Birch Swinnerton-Dyer conjecture: Introduction

This talk is an graduate-level introduction to the Birch Swinnerton-Dyer conjecture in number theory, relating the rank of the Mordell group of a rational elliptic curve to the order of the zero of its L series at s=1. We explain the meaning of these terms, describe the motivation for the conjecture, and discuss some of the progress that has been made on proving it.
(This topic was suggested by several viewers. I have not yet decided whether this is a one-off talk or the first of a series.)

[@power9k470]

Sir Borcherds sounds like an old wise man from a mythological story who is about to grace you with his deep wisdom and knowledge.

[@downinthehole]

I was actually rewatching the LotR movies just now and this comment made me conflate him with Gandalf in my mind

[@omargaber3122]

Professor ,You are an amazing treasure Where were you hiding before ?! thank you very much🌷

[@kaiderkraisel8409]

Great lesson (as usual). However, I think that the results of Elkies et al. actually show that the current record-holder, in terms of rank, is of rank at least 28 whereas the highest rank explicitly known is 20.

[@TheAcer4666]

I'm a little confused at the notation at 12:30 - the naming of the complex number alpha_p makes it seem like it is dependent only on p, but surely not all elliptic curves have the same number of points mod p? Is alpha also dependent on the elliptic curve?

[@joaquincuelho890]

Thanks for the amazing explanation on this difficult problem! Which steps do you recommend for upload new perspectives as a non- expert on the field? I think there are easy ways to demonstrate difficult problems like this, that should need to be considered for new discoveries, maybe

[@EdwinSteiner]

Two quadratic curves downvoted this video.

[@redfullpack]

this is more educational than all the nonsense paranormal mysteries

[@dmitripanov9980]

Dear Richard, thanks for the lecture! I am trying to understand what exactly is the definition of a modular curve. On wiki there is the following definition: "A curve C, over Q is called a modular curve if for some n there exists a surjective morphism φ : X_0(n) → C, given by a rational map with integer coefficients". I wonder what is the meaning of the last 6 words. What are the coordinates (or coordinate functions) that one uses on X_0(n) with respect to which the rational map has integer coefficients? Would you advise some place where to read about this?

[@richarde.borcherds7998]

I'm not quite sure what the phrase about integer coordinates is there for. A good place to read about modular curves is the book "A first course in modular forms" by Diamond and Shurman.

[@vishalmishra3046]

Prof Borcherds - Why did you skip ax^2 term in your video from elliptic curve equation y^2 = ax^2 + bx + c ? a is not always 0. As a popular example, a is equal to *486662* in the popular DJB *Elliptic Curve25519* .

[@MrChinos007]

Every elliptic curve defined over a field of characteristic different from 2 or 3 is isomorphic (sort of equivalent) to a curve of the form y^2=x^3+Ax+C where the term x^2 doesn't show up. This is called the (short) Weierstrass form of the elliptic curve. The elliptic curve you mention is given in Montgomery form: y^2=x^3+Ax^2+x because this gives certain computational advantages for cryptographic applications, but it is, in fact, isomorphic to a Weierstrass curve. This means that there is some curve of the form y^2=x^3+Ax+C that is essentially 'the same' as Curve25519.

[@fatimaalhoti7804]

Thanks Please Excuse me the example is (3,5) not (5,3)

[@truthteller4689]

Hi professor, very interesting!!! Something that popped into my head is that maybe rank 1 is associated with affine Lie groups and rank>1 is associated with hyperbolic lie groups. Another thing that popped into my head is that maybe the rank-N is associated N dimensional lattices. Also, if rank 28 is the largest so far, this might be related to leech lattice in 24D, string theory in 26D and exceptional Jordan algebra in 27D. Or the possible ranks could be one less than the factors of the Monster group. These are things I would investigate if I was any good at math 😁

[@migarsormrapophis2755]

yeeeeeeee