Huge news: An elliptic curve with rank at least 29 has been found!

[@clrkgmii]

i knew about this all along i just didn't want to tell you guys

[@exotic1405]

Yeah me too I'm just too humble

[@janzwendelaar907]

Bro has found an elegant proof for this, but the margins were too small to contain it

[@user-zz3sn8ky7z]

- Fermat

[@nehrilisoruz8182]

Sounds like aberkane's proof of Syracuse Conjecture to me 🤔🤔😶‍🌫️

[@JRazEd]

wake me up when they find a rank 30 elliptic curve

[@alvaro.lozano-robledo]

will do

[@johnchessant3012]

Thanks for bringing us this news! I find it fascinating that Elkies needs to assume GRH to prove that the rank is exactly 29; computing the rank is actually the subject of a different Millennium Prize problem: the Birch and Swinnerton-Dyer conjecture. It relates the rank to the numbers of points on the curve mod p, for all primes p.

[@leif1075]

How does anyone no matter how smart get bored and tired doing this?

[@drdca8263]

@@leif1075well, I suppose one gets tired eventually no matter what one is working on? People need to rest sometimes.

[@rtg_onefourtwoeightfiveseven]

I don't know much about number theory (clearly) so I don't know if this is possible, but imagine what a plot twist it would be if GRH were somehow equivalent to BSD.

[@o.s.h.4613]

@@rtg_onefourtwoeightfivesevenThey are actually very closely related. An L-function may be associated to an elliptic curve, and one can investigate whether or not the Generalized Riemann Hypothesis hold (namely, all non-trivial zeros should have real part equal to 1/2). Let m be the order of vanishing of the L-function at the central point s = 1/2. The Birch and Swinnterton-Dyer conjecture asserts that m equals the order of the Mordell-Weil group.

[@diegotorres2101]

GRH? I'm assuming riemann hypothesis, but idk what the G is

[@amos9274]

As an engineer I'm always stunned at how abstract, broad and interconnected higher maths is. Huge respect to anyone who takes it upon themselves to learn such a plethora of information to maybe push the boundaries just a little further. I wonder how long we can collectively keep this up though? If it continues like this, we will definetly hit a limit, where the sum of the knowledge that needs to be connected is just too "deep" for a person to learn in a single lifetime.

[@eikotokura1117]

One good thing about mathematics is, mathematics is modular, you don't need to know every detail about a theorem that other people established and verified by a lot of people, with more and more knowledge, you can use more powerful tools to find new results.

[@cleon_teunissen]

There are a couple of places where the auto generated transcript writes the word 'finally'. The transcript generator hears you saying 'finidely', and it doesn't have the context awareness to disambiguate

[@klausolekristiansen2960]

Also efficient for coefficent.

[@alvaro.lozano-robledo]

Sorry yes it's supposed to be "finitely generated" in all cases

[@cleon_teunissen]

@@alvaro.lozano-robledo I forgot to mention: as a human with near native fluency in english I automatically heard it as 'finitely'. That said: auto-generated transcription has become so good that you can use it as a measure of how easy or difficult it will be for human listeners to follow what you are saying. Presumably: for a sizable portion of your audience english is not their native language. It's not necessary to pronounce *every* syllable, that would make you sound robotic, but I think the auto-generated transcript is informative to see which syllables are necessary.

[@cleon_teunissen]

@@alvaro.lozano-robledo Oh, this one is quite the gem: 'elliptic curse'.

[@MatthijsvanDuin]

also near the start "to find over the rationals" rather than "defined over the rationals"

[@ikarienator]

So if we found another independent rational point on this elliptic curve, this proves the extended RH wrong?

[@HeavyMetalMouse]

That's what it sounds like, yeah. Though computationally that seems like it would be incredibly difficult.

[@rmsa1323]

29. rank 29. Unbelievable. Bravo vince.

[@TmOnlineMapper]

Always so funny how like half of recent mathematical discoveries or proves depend on the Riemann Hypothesis. Despite it being unproven. It finally being proven would have such an insane knock-on effect, it would be insane!

[@ffc1a28c7]

eh, there's tons of stuff that doesn't get talked about outside of specific fields that doesn't lol.

[@akirakato1293]

mathematicians already assume RH to be true so it won't open up new frontiers of math but the tools developed to prove RH will have a significant effect. the theorems that already depend on RH are mostly accepted as true anyways.

[@ffc1a28c7]

@@akirakato1293 honestly, (being totally biased as a model theorist), it wouldn't surprise me if a lot of the major hypothesises are independent (ie. both their statement and negation are true in a sense).

[@Wotsit27]

Alvaro, Fascinating! Please advise the publication reference for this discovery.

[@alvaro.lozano-robledo]

Paper is not yet available

[@Akranixx]

Whoa!! First place I heard of this! Wild news!

[@MUI_Noam12]

Didn’t understand much but I have the same name as one of the guys who discovered this so that’s cool!! :3

[@nykztv]

>ω< a fellow emoticon user and math enjoyer 🐾

[@MUI_Noam12]

@@nykztv heck yeah >v

[@playboibowlly2568]

This reminds me of the hunt for the biggest cactus in Minecraft, I’d watch a video about that if you like this sort of stuff (even if you don’t like Minecraft)

[@shruggzdastr8-facedclown]

Has it been determined that e-curves of ranks 25, 26, and 27 will be impossible to find (given that this list skips from 24 to 28)?

[@beaverbuoy3011]

Extremely interesting, thank you!

[@dang-x3n0t1ct]

please make a long format video, there short format is awful

[@JorgetePanete]

Your forced subtitles don't always match what you say.

[@jhgvvetyjj6589]

If an elliptic curve with rank of at least 192 gets found, would that crack the cryptographic security of secp192k1?

[@badhombre4942]

Most excited mathematicians have been since they found the rank 28.

[@rouvey]

Do we have other restrictions on the types of other finitely generated abelian groups that can be realized this way? E.g. do we know some groups that definitely can't correspond to an elliptic curce / do we know what the torsion part can look like? Sry don't have any familiarity with this topic but it seems very interesting.

[@rouvey]

Also is the reason for why we expect boundedness of the rank more group theoretic (like is there some special property that groups with rank higher than 21 are not expected to fulfill) or is the argument coming from a more purely algebraic geometric place?

[@rouvey]

I always find super interesting how one encounters so many relatively large "magic numbers" in number theory and algebraic geometry (and group theory as well of course). I've only had a few times where I got to understand where I got to understand where exactly such a magic number comes from and it's always been such a treat.

[@JohnDoe-ti2np]

For elliptic curves defined over the rationals, Barry Mazur proved that the possible torsion subgroups are either Z/nZ for 1

[@rouvey]

@@JohnDoe-ti2np Ah thank you very interesting!

[@Noconstitutionfordemocrats1]

What is rank?

[@epicgamerman2510]

Math Jim Carrey

[@kavindasari]

This is incredible

[@jamesbirmingham9951]

New elliptic curve drop!!!!!!!

[@chicojcf]

Wonderful video.

[@alvaro.lozano-robledo]

Thank you very much!

[@mangjitnijjhar1390]

Is this going to make PGP more secure?

[@__Brandon__]

It could, but we already have a good and standard curves. We likely won't recommend a new curve until at least one person can theorize a flaw in the current curves. It is exciting to have more curves to pick from especially big curves

[@Bruno_Haible]

Hardly. When you do cryptography with an object that is not well understood, there is a high risk that within 5 or 10 years, some researchers find new properties of the object, that allow to dramatically reduce the effect for breaking the crypto. Therefore it is better to take an object that has already been studied for many years.

[@Asdayasman]

It's already pretty good.

[@berlinisvictorious]

Oh shit thats wild man

[@stighemmer]

Math videos generally work better in landscape format.

[@necoluga7950]

i havent started studding this math but i still have a question none the less. what the implications of complex coefficients being imputed into the equation.

[@skylardeslypere9909]

I'm not sure that would be interesting. If A and B are complex and x is rational, then either x³+Ax+B has a non-zero imaginary part, which implies y cannot be rational, or Ax+B is real, but then we might as well just replace A and B by real numbers again.

[@necoluga7950]

@@skylardeslypere9909 thanks for answering my question, i honestly didn't know enough about the topic, i only knew the basics.

[@alvaro.lozano-robledo]

That's a great question. We do study elliptic curves over the complex numbers and we do study points with complex coefficients. But the most difficult problem is when both are rational!

[@portport]

Let's goooooo!!!

[@thehorizontries4759]

Ok but what is rank in this context I still have no idea what is being discussed

[@ronald3836]

If you look at the rational solutions of the elliptic curve, then they form an abelian group of rank 29, which means it contains 29 independent copies of Z. So it has very many rational solutions, one could say 29 dimensions.

[@alvaro.lozano-robledo]

Here is a thread about why this is a big deal (to me): x.com/MathAndCobb/status/1829912115278430266

[@davejacob5208]

as an absolute noob: what does this trait of the curve tell me about how it looks? does it just look like i would expect any ellipse to look like? or is it especially thin, wide, round, or extremely weird looking?

[@ronald3836]

An elliptic curve does not look like an ellips. If you look at the complex solutions, it looks like a donut. I think it is called an elliptic curve because these curves came up when studying the arc length of ellipses. The rank of an elliptic curve says something about the number of rational solutions of the equation. In a way the rational solutions form a group of "dimension" 29 (but "rank" is the correct technical term).

[@ronald3836]

And to be complete: an elliptic curve can be written as y²=ax³+bx²+cx+d for some values of a,b,c,d with a non-zero

[@davejacob5208]

@@ronald3836 yes, but does this ALSO tell me something about whether they look different from other eliptic curves? do all elliptic curves just look like donuts? are these with ranks of 29 or more look thicker, thinner, stretched or something like that?

[@ronald3836]

@@davejacob5208 their complex solutions all look like donuts. But if you look at the solutions with rational coordinates, then there are many more of those than for normal elliptic curves. You can "add" points on an elliptic curve basically by drawing a straight line between two points and finding the third intersection point (and mirroring in the x-axis). This addition operation works like normal addition, i.e. commutative and associative. If you look at the rational points of the elliptic curve with this addition operation, it forms a "group" which includes 29 copies of Z (the integers, which form an infinite cyclic group under addition) plus a small finite part.

[@ronald3836]

@@davejacob5208 elliptic curves looking like donuts means that they are typologically equivalent to a ball with one hole. The number of holes is called the genus. Elliptic curves have genus 1. Curves with genus 0 are basically quadratic equations in 2 variables and will either have infinitely many or none rational solutions. Curves with genus 2 or higher will have at most finely many rational solutions (Mordell's conjecture, proven in the 1980s by Gerd Faltings). Elliptic curves in a way have the "most interesting" arithmetic on them (and turn out to be very important in the proof of Fermat's last theorem). One of the "Millennium Prize" problems is to prove the Birch-Swinnerton-Deyer conjecture, which tries to link the rank of an elliptic curve to the behaviour of the curve's "L function" near s=1. I don't expect you to understand all this, since I hardly understand it myself, but just to show that this is a very rich area of mathematics. 😃

[@leif1075]

Why is this interesting, if I may ask?

[@bjornfeuerbacher5514]

It's interesting from a theoretical point of view, because elliptic curves are a very general type of curves with degree 3. While curves with degrees 1 and 2 are completely understood, the curves with degree 3 still have a multitude of open questions. This result is yet another tiny step into the direction of finally understanding them. From a practical point of view: elliptic curves are very widely used in cryptography, so a better understanding of them will probably lead to data being more secure.

[@akramqasem6322]

❤ great ‏‪

[@markphc99]

nice , but please get your head out of the way - I'm trying to read

[@__.__-_.]

cool

[@classicalmpk2681]

Use landscape format (i'd like to use more than 33% of my screen please), remove the close-up of your face or put it in a corner (it's pointless and blocks like half the screen, i'd like to read the content behind it please), remove the auto-generated captions (youtube already has that feature, don't force it onto us) You really, really don't need to make videos in short form if you're not uploading them as shorts, it makes them a lot less enjoyable.

[@alvaro.lozano-robledo]

You really, really don't have to watch my videos. This is the kind of entitled comment that made me be a TikTok content creator instead of KZbin. I post copies here as a courtesy for those who have asked me to post here because they do not use TikTok. By the way, the captions are usually really bad for math terms and math content so I try to fix them as much as I can, tho I often miss some words -- that's as much time as I can dedicate to go over the entire caption text.

[@nevascurded]

No reason to put your face on any of this.

[@MikkoAPenttila]

Ok , so what does it mean? 29 is the best number? 29 is the best prime?

[@NoNameAtAll2]

it's the new high score in another arbitrary measure

[@elizabethhenning778]

It's like setting a new 100m record by beating the old one by 0.03s

[@MikkoAPenttila]

@@elizabethhenning778 I get your analogy... but I think for the 100 meters 0.03s is quite a lot. 🙂

[@elizabethhenning778]

@@MikkoAPenttila It's solid but not earth-shattering, just like this result

[@MikkoAPenttila]

@@elizabethhenning778 A record is a record

[@samevans4834]

If you've gotten to the point that you're proving things with the generalized Riemann hypothesis, in my head, you're stepping outside the bounds of provability in ZFC. Obviously that's not rigorous at all, but call it a heuristic argument - the RH is so resilient against proof, and so many proofs depend on its provability (true or false), it feels very much like the sort of bottleneck that constrains other mathematical disciplines (the halting problem isn't a perfect comparison, but I hope it communicates the general idea). Thoughts?

[@ronald3836]

If RH is unprovable, then it is true because if the Nth zero is not on the critical line for some N, we could in principle verify that.

[@cipherxen2]

Why is this mathematics related video is in idiotic tiktok style?

[@alvaro.lozano-robledo]

Why is this comment written in the trashy style of the worst dumpster fire KZbin comment sections?

[@strawhaaton]

I may not agree with the video format, but I wholeheartedly support that reply LMFAO

[@error_6o6]

Why are these replies in the trashy style of reply sections?

[@elizabethhenning778]

Probably to make it easier to crosspost on TikTok. So what?

[@alvaro.lozano-robledo]

​@@elizabethhenning778 the video was created for TikTok and I share it here as a courtesy to people who do not use that app

[@נועםדוד-י8ד]

I don't get why is it interesting, the only thing revealed is that an EC with a rank of 29 exists. It's not surprising, or giving us any clue to discovering if the rank of an EC is bounded.

[@alvaro.lozano-robledo]

x.com/MathAndCobb/status/1829912115278430266

[@tankulator5042]

@@alvaro.lozano-robledo Unfortunately I cannot see the explanation without a twitter account. Could you paste the explanation elsewhere as well for non-twitter users?

[@alvaro.lozano-robledo]

​@@tankulator5042 it's an entire thread with many posts so it's not easily extracted from X

[@jonb4020]

I'm sure this is brilliant but one wonders if maybe these people could put their energies into things that actually help the human race and/or the world in general?

[@yasploofyh8358]

Advancement in math and/or physics helps advancement in other areas such as engineering, which can help advance society in many ways :)

[@jonb4020]

@@yasploofyh8358 Yes, you are right. 😊 I should be more tolerant! But I loathe anything to do with complex, abstruse, "pure" maths (probably for weird psychological reasons) and love anything to do with words and languages, so this sort of stuff just leaves me cold. If I could even begin to understand it maybe I'd feel different.

[@user-zz3sn8ky7z]

It does, physics and informatics are both fundementally dependent on having a strong mathematical foundation. If all our effort in math ceased and we just started focusing on other fields instead, we'd be able to progress for only few decades before hitting some form of a roadblock. And while yes, a lot of math ends up existing only for maths sake, the subject is so interconnected that eventually it would bite us in the ass if we only ever focused on the "practical" aspects, so the subject must be developed "naturally". Anyway, for instance these guys specifically are used for elliptic curve cryptography, which is pretty commonly used. The one of rank 29 is a massive overkill for that purpose, at least until some flaw is discovered, but it's not like some mathematician pulled the concept out of their ass

[@MatthewJohnson-hi2th]

@jonb4020 Could the same thing not be said about artists, authors, entertainers, philosophers, scholars of other fields, and many more professions? I don't think that as a species we should ever stop indulging our curiosity. The other replies are correct that this *could* end up being "useful" - but I actually don't care if it's ever useful in that way, at all. Maths is beautiful, and that's all there is to it in my view. I wouldn't want to live in a world where the only things available for people to do with their lives were "useful". But again, to be clear - nothing against people who care about things being useful :3 I just see this argument a lot and it makes me wonder how people feel about fields that will almost certainly never see useful applications.