The bridge between number theory and complex analysis
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- Опубліковано 31 тра 2024
- How the discoveries of Ramanujan in 1916, combined with the insights of Eichler and Shimura in the 50's, led to the proof of Fermat's Last Theorem.
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SOURCES and REFERENCES for Further Reading!
This video is a quick-and-dirty introduction to modular forms and elliptic curves. But as with any quick introduction, there are details that I gloss over for the sake of brevity. To learn these details rigorously, I've listed a few resources down below.
(a) ELLIPTIC CURVES
The book "Elliptic Curves: Number theory and cryptography" by Lawrence Washington is really good for self-study. It also has tons of numerical examples, making it good for self-study. The subject only really clicked for me after I read this book, so I'd highly recommend reading it.
Professor Alvaro Lozano-Robledo also has a wonderful UA-cam playlist which explains all the details of the subject rigorously: • An Introduction to the...
(b) MODULAR FORMS
For modular forms, a great book is "Modular Forms: A Classical And Computational Introduction" by Lloyd Kilford. It also has plenty of numerical examples and you can also code up a bunch of the sections as well, which makes it nice to work through.
(c) EICHLER SHIMURA THEORY (how to go from modular forms to elliptic curves)
The book "Elliptic Curves" by Anthony Knapp (see Chapter 11: "Eichler Shimura Theory") contains the main content of this video with the integrating and lattices and all that. This is a dense book, but it is really beautifully written. The first chapter contains an extended numerical example that illustrates how to go from modular forms to elliptic curves. I wouldn't read this book as an introduction, because it's very comprehensive and can be a little overwhelming. But rather, it's great as a second pass after reading the intro books I mentioned at the start.
(d) STRATEGY OF WILES' PROOF (how to go from elliptic curves to modular forms)
The book "Elliptic Curves, Modular Forms, and the Proof of Fermat's Last Theorem" edited by John Coates and ST Yau has a full rigorous explanation of Wiles' proof in the first chapter. It is very dense, and it requires a solid grounding in algebraic number theory (see the last video in this channel for resources to learn this). The chapter describes all the new techniques that Wiles invented essentially from scratch to tackle Taniyama-Shimura. The key to Wiles' approach was a technique called a "modularity lifting theorem". This is not easy reading: it is aimed at graduate students and researchers in number theory. But it is beatifully written and by far the clearest rigorous exposition of FLT I've seen so far.
If you really want to know: what are the 'curved arcs' from 2:40? The rigorous definition is: the "curved arc" is really a geodesic connecting two cusps that are equivalent under the action of Gamma_0(11). Equivalently, it is a homology class (with integral coefficients) in the modular curve X_0(11). These are the details you would find in "Elliptic Curves" by Knapp, see part (c) above.
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MUSIC CREDITS:
The song is “Taking Flight”, by Vince Rubinetti.
www.vincentrubinetti.com/
THANK YOUs:
Extra special thanks to Davide Radaelli and Grant Sanderson for feedback and helpful conversations while making this video.
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Intro: (0:00)
Eichler-Shimura: (2:04)
From Lattices to Number Theory: (3:21)
Counting Solutions: (5:00)
Taniyama-Shimura: (7:21)
It's hard to imagine that Ramanujan just stumbled upon modular forms. There was profundity in everything he touched, even when he didn't realize it.
He truly was a prodigy. Imagine what he might have discovered if he didn't die so young and senselessly.
@@Red-Brick-Dream an anomaly of nature is what I can best describe him from my baseless reminiscence of his works through books
I'm a middle-aged guy who's been studying this stuff at a very amateur level for the last couple of years. It's so much fun and feels like learning the secrets of the universe. I just wish I were young and could study this in grad school... if there's anything left to study. :)
Number theory (and complex analysis, by extension) has a LOT of unproven conjectures and open problems, there is definitely more to study!
Mathematics is a young man's game because older men think mathematics is a young man's game.
Mark my words: keep at your "amateur" study, otherwise one day you will find yourself saying, _"I'm an old man...I just wish I were middle-aged..."_ 🤓
There is a 75 year old man in my grad school. You’re definitely not too old.
I am also interested in the relation between number theory and complex analysis.. Can u please share ur telegram number Or email so that we can discuss about it??
Don't u DARE give up friend! I went back to school as a challenge at 36 & though I'm 52 now, I have 2 Math /Applied Stats Degrees & hoping to at least get a Master's once the $$ can be balanced out. It was demanding & I did FAIL /repaeat some higher Analysis courses, but it was all worth it...Math is simply too BEAUTIFUL for me to not keep trying...LOL
Ramanujan’s intuition for number theory was astounding and terrifying.
i feel like mine is just as good. just the low hanging fruit has all been plucked.
@@sharpnova2this is not low hanging fruit.Most math after the last 100 years isn't. Especially for its time.
@@sharpnova2 make sure to check that self-diagnosis dillema of humans
@@sharpnova2 💀
@@sharpnova2😭😭😭😭
This is the first time I've actually had the connection between elliptic curves and modular forms explained. I had always wondered about why elliptic curves were used in encryption, and learning of this connection explains why. I had understood that RSA used large primes, and that this was connected to a modular form. So this video helped me see the connection between RSA and ECC. Fascinating.
I absolutely love videos that are capable of giving me these sort of insights. I do read a lot of Wikipedia articles and other sites when looking into math, but just reading words and starting at an equation simply cannot replicate the sort of insights I make during the course of a video such as this.
I'm not an expert in this area, but my understanding is that elliptic curves are used in cryptography because they have a group operation x.y -> z which is easy to compute in one direction, but not in the other. The connection between elliptic curves and modular forms is not relevant to the use of elliptic curves in cryptography. (Wikipedia link: en.wikipedia.org/wiki/One-way_function#:~:text=An%20elliptic%20curve,compute%20k.)
This video explains one particular theme very well in which Ramanujan's 1916 paper called " On certain Arithmetical functions " had an impact on the eventual proof of Fermat's Last Theorem, through the works of Hecke , Mordell , Eichler and Shimura and later by Deligne who proved one of Ramanujan's Conjecture by associating to that Delta Modular form of Ramanujan a Geometric object called a Motive ( à la Grothendieck ).
I'd like to mention another theme from that 1916 paper which is even more directly related to the Wiles' proof of FLT.
On one hand where Ramanujan conjectured some direct properties of his Delta Modular form - like the fact that its coefficients are multiplicative, he further proved some congruences related to the coefficients of Delta which were extremely bizzare and completely unexpected at first. One such congruence which occurs in that paper was - ' tau(p) is congruent to 1+p¹¹ modulo prime 691 ' , here tau(p) is the pth coefficient in the expansion of delta.
It was Jean Pierre Serre who realized that there has to be some reason behind these congruences and their existence. To explain these, Serre discovered a huge set of ideas - he developed the notion of what's called p-adic and mod-p modular forms, related them to ' mod-p Galois Representations ' ( an extremely important tool in modern Number Theory ) , gave a new definition of what's called p-adic Zeta function which is itself related to an old approach of Ernst Kummer to prove Fermat's Last Theorem whenever exponent in the Fermat's equation is a ' regular prime ' and lastly , while he was trying to explain Ramanujan's congruences , Serre formulated what came to be known as " Serre's Modularity Conjecture " , he further deduced Fermat's Last Theorem directly from his Conjectures without having to take the middle step of using Shimura-Taniyama Conjecture and later it turned out if we actually do wish to take that middle step then a small part of Serre's original Modularity conjecture would suffice to prove the implication :- Shimura Taniyama => FLT , Serre called this small part " Epsilon Conjecture " and that's exactly what Ken Ribet proved thereby paving a way from Shimura-Taniyama to FLT.
Infact it doesn't end here yet , both - A sophisticated version of the theory of p-adic modular forms as well as a proved special case of Serre's conjecture, which Serre developed to explain Ramanujan's congruences was used by Andrew Wiles himself in his ' Modularity Lifting Criterion " which was the most important step in the proof of Shimura-Taniyama Conjecture.
So overall , the influence of Ramanujan on the proof of Fermat's Last Theorem is much more than we think it is.
Edit: By the way the full " Serre's Modularity Conjecture " is now a theorem of Chandrashekhar Khare.
Man, I love reading your comment even though I couldn't make out the head and tail of what you're commenting.
Thank you for this comment. It's very insightful and adds to this excellent video.
@Prakhar Pratyush do you know who discovered that the coeff of the modular form expansion are related to the number of solutions mod p of an elliptic curve, and how on Earth he or she came to that discovery?
@@zy9662 There's actually a long tradition of people trying to find correspondence like these which are often called " Reciprocity Laws " starting from the work of Gauss on Quadratic Reciprocity to the work of Emil Artin on his Artin Reciprocity law which gives a similar correspondence for Polynomial equations in one variable whose Galois Group is abelian, so it's not something out of nowhere.
The Reciprocity laws related to Elliptic Curves were a next step in this tradition first started by i guess Martin Eichler and Goro Shimura as explained in the video itself.
Given a specific sort of Modular form there's a way to attach an Elliptic Curve to it and it can always be done but the converse was formulated first by Yukata Taniyama and Goro Shimura and then refined by Andre Weil.
@@pursuingstacks yes, but the reciprocities of Gauss until Artin are more about mapping of groups, but identities between integer sequences are a different breath altogether. Modular forms had been around in number theory way before Eichler paper and I was wondering if it was Eichler who found the correspondance or maybe Hasse.
My master's thesis used a lot of modular forms. This was a really nice historical perspective and transitioned so cleanly from the definitions to their impact
Thank you! Another gorgeous video. Your content is always really well done, and strikes a really, really impressive balance between parsimony and depth of insight. Please keep up the great work! Looking forward to the next one!
Thanks Dan! Hope you enjoyed the video :)
@@Aleph0 hello sir ,i want to approach you please tell email......
I want to become a mathematician ...🙏🙏
As someone currently studying arithmetic geometry and proof of Fermat's last theorem, this is simply fantastic. You really captured some of the special meaning of the Eichler-Shimura theorem and modularity simply, and in just 10 minutes no less! I'm astonished!
1 year ago yt recommended me the video about derivatives, it's still hard for me but it sparked my interest in maths. Great job on the videos. I hope more people will see them in the future.
I love how you take these very deep technical ideas and show me just enough of them to go wow!
One of the most beautiful and aesthetically pleasing videos in have ever seen. I knew the maths, I knew the journey, but your video, both concisely, logically and especially visually was an absolute delight.
Thank you! This gives a nice non-technical intro into a seemingly difficult subject. I'm tempted to redo some of the calculations that you presented.
Wonderful, simple, and intuitive introduction to FLT and modular forms. Your videos never disappoint!
Hey, just wanted to say that you’re page is freaking awesome. Keep up the great work. That is all.
That was so freaking beautiful I'm astonished. Thank you so much for making this video. Wow, just wow.
Your channel won’t stop getting better. No other channel makes videos like yours, and any one that tried would be facing an uphill battle to compete with you.
Very simplified, but accurate. I hope these videos will stimulate some people to take up the so called hard maths. Very rewarding. Looking forward to more..
I love your detailed and visually appealing explanations. Great job!
The entirely unintuitive triumph at 6:12 literally made me shout in disbelief. It’s astounding that this isn’t an elaborate gag. Thanks for the great warm-up video.
I like how you show the genius mathematical intuition of these peoples (even if these ideas took them years to conceptualise).
Thank you for your videos!
Very nice! Your videos are very engaging. I have decided to support you in Patreon. Best wishes for you and your channel!
At this point of time, I think I can safely state that this channel is (at least in my opinion) one of the best mathematics related channels on UA-cam.
Btw awesome video!
Ramanujan is one those few people I just marvel at. He was on a unique plane from all others.
Very nice overview of a rather complicated and daunting relationship. I look forward to your self-study course on elliptic curves and cryptography.
It was a concise explanation that provided me with a high level overview of the subject. Thanks.
P
Your integration(pun intended) of history, small facts and you not fearing of losing viewers because of big maths is really refreshing. We need ofc those dumbed down channels but we also need this, this is like a proper documentary made by BBC that shows the history of a place, showing it's founders and stuff or an Animal planet show etc.
Such great job
This has hot to be one of the all time best math videos I have ever seen.This is the first time I have been able to understand the connection between modular forms and elliptic curves
Yes, I knew that RSA was connected to modular forms, and that elliptic curves are commonly used in cryptography, but I wasn't aware that they are both connected through modular forms.
Very nice historically grounded exposition. UA-cam puzzles me given that someone who is conversant with the fine details of many theorems mentioned is likely watching and yet there is in general no middle ground between light popularization and heavy research talks on this platform. Mathematicians are shy creatures, perhaps this furnishes an explanation, not to mention that UA-cam is somewhat of an ad riddled commodification machine that seems almost designed to repel serious mathematical discourse. Perhaps we can aim for a sort of digital common room in the distant future. Science shouldn't be confined to the limits of earshot.
!Yes
You've done it again man, amazing explanation
This is breathtakingly amazing! Thank you!!!
Your content is really unique. Thank you so much for sharing it. 🙂
Beautiful video, as always!
You uploaded after a long time!
I was hoping you’d go into HOW these conjectures were proven, but everything else was really well explained! Good work 👍
beautiful observation by sir srinivasa ramanujan 🙏
So f(q) is like a very sophisticated generating function. Amazing!
sublime clarity, thank you
well made video man,i really enjoyed it .
One of the most interesting channels.
That was beautiful. Thank you for your videos.
I've always wanted a video that gave a little insight into modular forms that didn't immediately soar above my head. They always come up in all sorts of places and nobody ever even tries to give the faintest intutition...
Thank you.
I'm actually doing my Ph.D. on a somewhat related topic (transfer operator techniques on hyperbolic surfaces, which can be linked to automorphic forms of which modular forms are a subset. There they appear as building blocks for eigenfunctions of the Laplacian). My impression as to why nobody seems to bothers to give an intuition about modular forms is that nobody really has one. People study them because they are quite peculiar things and seem to hint on profound hidden meaning whenever given the slightest of chances to do so. But apart from that, as far as I know, they remain quite mysterious things. They're like the Dark Matter of math.
@@lonestarr1490 They certainly are mysterious! The closest intuition I've found to modular forms being "natural" objects is that they are sections of line bundles on modular curves, and modular curves are moduli spaces of elliptic curves (which are definitely natural objects!). The way we study spaces in algebraic geometry is by understanding the functions that live on them, i.e. sections of line bundles (or vector bundles, more general sheaves etc.) so in this sense modular forms arise "naturally". Why this in turn has so many applications to complex analysis, hyperbolic geometry, sphere packings, quantum groups, etc. I have no idea, and I have to agree with your perspective that they're just the "Dark Matter of math"
Thank you, great video! Please make more of them : )
Great video indeed. Just watched a Terence Tao video few hours ago. I am lacking complex geometry and various maps knowledge. Elliptic curves occur in other number theory domains, for example congruent numbers, and even theta-congruent numbers, where the elliptic curve has a specific form. I am working on mathematical billiards and they occur there too.
Excellent 👏🏻. And thanks for bringing this up.
How can there be more than 2^2=4 solutions mod 2 at 5:45?
I am not sure if this is correct, but note that an elliptic curve includes the point at infinity - it works in projective coordinates.
Thanks!
@@garethma7734 You are correct, but additionally the equation is singular when reduced mod 2, 3, or 11, so there's some special-case handling needed for those primes.
5:53 Isn't the next prime 11 and not 13?
It could be that it's a modular form of level divisible by 11. The theorem excludes a finite set of primes, namely those dividing the "level" and the auxiliary prime of your Galois representation. However, the pattern holds for all remaining primes.
Indeed, it's the unique cusp form of weight 2 and level 11. www.lmfdb.org/ModularForm/GL2/Q/holomorphic/11/2/a/
See another similar qn here for a very thorough answer.
Your videos are really good!
This was a great video. On a side note which I think is super cool, I saw the picture of Conrad at the end and could have sworn it was my old Calculus 2 professor (Keith Conrad). Turns out they’re brothers, crazy. They’re nearly identical
This is a great video. I believe a lot of patters and corollaries we see in mathematical disciplines has to do with the nature of computation. Any multiplication or division operation in a non binary numbering system generates a symmetrical form of operations and ultimately creates prime numbers. It will be cool when people start to crack some of the toughest problems in the fields. It's good to see that curiosity isn't dead!
I'm curious where the 24th power in the first equation comes from. I know that specific equation only serves as background to help introduce the problem, but whenever I see a specific equation with seemingly random constants in them I always wonder where the constant came from.
It's a very important 24 :). The simplest definition of the modular discriminant function (the delta) is delta = g2^3 - 27g3^2, where g2 and g3 are essentially the coefficients of the elliptic curve attached to a modular form (or lattice). Then, g2 and g3 are essentially Eisenstein series; g2 is a sum of -4th powers, and g3 is a sum of -6th powers. So delta ends up being something like a sum of 12th powers. Those are 12th powers of tau, a variable in the upper-half plane; moving tau to q (the "nome") in the unit circle throws in another power of 2. Thus, an overall power of 24. (There's much more, like that delta is the first nontrivial *cusp* modular form, and appears with weight 12, which then becomes 24 after converting tau to the nome. And I think the 24 can also be traced back to the special lattices formed by {1,i}, which gives a factor of 4, and {1, e^{2pi/3}}, which gives a factor of 6. Then 4*6 = 24. I think 4 and 6 are also the nontrivial conductors of imaginary quadratic fields, or something like that.)
@@pinkalgebra Thank you for the easy to understand explanation! I was also wondering why in the world 24 of all numbers was used, but this makes a lot of sense.
@@pinkalgebra well my last comment got deleted for linking to, of all things, a stackexchange post about numerically solving this stuff in Java...
A lot of that initially went over my head, and I had to do a ton of digging through Wikipedia and other sites until I managed to piece together a semblance of understanding, but I'm beginning to understand what you're saying.
I really appreciate the reply, as it has led me to quite a few interesting things to read, and has definitely helped point me towards a better poverty understanding of the topic.
I mentioned in another comment here I'm self taught beyond pre-calc and the power rule for solving derivatives (and I have a tendency to not follow the standard order school trends to teach things, often diving into the deep end and trying to piece together an understanding of things). So while I still don't fully "get" the math, I can see where the numbers come from a bit better.
I love your channel.
I love your channel.
I love your channel.
It was really interesting. Thank you.
How can there be 5 solutions mod 2? There are only 4 possibilities for the pair (x, y), namely (0,0),(1,0),(0,1),(1,1). I'm confused...
Good point :) The solutions are viewed in projective space, so there’s also a “point at infinity”.
I agree @omri, and also, y = 1 does not work mod 2, since the equation reduces to y^2 = 0 mod 2.
This is very, very, very, very, very, very, very, very, very, very, very, very good!!!
Amazing as always ❤
@ 5:54 how are there 5 solutions mod(2)?? There are only 4 possibilities?
See another same qn. It involves points at +/- infinity.
In what way do the arcs correspond to the other lattice coordinate (other than the integral one).
Thank you for this great video.
Brilliant video. Make more on the same topic. Thank you
Beautiful
Thank you so much for your videos
Great video!
Awesome video!
Great explanation! I'm a bit confused at 5:43 though. For p = 2 how could there be 5 solutions if there are only 2^2 = 4 possible ordered pairs mod 2? Aren't the only solutions (0,0) and (1,0) since the right-hand side is always even?
I was looking at that too
I'm also confused about this :/
A few seconds before this, he also says there are 4 solutions for p=5. Then here in the chart he says 5 solutions for p=5. I’m lost.
I think the right elliptic curve is y^2+y =x^3-x (you may modified to make it the normal form) and the number of solutions should be modified to be +1(th infinity point is included)
Correction: y^2+y =x^3-x^2 should be the right curve
The legend is back 🔥🙌
Clear and concise! I'm a new subscriber :) Just curious, what is the 'so what?' of this finding? I'm sure it's a broad impact connecting modular forms to elliptic curves, and it's connection to Fermats last theorum, but what would be a prominent example of the use of this discovery?
Always wonderful
I don't understand the maths, but I sort of get this from your wonderful video!
Very nice video, thank you!
Great video! But I still don't understand how you connect the delta(q) (e.g. delta(q) = q[(1-q)(1-q^2)(1-q^3)...]^24), the resulting lattice to elliptic curve equation then the function f(q) = q - 2q^2 - q^3 +2q^4+ q^5+2q^6-2q^7-2q^9-2q^10+q^11+4q^13....). Where does the last f(q) polynomial expansion come from relative to the first? I get that the solutions (1+p-) (# of solutions mod p) reveal the same coefficients as this equation, but don't grok where it comes from as one steps through your video. Seems to pop out of nowhere? (e.g. the f(q))
did you have a lot of other videos on this channels that you deleted? or another channel?
At 6:30, would this also hold for p=11?
Will you make more videos about differential geometry?
Does multiplicativity only hold for coprime exponents? What's the meaning of the prime power coefficients?
(-2q²)(-2q²) ≠ +2q⁴
(-q³)(-q³) ≠ -2q⁹
EDIT: probably also related to the number of solutions in a ring. That would also explain the multiplicativity, comes from ring factorization. But which ring? Cyclic? Galois field?
Such a great topic! I've been just thinking about how you can tell the GCD of two numbers by looking at where two sine waves with the period of those two numbers meet at 0!
That has got to be one of the least satisfying ends to an aleph null video ever. Where can I find more?
Also, if I may ask a few questions, what is your specific area of study? What schools did you attend? And where do you find your articles and sources for your videos? Also how did you get so good at explaining these harder or more advanced topics?
There are numerous textbooks on modular forms and elliptic curves. I started with Tom Apostol's number theory books, which has modular forms as a subject, but doesn't really do modular forms. There's a book by Lawrence Washington on elliptic curves that I'm finishing up now, which is probably my favorite intro to elliptic curves, and does only a little bit on modular forms (though a lot on the Weierstrass p-function). So those are some places to find more. You can also just read the Wikipedia and/or Mathworld articles on the things mentioned. Have fun!
@@pinkalgebra thank you very much!
beautiful 💕!
but how can we find the no.of solutions mod p 🤔?
Great video, as usual! I have been fascinated with modular forms since I started reading about them in popular summaries of FLT and Wiles' proof. What are the steps towards getting deeply acquainted with them mathematically? I imagine an undergraduate course in Complex Analysis would be the starting place, wouldn't it?
Some complex analysis is required, but not that much. A fair amount of number theory is required. I suggest checking out Tom Apostol's textbook on modular forms. I believe he gives you the pieces of number theory and complex analysis that you need, and you can freely skip over the more technical details of stuff like contour integration which is required for some proofs. The main skill required may be being good at manipulating infinite series and infinite products. Good luck and have fun!
i missed the part where we'd taken x and y to be some functions. where does this come from?
To me was very educative. Thanks a lot
Dear Aleph,
It's my impression or you removed some content of your channel? If yes, could you tell which videos are not available anymore?
Can the analogy with the x^2+y^2=r^2 (1) case be extended? So how many Pythagorean triples with the same r^2 do exist? What would be the analog „modular form“ if we take (1) to be the „elliptic curve“?
This is a good math channel
Where does the modular form you're using in the video as an example come from?
It's the unique cusp form of weight 2 and level 11. See www.lmfdb.org/ModularForm/GL2/Q/holomorphic/11/2/a/
Also Diamond and Shurman's _A first course in modular forms_ has an exercise showing that S_2(Gamma_0(11)) is 1-dimensional (Exercise 3.5.4(b) in the chapter on dimension formulas).
There is actually another connection between a lattice and number theory. If you define a linear, recursive congruence of the form r_{i + 1} = (a*r_i + c) mod M, where a, c and M are constants, then plot successive elements of the sequence as points on a plane (x, y) = (r_i, r_{i +1}) you will see a lattice. If you choose c=1, a=4, M=9, and a starting value of r_0 = 3 you get the sequence 3, 4, 8, 6, 7, 2, 0, 1, 5, 3... and so on repeating. You then will have to plot (3, 4), (4, 8), (8, 6)... and so on. This example is taken from Computational Physics Problem solving with Python by Rubin H. Landau Manuel J. Páez
, and Cristian C. Bordeianu, chapter 4.2.1. It is presented as the linear congruent method for finding pseudo-random numbers. In the book it is given as a bad example of one, considering that the series repeats quite early. I do not know how you would relate this further with elliptic curves, nor if it has been done already.
So, a few questions that immediately spring to mind: Of course for any elliptic curve you can just make a list of all solutions mod all the p's and then write an f(q) with those coefficients, so...
1) what has to be done to prove that this is modular (like, is being modular any more special than looking like what you showed)
2) What has to be done to show that f(q), integrated over all the 'special arcs', gives a lattice which can be fitted periodically by a set of functions (x and y) satisfying the original elliptic curve ( y = f(x)) that it all came from?
In other words, are these things comparably difficult to prove as all the other difficult things that you mentioned people were trying to prove here, or have I missed some 'obvious' logic?
Did I miss something. When converting the equation for the elliptical curve to the modular equation, we find the coefficients by counting the solutions mod p to the elliptical equation. But when do we stop? E.g., in the example you have f(q) = q^2 - 48 q^3 + 1080q^4 .... is that supposed to be an infinite series or does it terminate at some exponent of q? Is there some n for which the elliptic equation has no solutions mod p, when n>p?
Great video! I am a bit confused on your creation of the lattice. It is clear to me that you are integrating the function over the geodesic of the hyperbolic plane, but you say in the description that you are only integrating those geodesics invariant with respect to the action of \gamma_0(11). Is this the fundamental group of the surface of genus 11? If it is, I find this very strange. Why 11?
The 11 is just due to the fact that the specific modular form has “level” 11. For a different modular form, you could get a different level. Level 11 means that the modular form satisfies a nice relation under the action of the group \Gamma_0(11), which is the group of 2x2 invertible integer matrices that are upper triangular modulo 11. This group also acts in the hyperbolic plane via mobius transformations.
great video
Incredible!
Thanks for sharing
I love this!
My legend is back :)
Wow… studying this closely.
Are there any relations or info regarding the intermediate coefficients?
For example at 0:43 you say -24*252 = -6048.
Then soon after, *skipping* -1472 you say -24*4830 = -115920.
And that's b/c the place/index of the coefficients, ordered, need to be coprime.
So what about other coefficients?
❤thank ya ,finally it makes sense to me
Maybe i am missing something, but what about the coefficient of q^11 ? (At 6:52 of the video)
Nice video, given I know nothing about this, I could follow it pretty well. I just need to clarify, what exactly is a modular form? It wasn't explained so clearly.
Awesome video! Within the next year, I'll be explaining FLT in detail on my channel over a very long series of videos, for those interested.
I may have to check your videos out. I've been wondering what the connection to the torus is, so hopefully your second video will be able to provide that insight.
@@Bobbias there's some wild stuff going on there. For FLT, I'll be doing a series of lectures much like for the Congruent Number Problem, but then I'll also turn everything into a much longer series of refined, shorter videos (10-20 min each) with a little higher production quality correcting any mistakes from the lectures.
Yaaay new video!
I've seen videos by Richard Borcherds about this stuff. I didn't understand those either.