Computing canonical heights on Jacobians [Elektronische Ressource] / von Jan Steffen Müller

Computing canonical heights onJacobiansVon der Universita¨t Bayreuthzur Erlangung des Grades einesDoktors der Naturwissenschaften (Dr. rer. nat.)genehmigte AbhandlungvonJan Steffen Mu¨lleraus Gießen1. Gutachter: Prof. Dr. Michael Stoll2. Gutachter: Prof. Dr. Victor FlynnTag der Einreichung: 6.10.2010Tag des Kolloquiums: 16.12.2010iiiiiAbstractThecanonical height isan indispensabletoolfor thestudyofthearithmeticofabelianvarieties. Inthisdissertationweinvestigatemethodsfor theexplicit computation of canonical heights on Ja-cobians of smooth projective curves. Building on an existingalgorithm due to Flynn and Smart with modifications by Stollwe generalize efficient methods for the computation of canonicalheights on elliptic curves to the case of Jacobian surfaces. Themain tools are the explicit theory of the Kummer surface asso-ciated to a Jacobian surface which we develop in full generality,buildingon earlier work dueto Flynn, and a careful studyof thelocal N´eron models of the Jacobian.As a first step for a further generalization to Jacobian three-folds of hyperelliptic curves, we completely describe the asso-ciated Kummer threefold and conjecture formulas for explicitarithmetic on it, based on experimental data. Assuming the va-lidityofthisconjecture, manyoftheresultsforJacobiansurfacescan be generalized.