An explicit theory of heights for hyperelliptic Jacobians
Final Report Abstract
The ultimate goal of this project was to improve and extend methods for solving diophantine equations of the form y^2 = f (x), where f is a polynomial, in integers or rational numbers. Equivalently, we are interested in the integral or rational points on the curve defined by the equation. Curves of this type are said to be hyperelliptic. Most of the available methods make use of the fact that the curve can be embedded into its Jacobian variety. This is an abelian variety of dimension equal to the genus of the curve (in our case, the genus is roughly half the degree of f ) and thus carries the helpful structure of a group. To make use of this embedding, we need to know enough about the group of rational points on the Jacobian variety, which can be described by specifying finitely many generators. The theory of canonical heights is an indispensable tool when studying abelian varieties defined over number fields. Besides numerous theoretical applications, one needs to be able to compute the canonical height of a given rational point and to enumerate the set of all rational points of bounded canonical height in order to compute generators for the group of rational points of a given abelian variety. This is one of the fundamental tasks in the algorithmic theory of abelian varieties and is required, for instance, to numerically verify the celebrated conjecture of Birch and Swinnerton-Dyer for concrete examples. Such generators are especially interesting when the abelian variety in question is the Jacobian variety of a hyperelliptic curve. If, in this situation, we have generators for the group of rational points available, then there are efficient algorithms to compute the rational points on the curve with height below a prescribed bound, and the full set of integral points. Before the start of the project, the explicit theory of (canonical) heights on Jacobians of hyperelliptic curves had been mostly restricted to curves of genus 2 or 3. The main reason for this restriction was that an explicit theory for the so-called Kummer variety, a quotient of the Jacobian, was only available for genus at most 3. Such an explicit theory is required, because the canonical height is defined on the Kummer variety. In this project, we extended several known results for genus 2 and 3 to genus 4, starting with an explicit theory of the Kummer variety. On the one hand, this yields explicit formulas and efficient algorithms for genus 4, with applications as described above. We have implemented these algorithms. On the other hand, these explicit formulas suggest generalizations to arbitrary genus.
