One of my favorite theorems!
A primer on the Hasse-Weil bound, a beautiful theorem about elliptic curves.
Today I wanted to share one of my favorite theorems. It’s a theorem called the Hasse-Weil bound.
In this issue, I’ll give a brief primer on elliptic curves and the Hasse-Weil bound.
What are elliptic curves?
An elliptic curve is an equation of the form
for constants A and B. Here an example:
The fundamental question people ask when studying elliptic curves is:
How many rational solution does this elliptic curve have?
That is, how many pairs of numbers (x,y) are there that satisfy the equation of the curve?
This is a very difficult problem.
For example, below we’ve highlighted the rational solutions to two elliptic curves as red dots.
![](https://substackcdn.com/image/fetch/w_720,c_limit,f_auto,q_auto:good,fl_progressive:steep/https%3A%2F%2Fsubstack-post-media.s3.amazonaws.com%2Fpublic%2Fimages%2F9e7db228-144f-4ad4-96d8-b42472c358d4_630x470.webp)
![](https://substackcdn.com/image/fetch/w_720,c_limit,f_auto,q_auto:good,fl_progressive:steep/https%3A%2F%2Fsubstack-post-media.s3.amazonaws.com%2Fpublic%2Fimages%2F77aef54c-d95a-470c-b961-f52c1dd6d34e_630x470.webp)
The first curve has finitely many rational points, while the second has infinitely many.
This is weird because:
The two curves look basically the same.
The two curves have basically the same defining equation.
And yet, they have different numbers of rational points.
So what do we do?
Here comes the first main tool people use to tackle this problem.
Instead of solving an elliptic curve over the rationals, solve it mod p, where p is a prime number.
Why is this is preferable?
Although finding rational solutions for an elliptic curve is hard, finding solutions mod p is easy. It’s just a finite search. Computers can do it very quickly.
Here is a graph for the number of solutions to an elliptic curve equation mod p for many values of p.
The above graph showed that the number of solutions mod p to an elliptic curve is approximately equal to p+1.
How bad is the error?
Unfortunately, the above approximation doesn’t mean very much unless we can get a handle on how bad this estimate is. Precisely, we can ask: what is the difference between p+1 and the number of solutions mod p?
Written in symbols, what is the below error term?
Here is a plot of the error terms below:
The graph forms a cone!
The equation for the upper line in the graph is 2 *sqrt{p} and the equation for the bottom line is -2*sqrt{p}. So we’ve arrived at the following fundamental inequality:
This is called the Hasse-Weil bound, and it’s a fundamental inequality for elliptic curves.
Problem of the Week
Here is the challenge problem for this week:
There are several circles of total length 10 inside a square of side 1. Show that there exists a straight line which intersects at least four of these circles.
If you have a solution to the above challenge problem, submit it here for a chance to be featured in the next issue of this newsletter.
Solution to last week’s problem
See here for the solution to last week’s problem.
Special thanks to Atonu Roy Chowdhury (Bangladesh) and Olivier Massicot (Champaign, IL) for submitting a solution to this challenge problem.
Thanks for reading and happy learning! See you next Saturday,
Adithya