Just a comment about the history. I have NOT read any of the original sources at all, so definitely 'fwiw', but: I had always been under the impression that Mordell had 'only' dealt with the rational points on an e.c. over the rationals, and that Weil introduced the machinery to handle a.v.s over number fields - this seems to match Wikipedia's description of the history, and does match Wolfram World's "... For elliptic curves over the rationals Q, the group of rational points is always finitely generated [...] was proved by Mordell (1922-23) and extended by Weil (1928) to Abelian varieties over number fields." Meanwhile, Manin's Appendix II of Mumford's AV's claims that Lang's contribution was to deal with the case of the base field being of finite type over the prime field. On the other hand, some of the internet believes that Neron did this...
What a coincidence, I'm studying elliptic curves these days. 11:53 "If we multiply by a positive integer that's an isogeny". I don't get this. The map 2× : E(Q) → E(Q) in general sholdn't be surjective since the Weak Mordell theorem states E(Q)/2E(Q) is only finite, not zero.
The multiplication by 2 map is surjective on "the entire elliptic curve," but not on the rational points. For any point p on E, there is another point q on E with 2q=p, but we cannot guarantee q is rational if p is.
@@mm18382 If A is generated by a1,a2,....,an, then A/2A is generated by [a1],[a2],....,[an]. But now [ai] has order 2, and so there are at most 2^n different elements in A/2A, namely 0 [a1],[a2],....,[an], [a1]+[a2],[a2]+[a3],....,[a(n-1)]+[an], [a1]+[a2]+[a3],...... .... [a1]+[a2]+....+[an]
Thanks for the reply, I don't agree, though, because A (and crucially A/2A) need not be commutative So a priori you can have an infinite sequence [a1], [a1]+[a2], [a1]+[a2]+[a1], ... Edit: E(Q) is commutative 😃
Woah! I was looking for something like this because Mordel-Weil has some applications to F theory but was not able to understand it. This will definitely help.
I’m sorry guys, I don’t know where else to ask this and y’all seem pretty smart, so Could you interpret F(a) in the FTC the following way: Let's say G(T) is the function that "tracks" the area of f(x) from a point "a" exactly so G(a)=0. Let's say F(T) is some antiderivative of f(x). Then F(T)=G(T)+C. Then, G(T)=F(T) -C. But at point "a" we get: G(a) =F(a) -C F(a)=С. So therefore G(T)=F(T)-F(a)??? So F(a) just happens to be that constant C that separates F(T) and G(T) because of the fact that G(a)=0?
Try studying one of - Beachy & Blair's "Abstract Algebra" or - Herrmann & Sally's "Number, Shape, & Symmetry" or - Birkhoff & MacLane's "Modern Algebra", and then rewatch his group-theory video. You might also like to look at two beautiful books: - Niven, Zuckerman, and Montgomery, "Introduction to Number Theory" or - Silverman & Tate's "Rational Points on Elliptic Curves" (based on lectures to [undergraduate] students at Haverford or Swarthmore, I believe). If you have a masochist kink, try Serge Lang's "Algebra".
When you upload, I will drop everything to watch these videos :)
Here, looks like you dropped something
🎉❤
dont drop the soap
Person like you are uploading it is really pleasure to see. ❤❤
Drake K-Dot beef: ❌️
The best prof dropping new vid: 🎉🎉🎉❤
Tryna strike a chord and it's probably a-minooooooooooooooooooooooooooooooooooooooooooooooooooor
Prof thank you so much for your contributions for ppl who can't afford expensive courses.
Thank you professor
Just a comment about the history. I have NOT read any of the original sources at all, so definitely 'fwiw', but: I had always been under the impression that Mordell had 'only' dealt with the rational points on an e.c. over the rationals, and that Weil introduced the machinery to handle a.v.s over number fields - this seems to match Wikipedia's description of the history, and does match Wolfram World's "... For elliptic curves over the rationals Q, the group of rational points is always finitely generated [...] was proved by Mordell (1922-23) and extended by Weil (1928) to Abelian varieties over number fields." Meanwhile, Manin's Appendix II of Mumford's AV's claims that Lang's contribution was to deal with the case of the base field being of finite type over the prime field. On the other hand, some of the internet believes that Neron did this...
What a coincidence, I'm studying elliptic curves these days.
11:53 "If we multiply by a positive integer that's an isogeny". I don't get this. The map 2× : E(Q) → E(Q) in general sholdn't be surjective since the Weak Mordell theorem states E(Q)/2E(Q) is only finite, not zero.
The multiplication by 2 map is surjective on "the entire elliptic curve," but not on the rational points.
For any point p on E, there is another point q on E with 2q=p, but we cannot guarantee q is rational if p is.
Any nonzero morphism is surjective, but only on the algebraic closure.
Another question: in 3:28, why E(Q) finitely generated implies E(Q)/2E(Q) finite?
@@mm18382 If A is generated by a1,a2,....,an, then A/2A is generated by [a1],[a2],....,[an]. But now [ai] has order 2, and so there are at most 2^n different elements in A/2A, namely
0
[a1],[a2],....,[an],
[a1]+[a2],[a2]+[a3],....,[a(n-1)]+[an],
[a1]+[a2]+[a3],......
....
[a1]+[a2]+....+[an]
Thanks for the reply, I don't agree, though, because A (and crucially A/2A) need not be commutative
So a priori you can have an infinite sequence [a1], [a1]+[a2], [a1]+[a2]+[a1], ...
Edit: E(Q) is commutative 😃
Mathematics videos for mathematicians. Thank you, Sir!
Woah! I was looking for something like this because Mordel-Weil has some applications to F theory but was not able to understand it. This will definitely help.
Hey professor, if you're reading this, do you plan on covering Graph Theory at all? If you have any insights on this topic I'd love to hear them.
Yes professor please upload from very basic level too. We want such series from you. Some of them very advanced to understand.
yeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee
I’m sorry guys, I don’t know where else to ask this and y’all seem pretty smart, so
Could you interpret F(a) in the FTC the following way: Let's say G(T) is the function that "tracks" the area of f(x) from a point "a" exactly so G(a)=0. Let's say F(T) is some antiderivative of f(x). Then F(T)=G(T)+C.
Then, G(T)=F(T) -C. But at point "a" we get:
G(a) =F(a) -C F(a)=С.
So therefore G(T)=F(T)-F(a)???
So F(a) just happens to be that constant C that separates F(T) and G(T) because of the fact that G(a)=0?
Prof. Do you teach from very basics ? Please
I tried watching Group Theory lectures, they were very advanced for me.
it would behoove you to google 'fields medal'
Try studying one of
- Beachy & Blair's "Abstract Algebra" or
- Herrmann & Sally's "Number, Shape, & Symmetry" or
- Birkhoff & MacLane's "Modern Algebra",
and then rewatch his group-theory video.
You might also like to look at two beautiful books:
- Niven, Zuckerman, and Montgomery, "Introduction to Number Theory" or
- Silverman & Tate's "Rational Points on Elliptic Curves" (based on lectures to [undergraduate] students at Haverford or Swarthmore, I believe).
If you have a masochist kink, try Serge Lang's "Algebra".
ich liebe dich
Poggers