Mark Bauer (University of Calgary):
Point Counting on Picard Curves
While many quite effective techniques are available for determing the order of the Jacobian of elliptic and hyperelliptic curves, general purpose algorithms for determining group orders
are not sufficient to calculate the order of a Jacobian of a non-hyperelliptic curve which is of cryptographically interesting size. We present an algorithm which may be used for
another class of curves, Picard curves. While this algorithm is not quite able to reach cryptographic sizes, it is pretty close. We will discuss how the techniques involved arise, and
how they can be extended to arbitrary curves of genus three whose endormorphism group contains an element of order 3.