## Part C Elliptic Curves, Week 2

I mention here a few topics that came up in the class on Thursday, for the benefit of the students who take the Tuesday class.

1. We discussed the Hasse bound: If $E$ is an elliptic curve over a finite field ${\bf F}_q$ with $q$ elements, then the cardinality $|E({\bf F}_q)|$ of the set of points with coordinates in ${\bf F}_q$ satisfies the following inequalities:

$q+1-2\sqrt{q}\leq |E({\bf F}_q)|\leq q+1+2 \sqrt{q}.$

For example, an elliptic curve over ${\bf F}_{11}$ will have at most 18 points in ${\bf F}_{11}$. However, it is still an interesting problem to actually *find* elliptic curves with points near the upper bound. For example, can you write down an elliptic curve over ${\bf F}_3$ with 7 points? How about an elliptic curve over ${\bf F}_5$ with 10 points?

2. If you fix an elliptic curve $E: y^2=x^3+Ax+B$ with $A,B$ integers, it is interesting to consider the *distribution* of $|E({\bf F}_p)|$ as $p$ varies. For example, sheet 1 showed that those of the form $y^2=x^3+B$ have $p+1$ points in ${\bf F}_p$ for all $p\equiv 2 \mod 3$ such that $p$ doesn’t divide $3B$. (Why do we need this condition?). What happens for such elliptic curves when $p\equiv 1 \mod 3$?

This question for a general elliptic curve was the subject of the celebrated *Sato-Tate Conjecture*, resolved a few years ago by Laurent Clozel, Michael Harris, Nick Shepherd-Barron, and Richard Taylor.

3. For a point $P$ on an elliptic curve it might be interesting to search for points $Q$ such that $2Q=P$. How might one go about finding these?