Universiteit Leiden Mathematisch Instituut
Geometry Seminar

Wednesday, 18 June 2008

Peter Bruin: Computing the stability of a torsion line bundle on a curve over a number field

Abstract:

Let C be a smooth projective curve over a number field K, and let L be a line bundle on C such that some tensor power of L is trivial. Fix a rational point O on C. By the Riemann–Roch formula, there is a minimal integer d such that L(dO) has non-zero global sections. This d lies in the interval [0, genus(C)] and is called the stability of L with respect to O. We explain how d can be determined by means of Riemann–Roch computations on the reductions of C at finite places of K, provided certain arithmetic invariants of (a semi-stable model of) C figuring in Arakelov intersection theory can be bounded effectively. These invariants include the Faltings height of C, the self-intersection of its relative dualising sheaf, and the maximal value of the canonical Green function on the Riemann surface C(C).