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.
This is by far the hardest video I've ever had to make: both in terms of learning the content and explaining it. So there a few questions I don't have answers for. If you're up for it, feel free to answer these as a KZbin comment or on Twitter (@00aleph00)!
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):
• Introduction to Modula...
Keith Conrad’s Notes on Modular Forms:
ctnt-summer.math.uconn.edu/wp...
“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!
SAGE RESOURCES:
“Sage for Undergraduates”: Gregory Bard’s Free Online Book on SAGE : www.gregorybard.com/Sage.html
Download SAGE: www.sagemath.org/download.html
Proof of the Hasse-Weil Bound on Terry Tao’s Blog: terrytao.wordpress.com/2014/0...
OTHER VIDEOS ON THESE TOPICS:
Numberphile Playlist: • Fermat's Last Theorem ...
Elliptic Curves and Modular Forms: • Elliptic curves and mo...
SOFTWARE USED TO MAKE THIS VIDEO:
SAGE for the code and the graphs
github.com/hernanat/dcolor for domain coloring
Adobe Premiere Elements For Video Editing
MUSIC:
Music Info: Documentary - AShamaluevMusic.
Music Link: www.ashamaluevmusic.com
Follow me!
Twitter: / 00aleph00
Instagram: / 00aleph00
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)

[@oleksiishekhovtsov1564]

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

[@brightsideofmaths]

I totally agree! Keep up the good work!

[@kumoyuki]

And since Kelsey left Infinite Series (which was subsequently dropped by PBS)

[@BRDRDRDAT]

except this guy is literally wrong about a lot and rigorous about nothing

[@dapdizzy]

There are a few profound channels that do not require hype to exist. I watched some of them with a strong lasting feeling of perfection. Just don’t believe only the “best” is worth your attention. You never know where you will find the treasure.

[@TheAgentJesus]

@@BRDRDRDAT wrong about what, specifically?

[@functor7]

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.

[@068LAICEPS]

and why are they use in encryption?

[@functor7]

@@068LAICEPS Elliptic curves are used in cryptography because you can get similar security as RSA, but with smaller key lengths. Broadly, this is because elliptic curves have more complex arithmetic and so there is more to consider when cracking them. For instance, over a specific finite field, there are many different elliptic curves, but only one multiplicative group (which RSA uses). What is really interesting is Supersingular Isogeny Key Exchange, which encodes things using graphs made from collections of elliptic curves with specific properties that make these graphs hard to crack. Totally different from normal cryptography, but way cool and it is safe from quantum computing attacks.

[@christiansmakingmusic777]

Not to wax philosophical, but what is the ultimate goal of the “program”? I appreciate the reply, and your name (functor7) clearly is intended to mask your identity. Where is the real world payoff, and, isn’t it true that at best elliptic curves must be but one of an infinite number of possible windows into the information behind all of mathematics?

[@wabahaba2108]

@@christiansmakingmusic777 In theoretic mathematical science, I think that generelly the "pay off" from discovery is the discovery itself. Proving Fermats theorem for example has no applications outside pure math, as far as I know. I am unsure what you mean with "information behind all math" but my take away from functor7s answer is that eliptic curves are interesting as they connect two different fields of math. I think you misinterpret if you read it as a claim that they hold some more special role in mathematics. In the same vein, the Langland program is described on wikipedia as a collection of conjectures regarding connections between geometry and number theory. I am a physics undergrad so by no means an expert on math, but I think that generally, the vast majority of cutting edge math research is completely unrelated to any practical application whatsoever. Further, in contrast to physics where there is a strive towards a "grand unified theory", I don't really think there is any similar strive in math. I recommend looking into Gödels incompletes theorem if you are interested in these things, which proves that there can be no self contained full description of a mathematic system that is free of logical inconcistencies, one can always formulate statements that must be true if the system is to be logically concistent but that cannot be proven by the axioms in the system, so that in effect, you will need an endless set of axioms (and a field of math might diverge into to different fields depending on in what way you choose to extend it).

[@christiansmakingmusic777]

@@wabahaba2108 It was a physicist who first showed me Godel. Godel and Chaitin show us our limits and should humble us. There are many social and psychological issues affecting cutting edge maths. I have no problem with people choosing their hobbies. Something seems off when we as a culture choose to create any kind of air of mystique around unsolved questions which will never be more than an interesting trivial pursuit question in the future. Mathematicians are given too much cultural license because they are “smarter” than the rest of us. There is really nothing difficult to understand about the proof of FLT, if it were fully explained with the intent of making it transparent. But, the five people who could do that choose not to. I don’t have time or desire here to talk about the cultural problems that have justified the current state of affairs, but many of them are not unique to math, but are found in all disciplines of higher study. Physics is no exception.

[@aky68956]

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

[@Aleph0]

thanks so much!

[@k-theory8604]

How can this be the best video, when their video on cohomology is the best video?

[@hyperduality2838]

@@Aleph0 Elliptic curves link up with hyperbolic geometry in space/time diagrams in special relativity, physics that is why they are important. Elliptic curves are dual to modular forms, synthesize Janus holes/points. Elliptic or spherical geometry is dual to hyperbolic geometry. Positive curvature is dual to negative curvature -- Riemann, Gauss geometry. Dark energy is dual to dark matter. The big bang is a Janus hole/point (two faces) -- duality.

[@MarcusAndersonsBlog]

@@Aleph0 Thank you for your generous invitation to comment in video reply, and I'll take that up if time permits. For now, I'll briefly answer your questions here. BTW, Nice treatment on Wiles proof of FLT. 1. Contrived? Occam's Razor dictates: the simplest of competing theories be preferred to the more complex. In that sense, the Taniyama-Shimura-Weil conjecture is vulnerable to a less complex proof. Wiles solution is profound for those who deem it so. I'm with you on this, I'm not satisfied. 2. You surely already know this - Elliptic curves arise from x^n + y^n = r^n which is the equation of an ellipse except when n=2. This is FLT. 3. In the coxtext of FLT this is not relevant. Pass. 4. a) No. b) Unrelated to FLT, so pass. 5. This is the problem with the Wiles approach - it goes to unnecessary extremes. Nothing in nature is this complex. Complexity is an identification of incorrectly rejecting a simpler solution. The greater the complexity, the greater the departure from Occam's Razor. 6. Yes, but you need to be 'in-the-grove'. Most people have other things to do with their time. Like me.

[@cbob213]

@@MarcusAndersonsBlogLOL….Clearly…

[@shomarzzz9765]

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

[@trajanhammonds8507]

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.

[@hyperduality2838]

It is much easier to answer some of these questions using physics:- The time domain is dual to the frequency domain -- Fourier analysis. Stability is dual to instability -- optimized control theory. The initial value theorem (IVT) is dual to the final value theorem (FVT) -- Optimized control theory. The Laplace transform links control theory with modular forms or complex numbers. Elliptic curves link up with hyperbolic geometry in space/time diagrams in special relativity, physics that is why they are important. You can model light cones (photons) with elliptic & hyperbolic curves. Elliptic curves are dual to modular forms, synthesize Janus holes/points. Elliptic or spherical geometry is dual to hyperbolic geometry. Positive curvature is dual to negative curvature -- Gauss, Riemann geometry. Curvature or gravitation is therefore dual or gravitational energy is dual. Energy is dual to mass -- Einstein. Dark energy is dual to dark matter. Apples fall to the ground because they are conserving duality. Potential energy is dual to kinetic energy. Action is dual to reaction -- Sir Isaac Newton. The big bang is a Janus hole/point (two faces) -- duality. "Perpendicularity in hyperbolic geometry is measured in terms of duality" -- Professor Norman Wildberger. "Reflections preserve perpendicularity (duality) in hyperbolic geometry" -- Professor Norman Wildberger. Space is dual to time -- Einstein. Gravitation is equivalent or dual to acceleration -- Einstein's happiest thought, the principle of equivalence (duality). "Always two there are" -- Yoda. Homology is dual to co-homology. Union is dual to intersection. Integration is dual to differentiation, convergence is dual to divergence. The dot product is dual to the cross product -- Maxwell's equation. Vectors are dual to co-vectors (forms). Concepts are dual to percepts -- the mind duality of Immanuel Kant. The intellectual mind/soul (concepts) is dual to the sensory mind/soul (percepts) -- the mind duality of Thomas Aquinas. Absolute time (Galileo) is dual to relative time (Einstein) -- time duality. There is a pattern of duality in fundamental physics which is being ignored! Duality creates reality!

[@raypraise]

@@hyperduality2838 Great !

[@hyperduality2838]

@@raypraise Duality is a pattern hardwired into the physics. Once you accept it you can ask the question what is dual to entropy? Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics! Teleological physics (syntropy) is dual to non-teleological physics (entropy). Randomness (entropy, divergence) is dual to order (syntropy, convergence). The bad news is that teleo-phobia dominates current thinking in main stream physics, this means that target tracking is not allowed. All observers track targets -- teleology! Syntropy is heresy as in Galileo levels of heresy. Duality: two sides of the same coin. Antinomy (duality) is two truths that contradict each other -- Immanuel Kant. Mathematics is also dual at a fundamental level:- Y = X. Y is equal to X, Y is the same, similar, equivalent or dual to X. Y is dual to X. All mathematical equations are dualities, they compare/equate one quantity with another.

[@qewqeqeqwew3977]

@@hyperduality2838 No, the "dual" quantity to entropy is simply temperature. volumepressure, entropytemperature, particle numberchemical potential, magnetic fieldmagnetization etc.

[@hyperduality2838]

@@qewqeqeqwew3977 The word entropy means "a tendency to diverge" or differentiate into new states, reductionism (science). The word syntropy means "a tendency to converge" or integrate into a whole, holism (religion). Divergence is dual to convergence, differentiation is dual to integration, reductionism is dual to holism, division is dual to unity. Syntropy is dual to increasing entropy! "Science without religion is lame, religion without science is blind" -- Einstein. Science is dual to religion -- the mind duality of Albert Einstein. What is converging is the fact that you are making predictions, projections to track targets, goals, objectives and aims -- teleology. All observers track targets. Target tracking is a syntropic process. Teleological physics (syntropy) is dual to non-teleological physics (entropy). Good news is dual to bad news. The bad news is that teleophobia currently dominates thinking in physics so you will not hear about this new law of physics for quite a while. Teleophilia is dual to teleophobia. Gravitation is equivalent or dual to acceleration -- Einstein's happiest thought, the principle of equivalence (duality). Energy is dual to mass -- Einstein. Dark energy is dual to dark matter. "Always two there are" -- Yoda. Once you understand the 4th law you will begin to realize that there is a 5th law:- The conservation of duality (energy), energy is duality, duality is energy, all energy is dual in physics. Potential energy is dual to kinetic energy -- gravitational energy is dual. Apples fall to the ground because they are conserving duality. Action is dual to reaction -- Sir Isaac Newton. Mind (the internal soul, syntropy) is dual to matter (the external soul, entropy) -- Descartes.

[@rishabharora4652]

Damn this channel is seriously underrated!!

[@Aleph0]

Thanks for stopping by!

[@TyronTention]

Definitely agree. I didn't even find this channel until yesterday.

[@technowey]

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

[@T3sl4]

As an electrical engineer, it's not even that hard. (Admittedly, if you don't work with signals, Fourier transforms may seem magical at first!) As a pun, it's a clever use; and when transformed, only continues to multiply. ;-)

[@hyperduality2838]

The time domain is dual to the frequency domain -- Fourier analysis. Stability is dual to instability -- optimized control theory. The initial value theorem (IVT) is dual to the final value theorem (FVT) -- Optimized control theory. The Laplace transform links control theory with modular forms or complex numbers. Elliptic curves link up with hyperbolic geometry in space/time diagrams in special relativity, physics that is why they are important. You can model light cones (photons) with elliptic & hyperbolic curves. Elliptic curves are dual to modular forms, synthesize Janus holes/points. Elliptic or spherical geometry is dual to hyperbolic geometry. Positive curvature is dual to negative curvature -- Gauss, Riemann geometry. Curvature or gravitation is therefore dual or gravitational energy is dual. Energy is dual to mass -- Einstein. Dark energy is dual to dark matter. Apples fall to the ground because they are conserving duality. Potential energy is dual to kinetic energy. Action is dual to reaction -- Sir Isaac Newton. The big bang is a Janus hole/point (two faces) -- duality. "Perpendicularity in hyperbolic geometry is measured in terms of duality" -- Professor Norman Wildberger. "Reflections preserve perpendicularity (duality) in hyperbolic geometry" -- Professor Norman Wildberger. Space is dual to time -- Einstein. Gravitation is equivalent or dual to acceleration -- Einstein's happiest thought, the principle of equivalence (duality). "Always two there are" -- Yoda.

[@Aleph0]

This comment is a winner.

[@zoltankurti]

@@hyperduality2838 that's a bunch of nonsense about physics piled together mixed with half truths.

[@hyperduality2838]

@@zoltankurti Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics. The big question in physics is what is dual to entropy? Teleological physics (syntropy) is dual to non-teleological physics (entropy). Duality is the gateway to the 4th law of thermodynamics! Energy is dual to mass -- Einstein. Energy is a dual concept and everything in physics is made out of energy or duality. Energy is duality, duality is energy. Potential energy is converted into kinetic energy by duality (gravity) -- apples. Gravitation is equivalent or dual to acceleration -- Einstein's happiest thought, the principle of equivalence (duality).

[@luis5d6b]

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

[@Aleph0]

Thanks!! Glad you enjoyed it :)

[@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!

[@Aleph0]

Thanks Luca! Glad you liked it.

[@charlesrodriguez6276]

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.

[@dicemaster5483]

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' !

[@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!

[@12jgy]

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)

[@Aleph0]

Thanks for the comment! (It's always super fun to read your comments, they're really well thought out.) I like your explanation for elliptic curves; it makes sense that they're a natural extension of Diophantine Equations. I've read somewhere that we have methods to solve equations of degree four and higher using topology, but I don't know much more than that. So I suppose that, combined with your explanation, narrows the search down to cubic equations. And elliptic curves appear to be the grand-daddy of cubic equations, for reasons I can't fully grasp. (PS: Thanks for suggesting the idea for this video.)

[@paulfraux2405]

@@Aleph0 I think the fact we can give a group structure to any elliptic curves has some think to do with the interest mathematiciens have to this subject. As more structure mean more way to studies an object, more way to approch a problem. It is not a definitive answer, but I am pretty confident that this fact plays it roles here.

[@zy9662]

@@Aleph0 Elliptic curves are the most important cubic equations because their solutions in the complex numbers form a space with one hole, the rest of cubics doesn't have any holes so they are topologically equivalent to quadratics, and in fact there are changes of variables that allow you to transform a cubic to a quadratic precisely only when their solutions form a space without holes. So if you take topology as the fundamental way to classify objects, then the degree of the equation is not that fundamental and is kind of masking the topology. Now, at the end of the day the solutions of an equation are a set that satisfy specific relationships between them, so elliptic curves define sets whose relationships are really of a different kind and more complicated than the sets whose relationships are defined by cubics that are not elliptic curves, and we can see this in the clearest way by the use of complex numbers and topology, which was a cool discovery

[@Crazytesseract]

They can't understand Dr. Wiles' proof, yet atheists ask for proof of God's existence, as if God is obliged to reveal His existence to them. One needs qualification (just as in mathematics, so also here).

[@physnoct]

@@Crazytesseract "yet atheists ask for proof of God's existence, as if God is obliged to reveal His existence to them." god is not obliged to reveal his existence, but if there's a god, how can we tell? Also, a god can't communicate with us without revealing himself. Right? Maybe we can't prove god's existence or not, but any verifiable claims by a supposed divinity allows us to dismiss a false claim.

[@user-uo5ro9xh9n]

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. You can check pretty readable note by Milne for the detailed discussion: www.jmilne.org/math/CourseNotes/mf.html 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. 6. The modularity theorem, proved by Wiles and Taylor, is an important breakthrough in number theory. There are a lot of mathematicians that are trying to find a generalisation of Wiles' idea and/or use it in a different way. They all, I believe, understand Wiles' paper in sufficient detail :) Apparently, some people (who are not Andrew Wiles :) ) even give graduate level courses on the topic: patrick-allen.github.io/teaching/f20-modularity-lifting.html. It does sometimes happen that there is a paper which is so difficult and obcure that almost no one can make sense of it, such as infamous Inter-universal Teichmuller Theory by Mochizuki, but Wiles' result seems to be very removed from this extreme.

[@some1rational]

gold star comment 👍

[@Jooolse]

"and observing that the image of the line is bx+dy=k(ax+cy) y=(ak+b)/(ck+d)*x." It looks wrong or I didn't quite understand the analogy.

[@user-uo5ro9xh9n]

@@Jooolse the signs are wrong, yeah, should be cx-d. That’s because lines are elements of the dual space really, and so they transform under change of coordinates in a different way, I guess

[@sukhjinderkumar2723]

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

[@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!

[@sahhaf1234]

That is an extremely well thought-out and prepared video.. You really have a knack in explaining things..

[@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.

[@p_square]

If KZbin is a library then Aleph 0 is one of it's shelves. Amazing video.... Keep up the good work!

[@amaarquadri]

Great video! I just found your channel and really like your content so far. Keep up the awesome work!!!

[@Aleph0]

Thank you!! Glad to have you join us :)

[@parklane4696]

Beautifully explained. Great video.

[@Entropize1]

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.

[@hyperduality2838]

It is much easier to answer some of these questions using physics:- The time domain is dual to the frequency domain -- Fourier analysis. Stability is dual to instability -- optimized control theory. The initial value theorem (IVT) is dual to the final value theorem (FVT) -- Optimized control theory. The Laplace transform links control theory with modular forms or complex numbers. Elliptic curves link up with hyperbolic geometry in space/time diagrams in special relativity, physics that is why they are important. You can model light cones (photons) with elliptic & hyperbolic curves. Elliptic curves are dual to modular forms, synthesize Janus holes/points. Elliptic or spherical geometry is dual to hyperbolic geometry. Positive curvature is dual to negative curvature -- Gauss, Riemann geometry. Curvature or gravitation is therefore dual or gravitational energy is dual. Energy is dual to mass -- Einstein. Dark energy is dual to dark matter. Apples fall to the ground because they are conserving duality. Potential energy is dual to kinetic energy. Action is dual to reaction -- Sir Isaac Newton. The big bang is a Janus hole/point (two faces) -- duality. "Perpendicularity in hyperbolic geometry is measured in terms of duality" -- Professor Norman Wildberger. "Reflections preserve perpendicularity (duality) in hyperbolic geometry" -- Professor Norman Wildberger. Space is dual to time -- Einstein. Gravitation is equivalent or dual to acceleration -- Einstein's happiest thought, the principle of equivalence (duality). "Always two there are" -- Yoda. Homology is dual to co-homology. Union is dual to intersection. Integration is dual to differentiation, convergence is dual to divergence. The dot product is dual to the cross product -- Maxwell's equation. Vectors are dual to co-vectors (forms). Concepts are dual to percepts -- the mind duality of Immanuel Kant. The intellectual mind/soul (concepts) is dual to the sensory mind/soul (percepts) -- the mind duality of Thomas Aquinas. Absolute time (Galileo) is dual to relative time (Einstein) -- time duality. There is a pattern of duality in fundamental physics which is being ignored! Duality creates reality! Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics.

[@pursuitofnumbers]

I plan to go on this same journey . It's been a year now, how is the end of year 2? Do you still estimate it will take 6 years?

[@BLACKHOLESFTW]

Thank you for your work. Absolutely exceptional exposition!

[@frankjohnson123]

This video is just oozing style in addition to being informative, fantastic.

[@shairozsohail1059]

Thank you for this, I never expected to be able to learn about math concepts at this level from youtube. I think two possible topics to make a future video on might be the (1) BSD conjecture (2) graph isomorphisms and history of NP problems

[@bartroberts1514]

What a great channel. By substitution and simplification -- algebra and Occam's Razor -- the hardest and most complex questions unlock with beautiful clarity. I believe this drills down to the roots of the question of why elliptic curves and their relation to modular forms spark joy and surfaces that this substitution makes easy the hard with internally consistent logic and reducing the plurality of posits to only what is needed to address the general question.

[@h92o]

I like that people are taking the time to answer your questions with complete sentences. Good work!! Question 4 is interesting for me. I have had similar thoughts around this. Mapping 2 complex numbers representing x and y has 4 dimensional geometry going on. Plus the two parts of either complex number have conditions that cause them to interact when "i" has an even power thus altering the 4-space mapping. I am sure there is programming that can be used to solve this, but the visualization is lost when we use time as the 4th dimensional part of the visualization. We can't see adding -1 in the time line when "i" is raised to an even power. That doesn't work for any time based visualization geometries.

[@felipegomabrockmann2740]

this channel is my new discovery, it is so amazing!!!!!!

[@zathrasyes1287]

Great! Keep up the spirit in learning more deeply!

[@christiansmakingmusic777]

I hope all your questions get answered. Thanks, your presentation brings me a little bit closer to understanding the arcane world of FLT.

[@tnorton314]

Just discovered your channel, love this video. Great work! Thanks!

[@franciscomackenney7664]

I have just started studying math subjects. I find your videos very entertaining. I will come back every year to this one until I understand it completely. Thx for the awesome content

[@robertschlesinger1342]

Excellent video. Very interesting, informative and worthwhile video.

[@ralfaralf6805]

Wow! What a treasure i have found with your channel! Thx a lot for the great content

[@Aleph0]

Thanks for stopping by!! Glad you have you join us :)

[@KStarGamer_]

Please do a video on the Birch and Swinnerton-Dyer Conjecture. I would love to see it!

[@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.

[@DjordjeRomanic]

I enjoy listening to your math explanations. Keep up the good work!

[@mehdinategh8876]

I'm new in this field and the video was like a briefing over three books and some thretise I had to read plus the time I spent on asking google to get what it is all about. Great job

[@sukhjinderkumar2723]

a lot of effort. hats off.

[@beauthetford7608]

Hey really good job with these, you're explaining these pure maths topics extremely cogently, ones that other people often make more confusing

[@mrlimemil]

Learning about elliptical curves on my own was quite difficult for me as well, I found the book Elliptic Tales- Curves, Counting, and Number Theory by Avner Ash and Robert Gross particularly helpful in terms of getting an intuition on the BSD conjecture. It also directly addressed some of the additional questions you placed in the description, maybe not the last one though.

[@Aleph0]

Hey emil, thanks for the comment! I just checked out the book on google and it seems quite interesting and accessible. I'll add it to the description.

[@TheSandkastenverbot]

This was unbelievably good!!!

[@factsheet4930]

9:20 The only thing I can think of for the second question is, obviously, security reasons! ECC (elliptic curve cryptography) to be exact, and the fact that it seems harder to find the discrete log than even factoring. And so ECC can allow us to have shorter length keys as apposed to RSA thanks to some people who cared about elliptic curves!

[@mathemagics266]

This is my second time watching this honest video which I guess the most clear short video covering the basics of the theory

[@Ceratops17]

I have a seminar presentation on sphere packaging (based on Viazoskas 2017 paper) next week and while it doesn’t sound related modular forms play a very important role in the proof of that theorem, I highly recommend having a look into that paper if that topic is of interest to you it’s called “the sphere packing problem in dimension 8”. And a huge thanks for amazing videos like this :)

[@lkocevar]

Got into a bit of a struggle midway but I get the idea. Kudos to you explaining 300 pages in a 10min video man! An also REALLY well!

[@NAANsoft]

At last I had a chance of understanding what was needed to prove Fermat. Thanks!

[@philipschloesser]

Really enjoyed this video, too! Have you ever looked into the relation between elliptic curves, elliptic functions, and elliptic integrals? A fascinating topic I have always wanted to give a talk about (and prob will soon) My two cents about your questions: 1: it is a nice result in complex analysis that elliptic functions (i.e. doubly periodic meromorphic functions on C, i.e. meromorphic functions that factor through C/\Lambda for a lattice \Lambda) can be represented as rational functions in a special elliptic function, the Weierstrass p-function, and its derivative, in a way that this Weierstrass function gives a (Lie group) iso between C/\Lambda and an elliptic curve determined by \Lambda, and conversely, every complex elliptic curve can be obtained this way. This shows that lattices in C in a way classify the complex elliptic curves. In a similar way, you could also define modular forms as functions on the space of lattices in C in such a way that f(a\Lambda) = a^k f(\Lambda): every lattice can be spanned by two vectors and we can rescale the lattice in such a way that one vector becomes 1 and the other has Im>0, and the SL(2,Z)-symmetry of our original definition ensures that the resulting function is independent on our choice of generators... So, modular forms give us a way of classifying lattices in C, and thereby elliptic curves. 2: You give the Weierstrass form of elliptic curves, and I agree that this is weirdly specific. Thing is though, that the definition you gave in the beginning gives a few more hints as to why they might be so prevalent: smooth, projective, irreducible... Ok, we want nice things... Genus 1, ah, there we have it. Ok, so generally, the genus of a projective curve is roughly related to how much nonvanishing global differential forms it has. A projective curve is a curve in projective space, which has a notorious behaviour when it comes to the existence of global objects, for example, locally there are a lot of regular functions (i.e. ones that can locally be described by rational functions), globally one can show that every (globally defined) regular function is constant (true on every projective curve), and similarly one could ask about differential forms (i.e. 1-forms). Now, there is a wonderful result called the Riemann-Roch theorem that says (in this special case) that 2-2g = \dim(O_X(X)) - dim(\Omega^1(X)) = 1-\dim(\Omega^1(X)), i.e. that there are 2g-1 linearly independent globally defined differential forms on our curve. Note that this means in particular, that g=1 is the lowest genus for which we have *any* global differential forms, and that it also is the only genus in which we have exactly one. And this means that we can probably do nice (co)homology. You can also show (using not that hard arguments) that if you pick a base point (for our Weierstrass curve that would e.g. be the point at infinity), then this induces a bijection of your elliptic curve with its divisor group, which is an Abelian group, so you can pull back the group structure from Div(X) onto X, and tada, you get that X is an Abelian algebraic group without ever having had to worry about if your whacky group law with intersections of straight lines satisfies associativity... 3:? 4: This group action is usually referred to as action by means of Möbius transformations (but now ones that involves discrete parameters) and it comes about by (as I mentioned earlier) requiring that your value not depend on your choice of basis. Usually, one thinks of them in terms of their generators: S=(1 1 \\ 1 0) and T=(0 -1\\1 0), which you pointed out correspond to translation and circle inversion (unless I'm much mistaken). 5: Usually people like to work with meromorphic functions more than just with sequences, as can also be seen by the constant urge to find generating functions for everything. I don't really know why that is, either, but another aspect to this might be that it gives you other ways of proving equality: if you want to prove that two sequences are equal you need to show that every term is equal, if you want to show that two meromorphic functions are equal you need to show that they are equal on some set with an accumulation point, which might be much easier (idk). Also, there might just be interesting other applications in this because your L-series probably doesn't converge everywhere, but you can analytically continue the function to a much larger set... Idk, it just reminds me a bit how we used the Zeta function of an operator to define an infinite-dimensional equivalent of its determinant. 6: lol, not gonna try

[@bobtheblob728]

this was great! really appreciated the detail on something so high level. what would have made it even better is if there was a way to use FLT as an example of the modular form to elliptic curve transformation to draw the contradiction you mentioned. it's p unintuitive so would have appreciated applying the concepts you mentioned to the problem at hand, as much as is understandable to this audience. but again super cool to see a channel talking about stuff at this level so clearly and engagingly, looking forward to more stuff :)

[@carywalker7662]

I'm stunned. That was amazing. Thank you.

[@Aleph0]

Thank you Cary!

[@papapu5001]

for your question 4: Geometrically, you can view the group acting on the upper half plane a rotation or translation in a hyperbolic plane with upper half plane model.

[@MadScientyst]

What a uniquely, wonderful presentation on an extremely difficult topic! I actually understood a great deal even with my minimal post-Grad Math level knowledge. Please keep up the great work, (from a new Subsriber).

[@bernhardriemann1563]

Your channel is one of the best math-channels, i found on KZbin. Iam real sad, that there are no math-KZbinr in german, which can only get close to your work. Greetings from Germany, nice video 👍

[@MrGiuse72]

Dear Aleph 0, regarding 5) my opinion: the big fuzz about L functions and the apparent over-using is motivated by the underlying effort of taming the final beast: the Riemann Hypothesis. All efforts, all approaches, all new discoveries always seemed to bring us closer to the proof but sadly failed to get the ever eluding result. So every topic is a good field worth exploring in the aim to get more insights.

[@cboniefbr]

GREATEST VIDEO ON THE TOPIC I'VE SEEN SO FAR. sorry for caps, but the video is really that good.

[@Aleph0]

THANKS!! (Hope you found it fun to watch!)

[@sergioaugustoangelini9326]

Bro this channel is gold

[@TesserId]

Added this to my playlist for fractals. Thank you.

[@roymiles6942]

Keep up the videos, these are soo good!!

[@gunjanlakhlani]

Great respect to you! Great videos! Absolutely amazing stuff. I wish you all the very best :)

[@KGrayD]

Hey ! Great video, very creative work, I enjoyed it a lot ! Regarding some of your questions (I am not a number theorist, but I am familiar with this), here are elements: 2. Indeed, if you look at an elliptic curve as this weirdly specific equation, then it is pretty hard to understand why they would come up at all ! The thing is that elliptic curves are the simplest example of "non-elementary" (ie. linear algebraic) geometric object. If you look at algebraic curves (ie. curves defined by polynomial equations) in the plane, the simplest example is that of a line, then of a conic (which is still a linear algebraic object), and then you get elliptic curves. And, as a matter of fact, mathematicians first encountered elliptic curves through the study of conics (whence the name). They can also be characterized more invariantly as smooth projective curves of degree 3 or smooth projective curves of genus 1 so the equation is not really what is important (although in practice it is very important). 4. The action of SL2(Z) on complex numbers is very much related its action on R^2 (or rather C^2 here). I will get to it in a minute, but for a geometric interpretation let's just say that SL2(Z) is generated as a group by the matrices [[1 1][0 1]] and [[0 -1][1 0]]. The first one acts as on IH as z -> z+1 and the second one acts as z -> -1/z (a sort of inversion). [Note that the first one explains the translation symmetry, and the second one explains the fractal symmetry in the patterns you point out in your video] Hence, any element of SL2(Z) acts as a succession of translations and inversions. What does our inversion do ? Take a nonzero vector z of length r, consider the vector on the half-line (Oz) with length 1/r (this is what would classically be called the inverse of z), then take that vector's reflection along the imaginary axis. This new vector is -1/z. Now what does this have to do with matrices acting on vectors ? The idea is to view C as a subset of its projective line P^1(C). This projective line is the set of lines in C^2 passing through the origin or, in other words, the set of homothety classes of nonzero vectors in C^2. Denote the class of a vector (x,y) as [x:y]. By definition, we have for any nonzero t [x:y] = [tx:ty]. In particular, if y is nonzero, we have [x:y] = [x/y:1]. Hence, we have an identification between C and the classes [x:y] of P^1(C) such that y is nonzero, ie. P^1(C) = C U {[1:0]}. Now, if Im(z) > 0, and M = [[a b][c d]] in SL2(Z), we have M.[z:1] = [az+b:cz+d] = [(az+b)/(cz+d):1]. 5. Indeed, on the face of it, the Dirichlet series contains no more information than the sequence, but it might be a more appropriate way to organize the information contained in the sequence and it might suggest new directions of inquiry. In other words, it is a different way of thinking about your sequence. As other comments have pointed out, the more structure you have the better. A Dirichlet series has poles and residues, it has an abscissa of convergence, it might satisfy functional equations. All of these things are uniquely determined by the sequence, so in a sense, they are information contained in the original sequence, and sometimes they encode nontrivial phenomena, but they are not *readily accessible* information if you just think of your sequence as a list of numbers. You probably will not come up the quantity corresponding to eg. the residue of the Dirichlet series at a given pole if your do not have the Dirichlet series at your disposal, even though that residue might contain important arithmetic information. Assigning a "generating function" to a sequence you wish to understand is a very general and multi-purpose idea. There are many types of such generating functions and not all of them are pertinent in all situations, but when you find the right one for your problem, it can really go a long way. As it turns out, Dirichlet series are very well suited to problems pertaining to multiplicative number theory.

[@tetraedri_1834]

Question 4: the action of SL(Z) is by Möbius transform, which are conformal maps (i.e. analytic bijections) from half-plane to itself. In fact, if we took action by SL(R) instead, we would get every possible such conformal map. Geometrically, Möbius maps are rotations of Riemann sphere. Riemann sphere itself can be projected conformally to the extended complex plane by stereographic projection. So you can think of the action as mapping the complex plane to sphere by inverse of stereographic projection, rotating the sphere by action of the matrix(*), and then projecting it back to complex plane. (*)I don't remember how coefficients of the matrix correspond to the rotation, but there should be some nice geometric picture of that too.

[@fattimiv]

I'm late to the game but this is a great video. I recently gave up on academia (in applied maths) and have been looking for a project to occupy that part of my brain, so thanks for helping to inspire me :)

[@aryanbhatt5835]

I think Ariana, who also commented, seems much more qualified than me to answer this (she said something about Indra's Pearls), but I'll give it a shot. I vaguely remember that when I was trying to gain a geometric intuition for the action of SL(2,Z) on complex numbers, visualizing where modular forms sent the fundamental domain was helpful. I've only ever studied algebraic NT very tangentially, so I hope I'm not misleading you! Thanks for the great video!

[@Moinsdeuxcat]

Let me share my opinion on the two questions you asked at the end. 1. The number sequences (or L-series) are not the crucial thing here. Galois representations are. You're looking at a shadow, which is why it seems random. The true story is: modular forms can easily be mapped to elliptic curves, and the étale cohomology of elliptic curves is a Galois representation. We have tools to know whether a Galois representation comes from a modular form (based on p-adic Hodge theory). It is not that hard to prove that on a finite field Fp the "residual" étale cohomology of an elliptic curve does come from a modular form, but to prove that every elliptic curve is modular you have to prove that the Galois representation over Zp is modular, which is done by the powerful theory of deformations due to Mazur, which creates a geometrical object (a local ring, i.e. geometrically a "fat point" with a tangent space and stuff) whose points are Galois representation associated to elliptic curves, and another geometrical object whose points are modular forms. He then uses *geometrical methods* (computation of dimensions, of cohomology, etc.) to prove that these geometrical objects are the same, which I find awesome. 2. All degree 3 dimension 2 smooth projective algebraic varieties in characteristic not 2 or 3 can be reduced to that "seemingly random" equation (Weierstrass form). So these actually are the simplest possible algebraic objects beyond polynomials (dimension 1) and quadratic forms (degree 2). Which is why they are essential! In degree >3 everything gets even messier, so we focus on the simplest case that we don't think we know enough about ^^ Just saw the additional questions: 3. Yes, this is actually the easy part. The full theorem gives an isomorphism between two classifying spaces (a Hecke algebra and a deformation ring).

[@joyboricua3721]

I'm just an enthusiast and this video gives a very good way to dispel some of the mysteries of such a famous problem. BTW, I do think Fermat had the proof in his head... I did not know him, but like you said in the intuition part, I believe he did have an outstanding computational skill that gave him a piercing insight.

[@manueldelrio7147]

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.

[@Aleph0]

Thanks for the comment! Your answer for the first question seems very reasonable. The two sequences were made in their own respective fields, so it is a surprise that they match. Intuitively, I suppose this means that "elliptic curves are modular forms in disguise." What confuses me is that we could have made the sequences in so many other ways (for example, picking N_p instead of epsilon_p for elliptic curves, or by considering the Taylor Series Coefficients instead of Fourier Series Coefficients for modular forms), so the Taniyama Shimura Conjecture seems so dependent on this particular setup. But I do have to agree with you that they are "natural" choices of sequences - they don't require too much tinkering. Thanks again for your thoughtful comment!

[@trhacprdeli6733]

@@Aleph0 Elliptic curves are widely used in computer cryptography. This website is secured thanks to the TLS protocol and its asymmetric encryption algorithms. There are many algorithms that rely on different specific elliptic curves ... see for example this: en.wikipedia.org/wiki/EdDSA#Ed25519

[@AsvinGothandaraman]

@@Aleph0 There are a broader set of conjectures called the Langland's conjectures (one for each positive integer and any number field). For n=1, it's class field theory. For n=2 and the rational number field, it's Taniyama-Shimura. In other cases, it gets even more complicated. If this makes the Taniyama-Shimura conjecture seem "easy", whatever that might mean, perhaps it helps to keep in mind that no one believed it when it was first made! It was not at all an obvious thing to guess.

[@ssvemuri]

Fantastic work and presentation

[@spencert687]

I notice some people have already given some answers to Q4 regarding Mobius transformation, SL(2, C) on projective line, etc, but there is another take which I think is more natural in the context of elliptic curves and might shed some light on your Q1 (in everything that follows I consider elliptic curves over C; one-dimensional abelian varieties--as such it will not be as immediately relevant to the issue of Fermat's last theorem, which happens over Q, but my main point is that modular forms and elliptic curves are deeply related, without needing to make reference to the modularity theorem). Take your elliptic curve E and use the Abel map to identify it with its own Jacobian, which is a quotient of C by a lattice with two generators w1, w2. These are given by the period integrals of the (unique up to scalars) holomorphic 1-form on E. The modular parameter of E is just the ratio \tau = w2/w1. Of course, other generators w1', w2' generate the same lattice iff they are related to w1, w2 by an SL(2, Z) matrix in the "usual" way, i.e. matrix acting on these guys stacked into a column vector. Translating this into the action on \tau recovers the usual action of SL(2, Z) on upper half plane (and indeed because the overall sign of the matrix is irrelevant when taking the quotient it is really an action of PSL(2, Z)). A more invariant way to say this is that the periods w1, w2 require a choice of basis in the homology group H_1(E; Z), which can be chosen to put the intersection form into canonical form (symplectic basis of cycles). Different choices of basis which preserve the canonical form are related by the action of the symplectic group Sp(2, Z) which turns out to be isomorphic to SL(2, Z). The amazingly nice statement here is that the ratio \tau, up to the action of SL(2, Z), in fact parameterizes E as a complex manifold of dimension 1 up to isomorphism--there's a holomorphic bijection (with holomorphic inverse) between two E's iff their values of \tau are the same up to SL(2, Z). H/SL(2, Z) is then the moduli space of elliptic curves (really it is a `moduli stack' but this involves a lot of technicalities), describing the space of all equivalence classes of them, and modular forms are just nice functions on the moduli space (really, sections of line bundles over it, because they are not quite single-valued under SL(2, Z)). This also touches on Q1--while I am not a number theorist, Q enthusiast, or particularly knowledgeable about the modularity theorem, elliptic curves and modular forms are in general objects which are already very deeply related because of the relationship between modularity and moduli of elliptic curves. The modularity theorem, in one of its formulations, is the assertion that any elliptic curve over Q can be rationally parameterized by a "modular curve", which is essentially just a covering space of the moduli space of elliptic curves (while I am of course making statements about complex manifolds here, much of this translates into the algebraic world where the ground field can eventually be replaced by Q). So we can (somewhat opaquely) phrase this as the statement that, over Q, elliptic curves are somehow parameterized by families of elliptic curves! Again, I want to emphasize that I know very little number theory, or details of working over Q, modularity theorem, etc, but just want to highlight some of these basic connections which already exist without invoking modularity theorem. Hopefully this makes it seem like a deeper statement.

[@craftycurate]

One of the best videos I've seen on this subject, as I try to get a grasp of it, from a layman's perspective. (PS There are regular loud clicks in the background - maybe something up with your equipment?)

[@antosha4812]

Thank you so much for this! Insanely helpful

[@ToddWildey]

There are a lot of really great comments on here as responses to the questions posted in this video. I think calling out the relevance of SL_2(Z) as integral Mobius transformations is key, as they are isomorphic to the Lorentz group. SL_2(Z) also underpins important dualities in string theory, including transformations of the axio-dilaton. Perhaps the most motivating reason for elliptic curves in my mind is their relevance in F-theory. Two of the 12 dimensions of F-theory describe a torus/elliptic curve that the rest of the theory interacts with. Different points in the 10 dimensions of spacetime determine different tori in these two dimensions of F-theory. The number of D7 branes present at a given point in space time determines the effective torus described in these two dimensions. One could think of the presence of D7 branes as units of flux. Furthermore, the geometry of the Calabi-Yau manifold in F-theory is determined by the D7 branes as well. At a point in spacetime colocated with a D7 brane, the torus described in these two dimensions by the D7 brane becomes singular. The equation for this torus is an elliptic curve, with coefficients in C, where the coefficients have a dependency on the torus' position in the Calabi-Yau manifold. I'm by no means a subject matter expert in this, but I am passionately curious about math and physics. I found most of this information in an excellent article by Jonathan J. Heckman titled "Particle Physics Implications of F-theory." www.annualreviews.org/doi/full/10.1146/annurev.nucl.012809.104532

[@Entropize1]

Answers to your questions (short versions, and keep in mind I am still far from being an expert): 1. The L-function of an elliptic curve is a connecting piece. Due to properties of geometric series, logarithmic derivatives, and zeta-functions, these give us the best way to summarize all the information about the F_q solutions to an elliptic curve E (and I know you know we care about solutions to curves in math!) Once you begin investigating modular forms, which are interesting in their own right, it becomes apparent based on the forms certain expressions take that there ought to be a way to associate a modular form to your L-function, hence to your elliptic curve. 2. First of all, keep in mind quadratics are very simple to understand, so naturally we consider cubics next. You can show that every cubic can be written in the form y^2 = x^3 + ax + b via nothing but linear changes of coordinates, hence it suffices to consider those when working with cubics. Now, if you know anything about uniformization, you know that every Riemann surface (generic dimension 1 complex manifold; something we can actually visualize) is biholomorphic to a sphere with n-handles, the cylinder, the complex plane, the disk, or some annulus. Of those, clearly the spheres with handles (otherwise known as tori/torus) are the most interesting. You can somewhat surprisingly show that every ordinary torus (one hole) is biholomorphic to the projectivization of some elliptic curve, and that every projective elliptic curve admits a torus. Hence, asking why we should study elliptic curves is essentially asking why we should study toruses up to biholomorphic group isomorphism. But there are many other answers: elliptic curves arise naturally when studying the arc length of the ellipse, for one, and secondly, they turn to be exactly the smooth, projective algebraic genus 1 curves with specified points (all the most generic and simple stupid requirements you could ask a curve to satisfy in projective space).

[@sb-qx6ew]

With regard to Q4, the az+b/cz+d transformations (i.e. integral Möbius transformations) come in 3 flavours, called elliptic, parabolic, and hyperbolic, determined by their number of fixed points. These flavours roughly correspond to rotations, translations, and dilations, by conjugacy. This means, for example, that if g is hyperbolic, then there exists another Möbius transformation h with (z -> kz) = h^(-1)gh (where the operation here is composition rather than multiplication). However, these rotations, dilations, and translations are really hyperbolic (in the sense of negative curvature geometry, rather than hyperbolic Möbius transformations) versions of these things. In our case, applying to the complex plane, it's the rotations that look a little weird. There's a similar action that can be defined on the unit disk (i.e. the Poincaré disk), which makes rotations look more like what we'd expect, but translations and dilations go a bit weird

[@QuadriviuumTremens]

Superb video ! Let me answer to the first question at the end of the video : where do the link between Fourier coefficients of modular forms and number of points in elliptic curves mod p came from ? 1 - It have an equivalent for the degree 2 : the number of solutions of quadratic equations mod p are given by the Legendre symbol, and they respect quadratic reciprocity. But morever, Legendre symbols respect multiplication relations, and so they can create interresting functions when put into Dirichlet series (analogs of the zeta function). 2 - When extended to the complex plane, Dirichlet series respect a symmetry relation, and the proof of this relation can be made using the symmetries of modular forms. And so we noticed that Fourier coefficients of modular forms can be made into a Dirichlet series. 3 - When we start counting the points of elliptic curves mod p, we had already the technique of incorporating these number into Dirichlet series (so called Hasse-Weil zeta functions). And so it was natural to ask whether these numbers can also appear in modular forms. This is the modularity conjecture and the answer appears to be yes. 4 - All af this is part of a greater picture, trying to show a relationship between the number of rational points of algebraic varieties, Dirichlet series and Fourier expansions of modular forms is a very active field of research today : this is the Langlads program. Now answer the second question : why are elliptic curves so important ? 1 - Elliptic curves are not a random example of cubic curves : it can be shown that every cubic curve can be reduced into an elliptic curve using projective transformations. So it is a tool to classify and investigate the general cubic case. 2 - There is a geometric way to associate to every couple of points in an elliptic curve a third point. This add a group structure over its points. This group can be used in cryptography, so elliptic curves have a lot of practical applications.

[@pinklady7184]

I love your video presentation. I can follow your tutorials.

[@Laz3rs]

Thank you for breaking this down to something understandable. I'd glance at the wikipedia articles everytime and give up. Finally I have some small idea of what exactly is going on.

[@benji104]

As a group theorist (not a number theorist) I must say that this video is by far the best introduction to FLT! Much much better than Singh's book for instance (albeit addressed to different readers). Btw, the Taniyama-Shimura conjecture has been proven in full generality in 2001.

[@funktorsound]

"I have so many questions, but this video is too short to contain them"

[@jezsabugo1186]

Sounds like it was intentional for fitting the theme of Fermat 😂.

[@k-theory8604]

Love this video, your channel is an inspiration for my own! Just fyi, I'm fairly certain the correct pronunciation of Weil is "Vay". Keep up the amazing work!

[@cristianandrei3154]

Great video!!

[@mahbodkhooblar2302]

Even comments of this video are educative. Number theory was my greatest interest but unfortunately my advisor discouraged me and told "that is wasting time, " but I can't forget it's natural beauty, so study it just in free time, and makes me happy

[@coolguy7676]

Love ur channel bro, keep it up

[@RajeevKumar-fu9jo]

Will rate your channel at the same level as Socratica .. in explaining complex math concepts simply and beautifully !

[@Aleph0]

Thanks @Rajeev, that's very high praise!

[@eric3813]

I Really Love your videos, i wantend to learn about Elliptic curves And modular forms, And now, i am Even more excited to learn it. I am Really thankfull for your incredibly good Content, you became my Favorite math youtuber.😄

[@eric3813]

I am 15 years old And learn a looot from books And math is Really my Passion. ( i learned for instance about Fibre bundles, homology/homotopy, differential Geometrie, Complex analysis, Algebra And stuff like that) For your fourth question, i think These matrices define the Möbius Transformations that you can Imagine like some stretched or squiched or Rotatef verison of inversion. And inversion is easier to Imagine than These abstract equations, but it's still not veeery Visual. I might ne wrong but that's what i reimender from my Complex analysis course. I Hope i could help you a tiny bit. As i Said, Thank you for your Content!😊

[@Aleph0]

@@eric3813 Ah yes! Mobius Transformations are probably it -- I'm learning complex analysis this semester, so I'll hopefully see the connection then. Glad to have a fellow math enthusiast on board!

[@theflaggeddragon9472]

I can try to give some answers, but I'm also learning this material myself! 1. Wiles' theorem, or the Taniyama-Shimura-Weil conjecture is an instance of the very general Langlands program for the specific group GL_2(Q). Consider algebraic varieties over a number field or p-adic field (algebraic geometry) and automorphic representations of a reductive algebraic group (representation theory). Attached to these are L-functions. One interpretation of the Langlands conjectures gives a correspondence between L-functions of algebraic varieties (such as the Hasse-Weil zeta function of an elliptic curve, a 1-dimensional variety of genus 1), and automorphic L-functions (the Dirichlet L-function of a modular form is an automorphic L-function for GL_2(Q)). In this context, Wiles' theorem is much less "random". 2. Elliptic curves arise in several contexts. On the one hand, they are complex tori, the quotient of C by a lattice. Associated to this latter however is the Weierstrass P-function. This function and its derivative are related by a cubic equation which is the *minimal model* of said elliptic curve (see Silverman's book). It turns out that this function mapping lattices in C to irreducible cubics is bijective, and these lattices are parametrized by the modular curve SL_2(Z)\H (technically you need to compactify). Here's one instance already of SL_2(Z) arising in the context of elliptic curves. Regardless, elliptic curves are also geometrically quite simple: they are one-dimensional smooth projective varieties of genus 1, and such varieties automatically have a group structure; they are *abelian varieties* of dimension 1. This is itself remarkable, and makes the study of the groups of K-rational points for K a number field, p-adic field, finite field, etc a natural avenue of research. Moreover, the fact that 1-dimensional abelian varieties admit minimal models defined by cubic equations is quite remarkable, but the full "why" story of the role of elliptic curves in number theory still isn't well understood. One last point is that the Tate module of an elliptic curve is super tractable, and can be understood by an undergraduate even though it's really the Etale cohomology of an elliptic curve, and thus gives "every" Galois representation of the elliptic curve. 3. I'm not sufficiently knowledgeable to answer this. 4. The action of SL_2(R) (not SL_2(Z)) is transitive on H. The isotropy group of i is SO(2), so SO(2) can be though of as the quotient of this action. SL_2(R) and H are also "model geometries" for 3-manifolds. This interaction preserves this geometry somewhat. SL_2(Z) is a discrete subgroup of SL_2(R), and while I don't know a good visualization, quotients of H by SL_2(Z) and it's congruence subgroups define modular curves which are the moduli spaces of certain types of elliptic curves (see Diamond and Shurman Section 1.5). 5. I don't quite understand the question, but the fact that there is a modular form whose automorphic L-function is the Hasse Weil L-function for an elliptic curve is one form of Wiles' theorem. 6. Working on this every day! Not very close XD Cheers.

[@Aleph0]

Gosh... this is so thorough - thanks a ton! I can't say that I have enough mathematical background to understand all the details, but I will say a few things. First, that your Langlands Program explanation totally connected the dots for me! The Taniyama-Shimura conjecture seems quite significant when you put in that context, linking representation theory, number theory, and complex analysis, etc. For the comment about elliptic curves, Silverman's book does explain that you can always rearrange a cubic to this "standard form", which I suppose is motivation enough. But why cubic curves? Is it because degree 2 is too easy and degree 4 is too hard? There's gotta be some historical reason why these curves are important ... but I can't figure out why. In any case, thanks for taking the time to type out such detailed responses!

[@jordanweir7187]

thanks for the vid bro rly nice stuff

[@PaulMurrayCanberra]

4 - geometric picture for SL2(z): There is only one thing in this video that I recognised: z' = (az+b)/(cz+d) . It's the formula for the mobius transformation. en.wikipedia.org/wiki/Mobius_transformation . The subgroup of mobius transforms where all the coefficients are real map the upper half plane onto the upper half plane, and that's significant for modular forms.

[@hyperduality2838]

The Klein bottle contains two mobius loops or spinors:- kzbin.info/www/bejne/f2S9nZuulrmSgdE Elliptic curves are dual to modular forms, synthesize Janus holes/points. Elliptic or spherical geometry is dual to hyperbolic geometry. Positive curvature is dual to negative curvature -- Riemann, Gauss geometry. Dark energy is dual to dark matter. The big bang is a Janus hole/point (two faces) -- duality.

[@sonarbangla8711]

Thank you Aleph 0, for this extremely interesting video on modular forms. I will have to watch the video again, later. In the mean time I was wondering if you could throw some light on the topic viewed from Ramanujan's perspective, he did a lot of work, perhaps during the time Hasse and Weil worked on it. However, did you place you questions to Tao? I don't think I can answer your questions, but have an idea. Tendulkar and Murali are the two of the best cricketers, one is a batsman and the other is a bowler. If you were to decide who is the all time best, I think modular forms might be of some help, but I am not sure.

[@priyansusingh1272]

Good content with amazing explained...

[@Aleph0]

Thanks!

[@peterhall6656]

I actually know someone who did their PhD in number theory under Wiles so I will send him a link of this video. I think you have done a good high level explanation.

[@IshanBanerjee]

Amazing sir 👍

[@algebra5766]

This is a great video!

[@MrOlivm]

Elliptic curves are a stunningly productive subset of Diophantine equations/geometry, which have been: trivial, one off puzzles, or insurmountable / without solution. I’m biased, or with limited view, having learned about them mainly for factoring and cryptography.

[@baqachaothmane]

You've just earned a sub

[@av5431]

The channel is super awesome.

[@ethanjensen7967]

The Fourier coefficients of modular forms have a special importance beyond what is described in the video here. They are often multiplicative, satisfy a certain second order recurrence relations, and grow in a predictable way. For the j function in particular, the coefficients are connected to many seemingly disconnected areas of mathematics including the classification of finite simple groups.

[@WilliamDye-willdye]

Thank you for going over my head, but not out of reach.

[@euclid1618]

Dude so good!!!

[@rishiraj8738]

Was in a way waiting for some maths inspiration. Thanks 👍

[@Aleph0]

Hope elliptic curves leave you sufficiently inspired ;)

[@conor3850]

Well done