Millennium Problem: Birch and Swinnerton-Dyer Conjecture aka BSD Conjecture

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The embarrassing thought about BSD conjecture is that it poses a question just beyond what we have learnt in high school, yet we are unable to prove it! It belongs to the area of mathematics known as number theory where we prefer to find solutions to equations in rational numbers (ratio of integers).

In school, we learn to find solutions to equation in one variable by using the Rational Root theorem.

We also learn how to solve equations of two variables in degree 1.

But what about equations in two variables of degree 2 ?
Consider the equation
x² +y² =1
This is known as the unit circle.

The point (-1, 0) lies on the circle and consider the point P(x, y)


Using all the rational slopes and finding the second point of intersection we are able to find all the rational points.



However,  

x² +y² =3

has no rational solutions! (For fun try proving that!)

But what about cubic equations in two variables?
As it turns out we have no general way of finding all the rational solution in this case. This is what the BSD conjecture addresses!
Every cubic equations in two variables can be written as







Let’s compare the cubic case to the quadratic case to understand the differences

This brings us to the crucial concept of rank of an elliptic curve. The rank of an elliptic curve tells us how many points we need to generate all the rational points of the cubic curve. 

In the 1960’s, Birch and Swinnerton-Dyer decided to attack this problem by counting rather than use algebra and geometry by doing numerical experiments on a computer. They counted the number of solutions of many elliptic curves and related it to the rank of the elliptic curve.
However, they ran into a problem. How to count large infinite solutions for many of these curves? The answer was modular arithmetic.

Imagine a one hand clock which goes from 0 to 4 and we have to count till 20. We can count all the numbers in terms of 0 to 4 

Since we have 5 numbers it is called modular arithmetic 5.  This way we can reduce an infinite set of counting to a finite set.

Here is a question, How big is Np- the number of all rational solutions on an elliptic curve in modular arithmetic p? Well it will have p-1 points in x and y respectively which will give us p slopes and thus p points. Hence, Np  should be close to p.  

Now we have all the tools necessary to understand the BSD conjecture.

Birch and Swinnerton-Dyer Conjecture 



r = Rank of the Elliptic curve 
C = constant 
We take the product of Np⁄p where  p is less than some large X  for an elliptic curve. It should grow as 
(log X)r times a constant, r is the rank.  When r =0, it should have zero or finite solution Np⁄p   is constant and when 
r ≥ 1 it should have infinite or greater solutions. (Np should be greater than p)

Reminder:  If you can prove this, you get a million dollars !  

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