The bridge between number theory and complex analysis 103,348 viewsApr 14, 2022

The bridge between number theory and complex analysis 103,348 viewsApr 14, 2022 5.4K DISLIKE SHARE DOWNLOAD CLIP SAVE Aleph 0 128K subscribers 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. Help fund future projects: https://www.patreon.com/aleph0 An equally valuable form of support is to simply share the videos. 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 YouTube playlist which explains all the details of the subject rigorously: https://www.youtube.com/playlist?list... (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. ----- MUSIC CREDITS: The song is “Taking Flight”, by Vince Rubinetti. https://www.vincentrubinetti.com/ THANK YOUs: Extra special thanks to Davide Radaelli and Grant Sanderson for feedback and helpful conversations while making this video. Follow me! Twitter: @00aleph00 Instagram: @00aleph00 Intro: (0:00) Eichler-Shimura: (2:04) From Lattices to Number Theory: (3:21) Counting Solutions: (5:00) Taniyama-Shimura: (7:21) Chapters Intro 0:00 Eichler-Shimura 2:04 From Lattices to Number Theory 3:21 Counting Solutions 5:00 Taniyama-Shimura 7:21 208 Comments rongmaw lin Add a comment... Dan Greenwald Dan Greenwald 3 weeks ago 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! 94 Aleph 0 Bobbias Bobbias 3 weeks ago 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. 25 pinkalgebra pinkalgebra 2 weeks ago 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. :) 18 Prakhar Pratyush Prakhar Pratyush 3 weeks ago (edited) 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. 50 Unlucky Phi Unlucky Phi 3 weeks ago 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. 57 Bobbias Bobbias 3 weeks ago 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. 19 Yakari Dubois Yakari Dubois 3 weeks ago 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! 21 Aditya Khanna Aditya Khanna 3 weeks ago 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 13 The Flagged Dragon The Flagged Dragon 3 weeks ago (edited) 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! 7 Aleph 0 cmilkau cmilkau 2 weeks ago (edited) 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? 3 John Chessant John Chessant 2 weeks ago 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? 3 Andrey Cheremskoy Andrey Cheremskoy 2 weeks ago It was a concise explanation that provided me with a high level overview of the subject. Thanks. 4 Luke Palmer Luke Palmer 2 weeks ago I love how you take these very deep technical ideas and show me just enough of them to go wow! 1 Number Cruncher Number Cruncher 3 weeks ago (edited) 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. 8 Kristi Topollai Kristi Topollai 2 weeks ago (edited) By far my favorite math channel on youtube. Carefully chosen and beautiful topics, clearly explained, accompanied by historic facts and important implications. Everything a non math major would want to properly appreciate these profound topics. 1 Manuel del Río Manuel del Río 2 weeks ago 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? 1 Bobbias Bobbias 3 weeks ago As someone mostly self taught beyond pre-calc, just seeing x = sin theta, y = cos theta, and x^2 + y^2 = 1 helped me see the connection between sin/cos and the unit circle in another way of never noticed. 6 RoyaleFighter01 RoyaleFighter01 3 weeks ago 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 Youtube. Btw awesome video! 1 Rho Rho 3 weeks ago Wonderful, simple, and intuitive introduction to FLT and modular forms. Your videos never disappoint! 1 wokeupinapanic wokeupinapanic 2 weeks ago I was hoping you’d go into HOW these conjectures were proven, but everything else was really well explained! Good work 👍 1 Bemused Indian Bemused Indian 3 weeks ago 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.. 3 Johan De Aguas Johan De Aguas 3 weeks ago I love your detailed and visually appealing explanations. Great job! David Brown David Brown 3 weeks ago That was so freaking beautiful I'm astonished. Thank you so much for making this video. Wow, just wow. 2 M Jones M Jones 2 weeks ago 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? Sid Sid 3 weeks ago Very nice historically grounded exposition. YouTube 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 YouTube 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. 38 Gingres Gingres 2 weeks ago 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 1 Wilder Uhl Wilder Uhl 3 weeks ago 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? 2 Colin Johnson Colin Johnson 3 weeks ago Hey, just wanted to say that you’re page is freaking awesome. Keep up the great work. That is all. Jit Th. Jit Th. 12 days ago 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 Zathras Yes Zathras Yes 2 weeks ago This is breathtakingly amazing! Thank you!!! 1 Tan Soon Tan Soon 3 weeks ago 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. 1 elfumaonthetube elfumaonthetube 3 weeks ago Thank you, great video! Please make more of them : ) Scraps Scraps 3 weeks ago You've done it again man, amazing explanation Julio Ezequiel Julio Ezequiel 3 weeks ago Your content is really unique. Thank you so much for sharing it. 🙂 David Scott David Scott 3 weeks ago 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 CurlBro15 CurlBro15 3 weeks ago 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? 1 Akshay Rahate Akshay Rahate 3 weeks ago I am trying to relate this concepts with Simon Pampena's "The Heart of Fermat's Last Theorem"... can you create a follow-up video on that. Thanks. 2 Tbop3 Tbop3 3 weeks ago I recommend the documentary on the solving of Fermat's last theorem. Very moving. 1 Richard Groller Richard Groller 3 weeks ago so cool. the math is like universal poetry. it creates and solves universal riddles by rhyming numbers and variables in special, unique lines of sequences. 2 Captainsnake Captainsnake 3 weeks ago 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. Syed Kazmi Syed Kazmi 2 weeks ago Brilliant video. Make more on the same topic. Thank you 1 Sergey Sergey 3 weeks ago Beautiful video, as always! 3 Anthony Vasaturo Anthony Vasaturo 3 weeks ago 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. 3 Alexander Arnold Alexander Arnold 3 weeks ago Thanks for the video, every other math video on this seems afraid to try to even describe the actual concepts lol Stephen Blackstone Stephen Blackstone 3 weeks ago (edited) 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. 1 Anton Kucenko Anton Kucenko 3 weeks ago The paper "On the modularity of elliptic curves over Q: wild 3-adic exercises" is about 100 pages. It will make good sense when it can be written easily over a few pages. 2 Bigbad bith Bigbad bith 2 weeks ago I don't understand the maths, but I sort of get this from your wonderful video! 1 Amin Assadi Amin Assadi 2 weeks ago To me was very educative. Thanks a lot 1 Cohomological Cohomological 3 weeks ago It was really interesting. Thank you. 2 Bible Bot Bible Bot 3 weeks ago Can you make a video on the symmetric product and algebra? 2 Fazil Najeeb Fazil Najeeb 3 weeks ago Excellent 👏🏻. And thanks for bringing this up. Fred Anderson Fred Anderson 3 weeks ago So f(q) is like a very sophisticated generating function. Amazing! 3 J G J G 2 weeks ago That was beautiful. Thank you for your videos. 1 master shooter64 master shooter64 3 weeks ago Will you make more videos about differential geometry? 1 Anibal Marrero Anibal Marrero 3 weeks ago Very nice! Your videos are very engaging. I have decided to support you in Patreon. Best wishes for you and your channel! 1 Giorgio Rusbanofski Giorgio Rusbanofski 2 weeks ago (edited) 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. 1 Perseus Georgiadis Perseus Georgiadis 1 day ago I'm gonna reserve this video for when I have the intellectual capacity to understand wh