The BSD Conjecture, part 1 (RH Saga S1E7)

[@jtauber]

Wow, I can't believe I only just worked out the labels for the example L-functions: P, E, A, K spell PEAK as in PeakMath. These sorts of easter eggs are one of the many reasons I love this series so much.

[@elfumaonthetube]

Can I say that I've been watching this RH saga over and over more than I ever watched Start Trek TOS? Thanks for all of these extremely interesting videos!

[@PeakMathLandscape]

Thank you, your support means a lot to us!

[@d-nize]

The way you explain and your Passion let me think that you will come up with a grande Final. Please Upload More asap. I can Not wait. :D

[@barrdetwix1894]

I'm just in the middle of reading Elliptic Tales which works its way up to BSD, so this episode comes perfectly timed :)

[@amritawasthi7030]

These videos are making it look like that we're connecting all of the branches of mathematics with number theory like algebra, analysis (real, complex, functional) and linear algebra. This is so goood. Thank you very much for this. Keep making it.

[@timbotemon]

Thanks for another wonderful episode

[@randalllaplante358]

I love immersing myself in these concepts. Not smart enough to comprehend all, but a joyful workout for my mind.

[@maccip1]

I can see the beauty, but i'm not smart enough to decrypt it. I's sad, but also very enlightening to know that someone have the power to do it. Those things may be the deepest symmetries of our conceptual mode of thinking. For me, following this series is like listening Chopin, like connecting to a pure form of art. I may not understand much, but for sure i'm getting more rich trying. Thanks for doing this! Sorry for my bad english doesnt allow me to be more expressive in the way i appreciate your video series.

[@bryandrury3220]

Extraordinary, thank you

[@johnl4885]

I kept expecting to see the group operation for points on an elliptic curve. Small steps is more fun. Well done series, thank you for your deep insights.

[@TheDannyAwesome]

I love the new software you're using for presentation

[@alexakalennon]

Thank you for breaking down this deep, fundamental parts of mathematics for amateures. It´s so exciting.

[@airfluxe2095]

Really interesting explanation about diophantine equations.

[@chrissch.9254]

Again, an excellent video - keep on your fantastic work! :-)

[@petrospaulos7736]

Next level to yt math videos. Thanx!

[@serajmd3256]

I am exciting to next video on BSD Conjecture

[@PeliQuiz]

I am the Number 1 Fan of this Channel... 😊

[@HadiLq]

Thanks. It's wonderful.

[@dontwannabefound]

I’m Gonna have to watch this again

[@schweinmachtbree1013]

erratum at 8:16 : Fermat equations have the trivial solutions which are integral points

[@jeremypayne6307]

Hangin’ for next ep…🙏

[@cowgoesmoo2]

I think youtube algo liked it more when you called episode 1 The Dream and made it more attractive to clickers.

[@bobtannous5464]

Very good presentation. Thank you. But i do have a question. What would the expression of zeta function be for real(s)

[@leonalmeida2411]

I love your work and your dedication, but this series is resulting a bit out of my comfort zone. Do you have any suggestions of some book that can lay the foundations I need to better understand this argument and maybe explore it a bit on my own?

[@PeakMathLandscape]

Have you seen the new elliptic curves project by Aleph 0? Not sure if I can add links to other videos here, but search for the recent video called "An announcement!" at the Aleph 0 channel. Other than that, maybe the Elliptic Tales book?

[@DeathSugar]

Does that mean that folk, who studies thoroidal manifolds is basically have a deal with some eliptic curves but from topology perspetive?

[@TheDannyAwesome]

Hi Andreas, I'm confused on a point and was hoping someone could explain it to me. I've looked up y^2 = x^3 + 1 on the LMFDB and it says it has analytic rank 0. By the BSD conjecture, it should have algebraic rank 0? But isn't that the number of generators for its set of rational points? It definitely has rational points, so it can't be a group with no generators...?

[@PeakMathLandscape]

This curve has 6 rational points (including the point at infinity), so this is a finite set. In other words, these points are all torsion points, aka points of finite order. For the rank to be larger than 0, you must have an infinite set of rational points. Does this make sense?

[@TheDannyAwesome]

@@PeakMathLandscape Thank you so much. I had forgotten that you don't include the torsion points. It's the number of generators of infinite order that constitute the algebraic rank!

[@hvok99]

Maybe a silly question but what is the reason for the labeling of the coefficients in the Weierstrass equation? Why no a5?

[@PeakMathLandscape]

Check this: math.stackexchange.com/questions/743473/reason-behind-standard-names-of-coefficients-in-long-weierstrass-equation

[@hvok99]

@PeakMathLandscape Oh thank you, that took some work to chew on and understand but I think it is clear. in short it is because no coefficient in the Weierstrass equation has a power of 5 under a standard change of variables, and the other terms are indexed with the order of their powers.

[@rtravkin]

How come you said that x^3 + y^3 = 1 does not have any integer solutions when there are two: (x,y) = (1,0) and (x,y) = (0,1)? (And the projective completion given by the homogeneous "Fermat equation" has one more point "at ∞", (1,-1,0).)

[@PeakMathLandscape]

You are totally right of course, that was sloppy. It should have been no positive solutions.

[@Axacqk]

16:50 Tries "any x, like x=6". Checks if the solutions are rational. Nope, but delta is a product of two Mersenne primes!

[@tomkerruish2982]

I don't think L should be a Hilbert space, but rather a Z-module. It seems unlikely to me that we could define a fractional direct sum of two L-functions without invoking some (necessarily arbitrary) branch cuts; e.g. ½L_X for any L_X would need to be a function whose values satisfy ½L_X(z)² = L_X(z) for all z. If that weren't bad enough, we would need to do this for every possible exponent, including complex.

[@faisalsheikh7846]

Incredible❤sir

[@fburton8]

14:20 What happened to a5?

[@PeakMathLandscape]

There is no a5. The reason for this convention has to do with degrees: math.stackexchange.com/questions/821187/history-of-the-coefficients-of-elliptic-curves-why-a-6

[@fburton8]

@@PeakMathLandscape That's a really helpful pointer, thank you!

[@caspermadlener4191]

"Fermat equation does not have any integral points" 0³+1³=1 1³+0³=1

[@PeakMathLandscape]

Good point! The rigorous statement should be no solutions in positive integers :-)

[@mathildaawn6010]

Turns out, Wiles was wrong. :(

[@RiotSociety666]

😂😂😂

[@eoghanf]

I have a proof of the BSD conjecture but unfortunately this YT comment is too small to contain it