easy to find. But the question of whether they are elements of infinite order, and if there are, how many there are, or how many generators you need, all these questions are difficult in generals for an elliptic curve. And so, my favorite theorem actually—so the way I ended up coming up with the idea of talking about the congruent number problem, is actually Mordell’s theorem. So I really like Mordell’s theorem.

KK: And that theorem’s not at all obvious. I mean, so you sort of, I'm not sure if I’ve ever even seen a proof. I mean, I remember, this is one of the first things I learned in algebraic geometry, you draw the picture, you know, of the elliptic curve. And the group law, of course, is given by: take two points, draw the line, and where it intersects the curve in the third point is the sum of those things, right—actually, then you reflect, it’s minus that, right? Yeah. Those three points add to zero. That's right.

EL: We’ll put a picture of this up. Because Kevin's helpful air drawing is not obvious to our listeners.

ML: That’s right.

KK: Yeah. And from that, somehow, the idea that this is a finitely-generated group is really pretty remarkable. But the picture gives you no clue of where the where to find these generators, right?

ML: Well, their first issue, actually, is to prove that this is an associative law. So that statement is annoyingly complicated to prove in elementary ways.

KK: Yeah, commutativity is kind of obvious, right?

ML: But, yes, already to prove that it's a group in the sense that associativity, yeah. And then Mordell’s theorem, actually, it follows, it does some descent. So it follows in the spirit of Fermat’s descent, actually. But I mean, in a more complicated context. But it's very beautiful, yeah.

So as I was saying, the number of generators that have infinite order, that's called the rank, and already knowing whether the rank is zero, or what the value is, that's a very difficult question. And so in 1965, Birch and Swinnerton-Dyer came up with a conjecture that relates the rank to the order of vanishing of a certain function that you build from the elliptic curve. It’s called the L-function. So, in principle, with this conjecture, one can predict the value of the rank. That doesn't mean that we can find easily the generators, but at least we can answer, for example, whether there are infinitely many solutions or not and say that.

EL: Yeah.

ML: So basically, that's kind of the most exciting conjecture associated to this question. And I mean, it goes well beyond this question, and it's one of the Millennium Problems from the Clay.

EL: Right. Yeah. So it’s a high dollar-value question.

ML Yes. And it's interesting, because for this question, it is known that—if the L-function doesn’t vanish, then the rank is zero. So it's known for R[ank] zero, one direction, and the same for R[ank] 1. But not much more is known on average. So this very recent result, relatively recent result by Bhargava and Shankar, where they prove that the rank for, if you take all the elliptic, curves and order them in a certain way, the rank on average is bounded by 7/6. And so that means that there is a positive proportion of elliptic curves that actually satisfy BSD. Okay. But I mean, the question would be what BSD tells us about the original question that I posed.

EL: Right, yeah, so when we were chatting earlier, you said that a lot of questions or theorems about congruent numbers were basically—the theorems were proved as partial solutions to BSD. Am I getting that right?

ML: Yeah, okay. Some progress that is being done nowadays has to do with proving BSD for some particular families, I mean for these elliptic curves are attached to congruent numbers. But if I go back to the first connection, so there is this famous theorem by Tunnell that was published in 1983 where he basically ties the property of being a congruent number to two quadratic equations in three variables having one having double the solutions as the other somehow. So Tunnell’s result came, obviously, in ’83, so much earlier than most advances in BSD.

EL: Okay.

ML: And basically what Tunnell gives is like an algorithm to decide whether a number is congruent or not. And for the case where it’s non-congruent, actually it is conclusive, because this is a case, okay, so it depends on BSD, but this is a case where we know. And then the problem is the case where it will tell you that the number is congruent. So that is assuming BSD. So for now, like I said, many cases will just be the cases that—for example, there is some very recent result by Tian, where basically he proved that BSD applies to certain curves. And so for example, it is known that for primes congruent to five, six, or seven modulo eight, they are congruent. So this is a result that goes back to Heegner and Monsky in ’52, for Heegner. So that's for primes. So that's an infinite family of numbers that satisfy that they are congruent. But every question attached to this problem has to do with, okay, can you generalize this for all natural numbers that are congruent to six, five, or seven modulo eight. For example, that’s some direction of research going on now.

EL: So you could disprove the BSD conjecture, if you could find some number that Turner’s [ed. note: Evelyn misremembered the name of the theorem; this should be “Tunnell’s”] theorem said was congruent, but was actually not congruent?

ML: Yeah, yeah. So you could disprove—say you find a number that is congruent to six mod eight that is not a congruent number, you disprove BSD, yes.

EL: All right. Yeah, so our listeners, I'm sure they'll go—I’m sure no one has ever searched a lot of numbers to check on this. So yeah, that's our assignment for you. So something we like to do on this show then is to ask our guest to pair their theorem with something. So what do you think enhances enjoyment of congruent numbers, the congruent number problem, the Birch and Swinnerton-Dyer Conjecture, all of these things?

ML: Well, for me it is really how I pair my mathematics with things, right? And so I would pair it with chocolate because I am a machine of transforming chocolate into theorems instead of coffee. I will also pair it with mate, which is an infusion from South America. That's my source of caffeine instead of coffee. So it's a very interesting drink that we drink a lot in Argentina, but especially in Uruguay.

KK: Do you have the special straw with the filter and everything?

ML: Yeah, yeah, I have the metal straw. So you you put the metal straw and then you put just the leaves in in your special cup, and you drink from the straw that filters the leaves. So yeah, that's right. And you share it with friends. So it's a very collaborative thing, like mathematics.

EL: So I’ve never tried this. Does this—what does it taste like it? I mean, I know it's hard to describe tastes that you've never actually tasted before. But does it taste kind of like tea, kind of like coffee, kind of like something else entirely?

ML: I would say it tastes like tea. You could think was a special tea.

KK: Okay.

EL: There’s a coffee shop near us that has that. But I haven't tried it yet.

KK: Oh, come on. Give it a shot, Evelyn.

EL: Yeah, I will.

KK: You have to report back in a future episode. Actually, I going to hold you to it the next time we meet. Before then, have some mate. Do you have a chocolate preference? Are you a dark chocolate, milk chocolate?

ML: Milk chocolate, I would say. I'm not super gourmet with chocolate. But I do have my favorite place in Montreal to go drink a good cup of hot chocolate.

KK: All right, I've learned a lot.

EL: Yeah.

KK: This is very informative. In fact, while you were describing the congruent number problem, I was sort of sitting here sketching out equations that I might try to actually solve. Of course, it wasn't an elliptic curves, it was sort of the naive things that you might try. But this is a fascinating problem. And I could see how you could get hooked.

EL: Yeah, well, it does seem like it just has all these different branches. And all these weird dependencies where you can follow these lines around.

KK: I mean, the best mathematics is like that, right? I mean, it's sort of kind of simple, it’s a simple question to ask, you could explain this to a kid. And then the mathematics is so deep, it goes in so many directions. Yeah, it's really, really interesting.

EL: Yeah. Thanks a lot. Are there any places people can find you online? Your website, other other things you'd like to share?

ML: Well, yeah, my website. Shall I say the address?

EL: We can just put a link to that.

ML: Yeah, definitely my website. I actually will be giving a talk in the math club at my university on congruent numbers in a couple of weeks. So I’m going to try to post the slides online, but they are going to be in French.

EL: Okay, well, that'll be good. Our Francophone listeners can can check that out.

ML: I really like some notes that Keith Conrad wrote. And actually, I have to say, he has a bunch of expository papers in different areas that I always find super useful for, you know, going a little bit beyond my classes. And so in general, I recommend that his website for that, and in particular, the notes on the congruent number problem, if you're more interested. And then of course, there are some, some books that discuss congruent numbers and elliptic curves. So for example, a classic reference is Koblitz’s book on, I guess it’s called [Introduction to] Elliptic Curves and Modular Forms.

EL: Oh, yeah, I actually have that book. Because as a grad student, my second or third year, I, for some reason—I was not interested in number theory at all, but I think I liked this professor, so I took this class. So I have this book. And I remember, I just felt like I was swimming in that class.

KK: I have this book too, sitting on my shelf.

EL: The one number theory book two topologists have.

ML: So for me, I got this book before knowing I was going to be a number theorist.

EL: Yeah. No, but it is a nice book. Yeah. But yeah, well, we'll link to those will have. Make sure to get those all in the show notes so people can find those easily.

ML: Yeah.

EL: Well, thanks so much for joining me.

KK: Yeah.

EL: Us. Sorry, Kevin!

ML: Thank you for having us—for having me, now I’m confused! Thanks a lot. It's such a pleasure to be here.

EL: Yeah.

KK: Thanks.

On this episode, we were excited to welcome Matlide Lalín, a math professor at the University of Montreal. She talked about the congruent number problem. A congruent number is a positive integer that is the area of a right triangle with rational side lengths.

Our discussion took us from integers to elliptic curves, which are defined by equations of the form y2=x3+ax+b. As we mention in the episode, solutions to equations of this form satisfy what is known as a group law. That is a fancy way of saying there is a way to “add” two points on the curve to get another point. The diagram Kevin mentioned is here: