The bridge between number theory and complex analysis

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.

(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.

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)

[ScottSobolewski]

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.

[pinkalgebra]

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. :)

[JG-zstr]

Ramanujan’s intuition for number theory was astounding and terrifying.

[Bobbias]

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.

[pursuingstacks]

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.

[adityakhanna]

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

[dangreenwald]

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!

[theflaggeddragon]

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!

[MrLidless]

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.

[unluckyphi]

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.

[justinkauffman]

Ramanujan is one those few people I just marvel at. He was on a unique plane from all others.

[LukePalmer]

I love how you take these very deep technical ideas and show me just enough of them to go wow!

[yakaridubois]

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!

[Number_Cruncher]

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.

[ScienceAppliedForGood]

It was a concise explanation that provided me with a high level overview of the subject. Thanks.

[bemusedindian]

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..

[DavidBrown-ndlz]

That was so freaking beautiful I'm astonished. Thank you so much for making this video. Wow, just wow.

[Ginebraconvention]

Very nice! Your videos are very engaging. I have decided to support you in Patreon. Best wishes for you and your channel!

[davidscott]

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

[visceralconfidence]

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.