## 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 is an elliptic curve over a finite field with elements, then the cardinality of the set of points with coordinates in satisfies the following inequalities:

For example, an elliptic curve over will have at most 18 points in . 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 with 7 points? How about an elliptic curve over with 10 points?

2. If you fix an elliptic curve with integers, it is interesting to consider the *distribution* of as varies. For example, sheet 1 showed that those of the form have points in for all such that doesn’t divide . (Why do we need this condition?). What happens for such elliptic curves when ?

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 on an elliptic curve it might be interesting to search for points such that . How might one go about finding these?

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