Endomorphisms of Jacobians
Abstract
Let X be a curve over a number field, with Jacobian J, and let End (J) be the endomorphism ring of J. The ring End (J) is typically isomorphic with ZZ, but the cases where it is larger are interesting for many reasons, most of all because the corresponding curves can then often be matched with relatively simple modular forms.
We discuss algorithms to numerically determine the ring End (J), and to rigorously confirm such numerical results, starting from a concrete defining equation for the curve X. This involves both methods for computing upper bounds on the rank of End (J) and the certification of the existence of endomorphisms of J with a given tangent representation. The resulting algorithms have acceptable running time when the genus of X is small.
This is an overview of a large project over many years that started in joint work with Edgar Costa, Nicolas Mascot, and John Voight, and was then continued in further collaborations and implementations.