Genus-2 curves with a given number of points

This is a report on joint work with Kristin Lauter and Peter Stevenhagen.

Broker and Stevenhagen have shown that in practice it is not hard to produce an elliptic curve (over some finite field) with a given number N of points, provided that the factorization of N is known. In his talk this week, Stevenhagen will show that the natural generalization of this method to produce genus-2 curves with a given number of points on their Jacobian is an exponential algorithm.

I will consider the related problem of constructing a genus-2 curve over some finite field such that the curve itself has a given number N of points. The idea of “explicit gluings” of pairs of elliptic curves leads to a solution for most values of N; I will discuss this, and other, applications of explicit gluings.

Speaker Details

Everett Howe has worked at the Center for Communications Research in San Diego, California, since 1996. His published research is mainly concerned with problems involving curves and Jacobians over finite fields and number fields. He is proud to be a coauthor of the work that produces the top hit when one Googles “worst limerick in the world”.