Update: This Discussion Group is postponed to Monday, March 8th.

Elliptic curves are, as any regular attendee of discussion groups knows by now, one of the richest and most interesting topics in mathematics.

Of particular importance over the rational numbers are elliptic curves with complex multiplication: they are those that come with additional endomorphisms. Over finite fields however, all elliptic curves have more endomorphisms than just the usual multiplication by maps: there is the beloved Frobenius that helps us count points and solve many other problems. But there are even more special elliptic curves over finite fields: supersingular curves. In this case, their ring of endomorphisms has the interesting structure of an order in a quaternion algebra, and so in particular is non-commutative.

All supersingular elliptic curves share many important properties, and it will be one of the aims of this talk to show the equivalence of many of these properties. These range from the description of the endomorphism ring to measuring -torsion, considerations of isogenies or just counting the number of points.

The talk will end with the consideration of the relation between elliptic curves with complex multiplication over and supersingular elliptic curves over finite fields; in particular, we deduce many easy estimates for the size of the torsion of an elliptic curve over (or over other number fields).

Come to MS.05 at 7:30 to hear about all this, and more! After which we reduce ourselves to the pub.