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?

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