Project Details
An explicit theory of heights for hyperelliptic Jacobians
Subject Area
Mathematics
Term
from 2017 to 2024
Project identifier
Deutsche Forschungsgemeinschaft (DFG) - Project number 372107645
The ultimate goal of this project is 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.Thus far, the explicit theory of (canonical) heights on Jacobians of hyperelliptic curves has been mostly restricted to curves of genus 2 or 3. One reason for the restriction to small genus is that one first needs an explicit theory for the so-called Kummer variety of the Jacobian, which at the moment is only available for genus at most 3.In the proposed project, we will extend the known results for genus 2 and 3 to larger genus, starting with an explicit theory of the Kummer variety. On the one hand, this will yield explicit formulas and efficient algorithms for genus at least up to 5, which should be essentially optimal for genus up to 3 and possibly beyond. We will implement these algorithms, thereby making the efficient computation of generators in moderate genus possible, with applications as discussed above. On the other hand, we expect that these explicit formulas will suggest generalizations to arbitrary genus (and possibly to non-hyperelliptic curves), which we will then attempt to prove. To this end, we will conceptualize some of the explicit results and proofs for genus 2 and 3, which will also lead to a deeper understanding of the theory of canonical heights.
DFG Programme
Research Grants
International Connection
Netherlands