Elliptic Curves and Modular Forms | The Proof of Fermat’s Last Theorem

Elliptic curves, modular forms, and the Taniyama-Shimura Conjecture: the three ingredients to Andrew Wiles’ proof of Fermat’s Last Theorem.

QUESTIONS:

1. The Taniyama-Shimura Conjecture seems really contrived. We made a weirdly specific sequence from elliptic curves. We made a weirdly specific sequence from modular forms. And behold, the sequences match! It seems manufactured to work. What’s profound about it?

2. Why do we care about elliptic curves of all things? It’s described by, again, a weirdly specific equation: why is it the darling child of number theory?

3. Does the Taniyama-Shimura conjecture also guarantee uniqueness? That is, does it say that for every elliptic curve there is a *unique* modular form with the same sequence as it?

4. We defined how a matrix from the group SL2Z “acts” on a complex number. Does anyone have a geometric picture for this? Does a matrix act on a complex number just like how it would act on a vector in R^2 (i.e: by rotating it)?

5. This is a more advanced question. Most elliptic curve books encode the sequence m_n of a modular form using something called a Dirichlet L-function, a generalization of the Reimann Zeta function. More precisely, instead of associating a modular form to a *sequence*, we associate it to a modified version of the Riemann Zeta Function, where the n_th coefficient of the series is the term m_n. (This is sometimes called the Hasse-Weil L-function of a modular form). This seems unnecessary. What is the benefit of doing this?

6. Does anyone understand Andrew Wiles’ paper? LOL

SOURCES I USED TO STUDY:

Keith Conrad’s Lectures on Modular Forms (8 part video series):

Keith Conrad’s Notes on Modular Forms:

“Elliptic Curves, Modular Forms, and their L-Functions” by A. Lozano-Robledo.
(The above book is very accessible! You only need basic calculus to understand it. You also need to know the definition of a group, but that’s pretty much it.)

“The Arithmetic of Elliptic Curves” by Joseph Silverman

HOMEWORK IDEA CREDIT goes to Looking Glass Universe!

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Intro: (0:00)
Elliptic Curves: (0:58)
Modular Forms: (3:26)
Taniyama Shimura Conjecture: (7:26)
Fermat's Last Theorem: (8:02)
Questions for you!: (8:51)

[oleksiishekhovtsov]

This channel is the greatest thing to happen to popular maths content since 3blue1brown popped onto the scene!

[aky]

This is the best math YouTube video. You didn't oversimplify and condensed the important stuff so nicely. Subscribe this guy.

[functor]

Elliptic curves are important because of Class Field Theory. The Kronecker-Weber Theorem characterizes all abelian extensions of Q using points on the unit circle - roots of unity. Hilbert's 12th Problem, or Kronecker's Jugentraum, asks if there is a similar way, using concrete objects like roots of unity, to characterize abelian extensions of any base field. Class Field Theory doesn't quite answer this, because it characterizes abelian extensions in terms of class groups rather than concrete geometry.

But there is a generalization to the Kronecker-Weber Theorem in this way using Elliptic Curves with Complex Multiplication. The values of the j-functions associated certain elliptic curves with complex multiplication can produce the abelian extensions of imaginary quadratic fields. And so, in this way, specific values of the j-function is analogous to the roots of unity. This indicates that elliptic curves contain within them sophisticated information about arithmetic.

The Taniyama-Shimura Conjecture fits within this narrative. In the general theory, class field theory is the dimension=1 case of something much larger, now known as Langlands' Program. There are two sides to Langlands Program, an arithmetic side made from Galois representations, and an analytic side and in the dimension=1 case this analytic side is made from Hecke characters. Class field theory links these two things. The Galois side naturally generalizes to any dimension, but the analytic side is a bit harder. In the dimension=2 case, the generalization to Hecke characters are modular forms. Connecting 2 dimensional Galois representations and modular forms is very tough. One thing that we can notice, however, is that we can attach to every elliptic curve a local 2-dimensional Galois representation. The idea is then that we can link modular forms to 2-dimensional Galois representations through elliptic curves. There are some interruptions to this line of reasoning and as with everything Langlands' Program there are only partial results, but it gives us a way to connect arithmetic objects to analytic ones. This generalizes even further, with Shimura Varieties being ways to extend this to higher dimensions.

In the end, elliptic curves live in a sort of middle ground. On one hand, they are a kind of arithmetic object with highly sophisticated information. On the other hand, elliptic curves are the simplest case for abelian varieties. We can then do fairly concrete computations with them, while using them in highly abstract ways.

[shomarzzz]

u are the only one who does proof explanotory
videos on this kind of topics

[trajanhammonds]

Here's an attempt to answer some of your questions

1. The sequence from modular forms is not that specific or unusual. You noted that inputing a certain matrix gives you the relation f(z+1) = f(z), and a natural thing to do with a holomorphic periodic function is to write down it's Fourier expansion and the most interesting thing about a Fourier expansion are the Fourier coefficients. The sequence for elliptic curves may seem uninspired, but they form the coefficients of the L-function or Dirichlet series for the elliptic curve. Modular forms have L-functions too, and you can obtain them by taking the Mellin transform. So really one version of modularity is to say that the L-functions of these two objects are the same, and that should really be surprising as looking at solutions mod p of some equation should have nothing to do with modular forms. Another part of the story not mentioned in the video is that to elliptic curves you can associate a Galois representation, a representation of the Galois group Gal(\bar{Q}/Q) and to a modular form you can associate a Galois representation in a completely different way, and so another version of modularity states that these Galois representations agree and there is a way to recover these sequences from the Galois representations.

2. Elliptic curves are genus 1 curves. The rational points of curves that are genus 2 or higher is well understood due to Falting's theorem which says the set of rational points for these higher genus curves is always finite. Genus 0 curves are also not that interesting, like the plane conics you see in grade school. The reason why elliptic curves are so interesting is because the set of rational points forms a finitely generated Abelian group called the Mordell-Weil group and this group has r copies of the integers Z. This number r is the rank and is in general not very well understood. If r = 0, your elliptic curve has only finitely many rat'l pts but if its bigger than zero then it has infinitely many. Trying to figure out when an elliptic curve has rank 0 or 1 is an interesting question and it's even conjectured that we should see higher ranks (>= 2) "0%" of the time. Questions about the rank are also related to Birch and Swinnerton-Dyer conjecture so that's also why elliptic curves are so fascinating.

3. Unique perhaps up to the right notion of isomorphism. I think there's a theorem that says if all but at most finitely many of your Fourier coefficients agree, then you actually have the same form, but I'm not sure.

4. I think a more interesting geometric picture is what you obtain by quotienting the upper half plane by this action. You get a Riemann surface. And when you consider congruence subgroups of SL_2(Z) and quotient the upper half plane by the action of these subgroups you obtain modular curves which have a lot of interesting structure. In fact, most interesting modular forms are not defined for all of SL_2(Z) but rather one of these congruence subgroups and that information is contained in the "level" of the modular form.

5. Hopefully I answered this question, but also it's completely natural in combinatorics for example to slap arithmetic sequences onto generating functions, and you should think of this as something similar. We have some sequence containing arithmetic data, let's give it a "generating function" and see what interesting properties it might have.

6. Not me lol

Lastly, when discussing the Hasse-Weil bound I'd be hesitant to say the intuition comes from computer simulations. (Though the BSD conjecture was originally formulated by looking at computer data). There's a simple heuristic to see why we should expect p+1 points on an elliptic curve over F_p. Indeed, if i have the equation y^2 = x^3 + ax + b, I can count the solutions over F_p directly with the sum_{x in F_p} ((x^3+ax+b)/p) + 1 where ((x^3+ax+b)/p) is not a fraction but rather the Legendre symbol (a/p) which is 1 when a is a nonzero square mod p, -1 when a is a non square and 0 when a is zero. Note that this sum exactly counts the solutions to y^2 = x^3 + ax + b because if x^3 + ax + b is a square mod p, the legendre symbol will record a 1, and with the +1 in the sum you will get two solutions (because both +y and -y give the same value of y^2). When x^3+ax+b is not a square mod p, the legendre symbol records a -1, and with the +1 in the sum you will get zero solutions. So if you expect the cubic x^3 + ax + b to behave somewhat uniformly, then you'd expect it to be a square half the time, and a nonsquare the other half of the time, so the legendre symbols should cancel. So in total, the sum would be just p. Throw in the point at infinity and you get p+1.

The error term of 2sqrt(p) is essentially the Riemann hypothesis for curves over finite fields, but it's not the only time we encounter "square root errors" in number theory. In fact, the fourier coefficients of certain weight k modular forms satisfy the bound |a_p| < 2p^{(k-1)/2}, where a_p is the pth Fourier coefficient. This was proven by Deligne in the 70s as part of his proof of the Weil conjectures. You can see that the modular forms associated to elliptic curves are weight 2, and you recover exactly the same bound as in the Hasse-Weil theorem.

[technowey]

I smiled when he mentioned "Fourier Series" and then asked, "How 'convoluted' could this get?" :)
Did he intend that pun?

[rishabharora]

Damn this channel is seriously underrated!!

[luisdb]

Your channel is one of the bets out there didactic, entertaining, deep, thanks again for all of your work.

[keyyyla]

The combination of creativity, formal correctness and entertainment is very fascinating. Keep up the great work, it is very rare to find such advanced pure mathematics topics on youtube at such a didactic level. so great man!

[СтепанНестеров-рр]

This video's style is very charming :)
I'm a grad student in number theory, and although I do not understand Wiles' paper, I would like to give a shot at your questions 1 through 5 ;)
1. I believe that the answer to that is your question 5. There is a close analogy between the rings Z and F_p[x]. In fact, the whole of elementary number theory relies on two principles: that Z is a principal ideal domain, and that the quotient fields Z/(p) are finite. There are two kinds of rings which algebraic number theory works for: rings of integers in number fields (these are not quite PID's in general, but they come close) and finite extensions of F_p[t]. Andre Weil was able to find and prove an analog of the Riemann Hypothesis for the latter rings, hoping to get insight in the Riemann zeta function. Weil's proposed analog for the zeta function of a ring F_q[x, y]/(y^2=x^3+ax+b) evaluates to (1-cq^(-s)+q^(-2s))/(1-q^(-s))(1-q^(1-s)), where c=(# of solutions)-q-1 is the error term which we are talking about. If you have an elliptic curve with integer coefficients a, b, then there is a further generalisation of the Riemann zeta function, which is, roughly speaking, the product of Weil's zeta functions of (y^2=x^3+ax+b mod p) over all primes p. One of the Millenium Prize problems, called the Birch and Swinnenton-Dyer conjecture, asks for a proof that the rank of the group of rational points on a curve is equal to the order to which this zeta function vanishes at the point s=1. So, conjecturally, the rank of our elliptic curve and many more invariants can be computed in terms of c_p alone. The significance of the Taniyama-Shimura conjecture is in that it is equivalent to the following statement:
Zeta function of an elliptic curve with integral coefficients has an analytic continuation to a meromorphic function in the whole complex plane and satisfies a functional equation similar to that of the Riemann zeta function. Basically, if you know that \sum c_m e^(2 \pi i m) is a modular form, then the equation f(-1/z)=z^kf(z) translates to the functional equation for the L(s)= \sum c_m m^(-s) of roughly the form L(2-s)=(something easy)L(s). The only known way of extending L(s) for an elliptic curve to the whole of C and even making sense of L(1) is through the functional equation provided by the Taniyama-Shimura conjecture. Analogous L-functions are defined for every algebraic variety over Q but the elliptic curve case is essentially the only case where this L-function is known to have an analytic continuation at all.
2. It isn't really ;) The popularity of elliptic curves is partially credited to their usefulness in cryptography and partially to the fact that their definition is very simple, and a nontrivial theory of elliptic curves can be introduced in a beginners number theory course with no prerequisites. There is absolutely nothing special or interesing about y^2=x^3+ax+b. Number theory, or more specifically, its subfield called arithmetic geometry, succesfully studies all curves, and partially succesfully some varieties of higher dimensions. If you read a more advanced textbook on elliptic curves, you will probably see a different definition, that an elliptic curve is a genus 1 nonsingular complete curve with a distinguished rational point. What happens really is a manifestation of the underlying classification of algebraic curves. An algebraic curve has a very important invariant, called its genus. An algebraic curve f(x, y)=0 over the field of complex numbers is said to have genus g if the set of all complex solutions of f(x, y)=0 (modulo technicalities such as including solutions at infinity and treating branches at singular points as different points) is homeomorphic to a sphere with g handles. Continuing over C, there is a standard form for curves of small genus. A curve of genus 0 is isomorphic to a line, a curve of genus 1 is isomophic to a curve y^2=x^3+ax+b, and a curve of genus 2 is isomorphic to a curve y^2=x^5+ax^3+bx^2+cx+d modulo the same technicalities. If you are willing to develop the theory of curves of genus 100, you can not resort to such standard forms and have to use general techiniques of algebraic geometry.
3. What Taniyama-Shimura conjecture gives you fails to be a one-to-one correspondence in every possible way. For one thing, if two elliptic curves are isogenous, meaning there is a nonconstant map from one to the other, then they have the same zeta-function, which would be written in terms of the same modular form. Not only that, the modularity theorem is stated in terms of very special forms, so called cusp newforms, which have a very technical definition which is difficult to sum up. And there is another parameter of a modular form that you didn't mention in the video, which is called its level. Basically, you are working with functions which are not modular with respect to every matrix from the jar, but only with respect to matirces whose lower left corner is divisible by the level N. Modularity theorem gives you a modular form of level N equal to the conductor of your curve and weight 2 with the correct L-function, but it is highly probable that you can find modular form of different level with the same L-series. It might be true that there is a unique modular form which is minimal in some sense, but I wasn't able to quickly find a reference where such matters are discussed. Maybe the form of minimal level is unique up to some obvious automorphisms, something like that.
4. The action that we have in the theory of modular forms seems to be very close to the natural action of matrices on the underlying vector space. 2 by 2 matrices act on the 2-dimensional vector space by sending a vector (x, y) to (ax+by, cx+dy). Since this action preserves lines through the origin, it descends to the projective line [x:y] -> [ax+by:cx+dy]. Everything we did was rewriting this in terms of inhomogenous coordinates [z:1] -> [az+b, cz+d]=[(az+b)/(cz+d):1]. So, in more geometric terms, if you forget that z is complex and substitute real values, than this is nothing more than taking a line y=kx in R^2, applying the linear transformation A=(a b | c d), and observing that the image of the line is bx+dy=k(ax+cy) <=> y=(ak+b)/(ck+d)*x.

[jgy]

That's pretty cool! This is a very complex topic (Andrew Wiles' paper for example is pretty much only really understood by the a very few selection of mathematicians that work on the topic) and yet I think you did a good job on whatever you could. I myself know very little about it, but I think I can still see its beauty. For your question, I think I have some sort of guess for the third one, I think number theorist are so interested in elipitic curves simply because it feels like some sort of natural continuation of the study of diophantine equations, since what you are doing by searching for rational points is pretty much just finding "rational" solutions to the equation, and since linear and conic diophantine equations are already relatively well understood, some sort of cubic would seem like the next logical step, thought why to study specifically this type of cubic is beyond me, maybe some sort of historical context, I really don't know. By the way, I see that you are now somewhat settling your style of presentation, which is pretty nice! I really like the one you used on this video, I find it a bit different than what other people are doing in execution, while at the same time being quite familiar in concept. As for suggestions of topics, I'd love to see something on ring theory, I have personally just been in love with it lately, so I like to see it if possible (maybe Noetherian Rings? ;) ). By the way, just a heads up, on the fifth question there's a little typo, you wrote "Reimann" instead of "Riemann". Anyway, great stuff, keep up the good work! (P.S. sorry for the big comments lately, I don't know if they are interesting or not, I just find them quite fun to make lol)

[charlesrodriguez]

Imma says this rn I am finishing up a math major and have some idea of the difficulty of the topics you covered here. This channel easily deserves a million subs. You are doing the math community a great service by providing such powerful, clear, and concise explanations to some of the most difficult problems in mathematics. Seriously, the content you put out is amazing.

[dicemaster]

Great video ! About question 4 : there is indeed a geometric interpretation for the action of a matrix of SL2(Z) on a complex number in the upper-half plane. The transformation z ---> -1/z could be called 'reflexion about a circle', or rather about a semi-circle since we consider it only on H. It is not a Euclidean isometry, it changes distances a lot, but preserves the circle |z|=1 and exchanges the inside and the outside of the circle. In general, a transformation z ---> (az+b)/(cz+d) is a composition of translations z ---> z + a, scale transformations z ---> bz, and this 'reflexion about the unit semi-circle' z ---> -1/z. In fact the best way to interpret matrices in SL2(R) is as hyperbolic isometries. The upper-half plane H is a model for the hyperbolic plane and elements of SL2(Z) are special cases of hyperbolic isometries PSL2(R). In this model the semi-circle |z|=1 is actually a line, and the transformation z ---> -1/z is a reflexion about a line. The real line (which does not sit inside H) is the 'horizon' or 'points at infinity'. Hyperbolic isometries can be classified depending on their number of fixed points, just like in Euclidean geometry: rotations, reflexions and translations. Given a matrix A you can read the type of isometry by taking its trace : rotation ( | Tr A | < 2 ), translation ( | Tr A | = 2 ) or reflexion ( | Tr A | > 2 ). In this picture, SL2(Z) is just the subgroup of hyperbolic isometries generated by the unit translation z ---> z+1 and the unit inversion z ---> -1/z. There is an associated tiling of the hyperbolic plane that helps visualize the symmetries of SL2(Z). It's a hyperbolic 'wallpaper group' !

[sukhjinderkumar]

man your honesty at will become great mathematican. i am currently studying, but someday i will answer these

[manueldelrio]

Great video, as usual! I am a mathematical ignoramus (I am starting from almost scratch and trying to catch up; done my bit with some popular math books too), but I would say for the two questions you make the following:
1) as for the matching sequences, isn't it the case that they weren't created on purpose to match? Rather, they were each autonomously made in each of their fields (modular forms and elliptic curves, respectively); in their fields they are a pretty natural product of some tampering. It was only later discovered (by Taniyama and Shimura, I think) that these two sequences matched in some consulted cases (leading to the conjecture that they matched for ALL cases). The match shows hidden connections in math, it wasn't artificially created to match.
2) As to Elliptic Curves, their importance, I think, lies in the fact that they are in a 'sweet' spot for research. Equations with less variables and/or degree are trivial for current math research; those with higher degrees and variables are intractable; elliptic curves has just enough difficulty (but also partial solvability) as to be adequate for working and investigating.

[NothingMaster]

Excellent presentation; you made an incredibly abstract subject highly digestible for the uninitiated audiences. 👏🏻

P.S. In order to start finding some answers to your questions, I believe you have to begin with Mazur's torsion theorem, spanning a complete list of the possible torsion subgroups of elliptic curves over the rational numbers; a most critical result in the arithmetic of elliptic curves. Look at Mazur’s paper on "Modular curves and the Eisenstein ideal", therein he offers a complete analysis of the rational points on a number of modular curves. Mazur's idea of Galois deformations, are among the key concepts necessary for understanding Wiles's proof of Fermat's Last Theorem. I’m also of the opinion that symmetries are what internally connect the arithmetic of elliptic curves to the inherent (and in the words of Mazur, inordinately symmetric) structures of modular forms.

[Entropize]

I am at the end of my first of an estimated 6 years learning the proof of FLT. I love high level overviews like these! Nice work.

[AndrewWyld]

Thanks for an awesome video! I am new to all of this, but I felt like I gained a glimmer of intuitive understanding of how this works. I will certainly be rewatching and reading around it.

I'm going to take a shot at saying what I think is impressive about creating these weirdly specific matching patterns in elliptic curves and modular forms! I think of it as like creating a sort of zip, with one half of the zip on elliptic curves and the other half on modular forms, that allows the two things to be fastened together perfectly, with every point of one fastened to exactly one point of the other. I think there are probably many ways one could arrange such a zip, but finding a way to integrate the zip teeth into the thing is what I find impressive. The apparent arbitrariness is part of what IS so impressive, in fact: it shows how much ingenuity was required to mount an effective zip on this object!

By analogy, the method of counting the rationals could be any way of filling the plane of numbers with a line of numbers; conventionally, we use diagonals, but we could equally well build up the edges of a squarish rectangle, radiating out from the origin. The arbitrariness does not detract from the elegance of finding a way to fit the two different things together.

I think it's much as you observed with the Fourier series: the fact that it's a Fourier series doesn't matter, as long as we get the coefficients! The coefficients are the zip teeth in this particular instance.

Having said all of this, I may be missing your point here! I hope my thoughts are of interest anyway. Thanks once again for a great video.

[MrBmarcika]

2. I remember hearing about a theorem, that under some irreducibility and/or smoothness assumptions degree 3 is the highest degree where infinitely many rational points can occur, so that's one reason. Also, since we see how many hard and interesting things one can prove with them, they make themselves interesting I guess. (Also, you can define a group structure on elliptic curves, they are the only ones with this property also)
3.If this sequence associated to the modular form is just the positive fourier coefficients, then sure, since the fourier coefficients are unique (I don't know what kind of sequences do we get from elliptic curves, so there may be some convergence problems here)
4. Actually it's not SL_2, it's PSL_2, there are a lot of very interesting geometry regarding these fractional linear functions, we can map any Jordan domain conformally onto any other Jordan domain with them, the main theorem here is the Riemann maping theorem. They don't act, as they do on vectors, the generating elements are the translations, multiplication by a complex number, and taking the reciprocal. The first two behave like you would expect, translating the vector, and multiplication, as with complex numbers, so rotation, and changing the length, but taking the reciprocal is a geometrical inversion, and a reflection afterwards with the real axis. My guess as to why we choose this group, is that it is the conformal automorphismgroup of the upper half plane, so the requirement, that f be "invariant" in the sense discussed in the video, is rather natural, since we only ask it to be invariant to ways, in which we can map the upper half plane onto itself, while preserving angles (which is what conformal means)
As for the others, I'm just a student, haven't learned much (or any, rather) ANT.
Great videos btw, criminally undersubbed channel, keep it up!

[lordepl]

You are doing a great job explaining maths for real and still simplified enough for some basic people to follow! Keep up with the great job!