Number Theory Hyperelliptic Curves
Lead Research Organisation:
University of Cambridge
Department Name: Pure Maths and Mathematical Statistics
Abstract
One aspect of Number Theory is to study the number of rational solutions to a set of polynomial equations defined over the rationals. One special type of polynomials is in the form of y^2=f(x) which defines a hyperelliptic curve. A well-known technique of tackling this problem is to consider the Jacobian variety of this hyperelliptic curve because the rational points on the Jacobian variety forms a finitely generated abelian group. Therefore, it is natural to consider the rank of this abelian group. The focus of my project is to find new ways to give an upper bound of the rank of the group of rational points on the Jacobian variety of a hyperelliptic curve of genus 2, that is when f is a generic polynomial of degree 6. More precisely, I am trying to generalise the method of Cassel-Tate paring on elliptic curves to genus 2 curves in order to improve the current upper bound on the rank achieved via 2-descent calculation.
Organisations
People |
ORCID iD |
| Jiali Yan (Student) |
Studentship Projects
| Project Reference | Relationship | Related To | Start | End | Student Name |
|---|---|---|---|---|---|
| EP/N509620/1 | 01/10/2016 | 30/09/2022 | |||
| 1951110 | Studentship | EP/N509620/1 | 01/10/2017 | 31/12/2020 | Jiali Yan |
| Description | The explicit rank of the group of rational points of an elliptic curve or the Jacobian variety of a higher genus curve is a center of interest in computational number theory. This is because explicit ranks are closely related to explicit solutions to certain algebraic equations. Throughout the development of the field, there are various ways to give upper bounds for the rank. One common method of obtaining an upper bound is via computing the n-Selmer group. However it is only feasible to compute the 2-Selmer group and the upper bound derived from it is often not very good. One main difficulty in computing higher Selmer groups is due to the involvement of the high degree number field coming from the n-torsion subgroup. Therefore, people look for methods to compute higher Selmer groups using the 2-Selmer group. Cassels-Tate pairing is developed with this principle in mind. Roughly speaking, if we can compute the 2-Selmer group and the Cassels-Tate pairing on the 2-Selmer group, we can recover the 4-Selmer group. Therefore, we potentially can avoid complicated number fields. This method has been implemented in various papers for elliptic curves. However, little has been done for Jacobians of higher genus curves. The most well understood higher genus curves are the family of the genus 2 curves. Hence it is natural to try to generalize the current algorithms of the explicit computation of the Cassels-Tate pairing on elliptic curves to the Jacobians of genus 2 curves. I, with the help of my supervisor, have successfully computed an explicit algorithm to compute the Cassel-Tate pairing on Sel^2(J) × Sel^2(J) given an arbitrary genus two curve with full rational two torsion points on its Jacobian. In particular I showed explicit examples where the pairing may indeed improve the rank bound obtained by descent calculations. |
| Exploitation Route | It is an interesting research topic with some great applications and connections to other areas. The result that I, with the help of my supervisor, achieved may lead to further studies in related topics. |
| Sectors | Other |