14G40 Arithmetic varieties and schemes; Arakelov theory; heights [See also 11G50, 37P30]
Computing canonical heights on Jacobians
Jan Steffen Müller
- The canonical height is an indispensable tool for the study of the arithmetic of abelian varieties. In this dissertation we investigate methods for the explicit computation of canonical heights on Jacobians of smooth projective curves. Building on an existing algorithm due to Flynn and Smart with modifications by Stoll we generalize efficient methods for the computation of canonical heights on elliptic curves to the case of Jacobian surfaces. The main tools are the explicit theory of the Kummer surface associated to a Jacobian surface, which we develop in full generality, building on earlier work due to Flynn, and a careful study of the local Néron models of the Jacobian. As a first step for a further generalization to Jacobian threefolds of hyperelliptic curves, we completely describe the associated Kummer threefold and conjecture formulas for explicit arithmetic on it, based on experimental data. Assuming the validity of this conjecture, many of the results for Jacobian surfaces can then be generalized. Finally, we use a theorem due to Faltings, Gross and Hriljac which expresses the canonical height on the Jacobian in terms of arithmetic intersection theory on the curve to develop an algorithm for the computation of the canonical height which is applicable in principle to any Jacobian. However, it uses several subroutines and some of these are currently only implemented in the hyperelliptic case, although the theory is available in general. Among the possible applications of the computation of canonical heights are the determination of generators for the Mordell-Weil group of the Jacobian and the computation of its regulator, appearing for instance in the famous Birch and Swinnerton-Dyer conjecture. We illustrate our algorithm with two examples: The regulator of a finite index subgroup of the Mordell-Weil group of the Jacobian of a genus 3 hyperelliptic curve and the non-archimedean part of theregulator computation for the Jacobian of a non-hyperelliptic genus 4 curve, where the remaining computations can be done immediately once the above-mentioned implementations are available.