L-functions and the Langlands program (RH Saga S1E2)

This is the second episode of the RH Saga.

We continue the journey into the world of L-functions, by focussing on two specific examples of motivic L-functions. These examples illustrate Langlands reciprocity, i.e. the conjecture that every motivic L-function is also an automorphic L-function.

The overall aim of RH Saga Season 1 is to map the landscape of L-functions, as a foundation for future in-depth exploration of some of the most immortal math problems of all time.

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*Chapters:*

00:00 - Intro
01:40 - Two examples of L-functions
07:08 - The LMFDB
11:50 - The mystery of K
23:31 - The mystery of E
44:33 - Final remarks on Langlands

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*Links:*

1. SageMathCell

2. LMFDB

3. The number defined by the polynomial x^2+1, which we refer to as "K"

4. The elliptic curve "14.a5" which we refer to as "E"

5. "Black box" for sequence "K" code in SageMathCell

6. "Black box" for sequence "E" code in SageMathCell

7. The point counting code in SageMathCell

8. Matthew Emerton's homepage

9. Langlands reciprocity survey by Emerton:

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*Errata:*

Around 42:30: In the formula for the generating series (modular form), there is traditionally a “q” in front of the product sign. This should be included IF you want the sequence indexed with a1 as the label for the initial term rather than a0.

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*Social:*

#RiemannHypothesis #F1Geometry #Mathematics #PeakMath #RHSaga #Langlands

[Tenraiden]

Your PACE is a rare gift! You don't simply recite from a script, or cut corners to fit the video into a specific duration, you TALK to the viewer and you wait for them to naturally absorb what you just said. It's perfect 👌

[dcterr]

Wow, what an amazing video! I received a PhD in algebraic number theory from UC Berkeley in 1997, but I never understood much of what you discuss in this video before, but now I do! Excellent lecture! Now I want to study L-functions in more detail!

[PeakMathLandscape]

Thanks for all the comments and encouragement! This is just the beginning.

To answer some recurring questions:
(1) All these Episodes will become openly available here at this YouTube channel.
(2) We as a team are very much in a learning process when it comes to making math videos, but as we get up to speed, the aim is to post a new video every two or three weeks, with exceptions for things like holidays.

We are also building an online community around L-function exploration (and later, exploration of the mathematical landscape as a whole). This course/exploration community is subscription-based, hosted at peakmath.org, and there you will find a place for asking questions about L-functions, hanging out and discussing the course content with us and with fellow explorers. We also have challenge problems and extended written course notes for each Episode, designed to take you from the basics of L-functions to a point where you can explore the immortal problems on your own. And you will get the videos a few weeks before they appear on YouTube.

[pourtoukist]

The magic of Youtube lies in this kind of videos. Having access to this knowledge and this quality of content for free is simply amazing!

[andyl.]

20:35 The construction of K made me rewatch 3b1b's "Pi hiding in prime regularities" video. I couldn't make much sense of that video the first time, but it's a pleasant surprise to meet gaussian integers and χ function again in this video.

[navjot]

I have a gut feeling that you have proved RH is true, and gonna shock the whole world in the last episode 😅

[LUCASTAVARESCARDOSO]

As someone who was looking for a clear exposition of the main ideas in the Langlands Program, you were the perfect match. Thank you so much doing that. As a mathematical physicist, I usually try to bring tools from distinct areas of math to understand and deeper explore a model or theory which originally had nothing to do with what I brought in, and when I succeed in making interesting connections, it exhilarates me profoundly. Lately, I have also come across a family of problems which I think the ideas in the LP apply perfectly. Please, continue your wounderful work.

[HanniSoftware]

I’m so happy I’ve found this channel, it’s amazing math content. I hope you keep going

[yoannmery]

Glad I did not spend too much time trying to guess the sequences at the beginning 😅

[mlmnmelkior]

honestly this seems interesting but- at every step I'm like "why ?? what's the link ?" because every time you present a new way of computing L-functions I can't understand how it works and for all I know this could be a gigantic prank - really I like your tone, the animations and how you present things in your videos, the only problem is that everything you say seems arbitrary, I would love if you were to explain a bit more how you get from one object to the other and WHY it works that way !!

[EdwinSteiner]

I can hardly believe that you gained such a large audience with just two videos! It's well-deserved since the videos are very good and much needed. Automorphic forms and their connections seem to be very beautiful subjects but almost all lectures one can find online are just atrociously bad and the material on Wikipedia isn't great either on these topics. These areas seem to attract people who are bad at pedagogy or disinterested in it. You are a welcome exception. I wish I knew how to attract so many views since I make videos on quantum mechanics and it seems really difficult to get any audience for longer videos on serious topics at more than a basic level.

[piotr]

Hi PeakMath. Thank you for the series and for such broad view on the subject. I am waiting for each episode. I hope that you will have time and motivation to finish the saga - all

[cblpu]

Incredible. I hope you go as far in depth as possible

[fibbooo]

Quadratic reciprocity is one of the few topics that I've tried to learn a dozen times and each time have come away without having gained anything. I wish myself luck fo this series!

[suzy]

Oh my god . These lectures are just pure gold. Thank you so much 😊👍

[Driancreid]

This is a great video, particularly because you keep forging ahead without explaining or justifying every point with complete rigour. In this way you can get more quickly to the interesting points before you lose the watcher. Well done from someone who specialises in Maths Education and Online Learning!

[baticadavinci]

Please don't make us wait another 10 years for the next episode.
I don't know a lot of math but I still enjoyed this stuff!

[tanchienhao]

This is an amazing series I can’t wait to see where it’ll lead us to

[quixoteyeah]

I'm barely hanging on, but these are great videos. Really well done. Thanks.

[TranSylvainie]

Perfection ! Can't wait for the adventure to continue !