Computational Algebra Seminar: Sijsling -- Endomorphisms of Jacobians

Speaker: Jeroen Sijsling (Ulm) 

Title: Endomorphisms of Jacobians 

Time & Place: 15:00-16:00, Thursday 17 September, SMRI Seminar Room 

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.