Explicit methods for Jacobi forms over number fields

This is a highly intra-disciplinary proposal connecting number theory, computational mathematics and representation theory. This proposal is designed to obtain new insights into the BSD conjecture through the use of theoretical and computational methods. The BSD conjecture describe a deep relationship between analytic and arithmetic properties of elliptic curves. More precisely the weak version says that the order of vanishing at the point 1, of the L-function associated with an elliptic curve E (the analytic rank) is equal to the rank of the abelian group given by the points of E (over some given field).

As yet the BSD conjecture is only proven completely for certain types of elliptic curves. In particular we know that it is true if the analytic rank is less than or equal to 1. However, there has recently been a lot of progress for versions of BSD on average by the 2014 Fields medallist M. Bhargava. Our most extensive knowledge about the BSD conjecture (as well as the original motivation for it) comes from numerical investigations.

Consider for the moment only rational elliptic curves, that is, they can be described by equations with integer coeffi cients. By a well-known construction of Eichler and Shimura we can associate such elliptic curves to certain complex functions (newforms). The converse of this construction is given by the so-called modularity theorem (former Taniyama-Shimura-Weil conjecture) which played an important role in Wiles' proof of Fermat's last theorem. From the modularity theorem we know that the L-function associated with the elliptic curve E is in fact equal to the L-function of the newform associated with E.

The problem of verifying BSD can then be transferred to that of computing L-functions of modular forms and in particular their values at 1. It turns out that if we have an elliptic curve E with associated newform f and L-function L(f,s) then we can obtain a whole family of curves by twisting so that the curve E twisted by a discriminant D is associated to the L-function L(fxD,s).

It is now known that there exists another complex function F such that the Fourier coefficients of F are given by the values of L(fxD,s) at s=1. If we compute the function F and its Fourier coefficients we might then be able to verify the BSD conjecture in specific cases.

The function F described above can either be given as a scalar newform of half-integral weight, a vector-valued modular form, or a so-called Jacobi form. In many ways the latter description is the most natural one. The relationship between the two functions f and F is called the Shimura correspondence.

In this project we aim at studying the BSD conjecture for elliptic curves over number fields instead of the rational numbers. In this case the modularity theorem is in general not known but only conjectured to hold. The Shimura correspondence in terms of Jacobi forms has so-far not even been studied in this setting. It is only with recent developments in the theory of Jacobi forms over number fields that it is possible to formulate precise conjectures about what the correspondence should look like.

The main goal of this project is to develop explicit methods and algorithms for Jacobi forms over number fields. In particular we will obtain dimension formulas for the spaces and develop algorithms which allow us to compute Fourier expansions and in the end obtain examples for BSD in the setting of elliptic curves over number fields.

One of the key points is that we associate the Jacobi forms (over number fields) with vector-valued Hilbert modular forms for the Weil representation. In this manner the Shimura correspondence can be realised as a correspondence between different Weil representations. Before reaching the main goal mentioned above we need a better understanding of Weil representations, lattices and finite quadratic modules over number fields. As part of the project we will therefore also focus on these objects.

The proposed project is concerned with fundamental research in the area of analytic number theory
and the results will have applications in arithmetic geometry. The project will have direct impact on
knowledge (scientific advances and techniques) and people (skills and career development). It will
have an indirect impact on society (national and international development) and economy (products
and procedures).

The impact on knowledge will be the provision of a new set of tools that will allow researchers to advance progress towards attacking the Birch and Swinnerton Dyer conjecture, one of the great unsolved problems in mathematics today. The beneficiaries will be mathematicians working in academia and elsewhere.

The impact on people will be through training of a researcher in a set of skills which are in short supply, that is the mix of both theoretical understanding and computational skills. Both they, and other junior researchers associated with the project, will benefit in terms of career development. And by extension, potential employers will benefit from these highly trained individuals. The PI will also benefit in terms of career development, helping him to establish his career in the UK and enhance his international profile.

The impact on society is more indirect due to the fundamental nature of the research. The aim is to increase interest in and understanding of pure mathematics by the general population, in particular to young people to encourage them to pursue mathematics as a career. Through both provision of easy-to-use free software and outreach activity, we hope to help realise this impact.

The impact on the economy is also indirect. The methods and tools developed in the proposal are of relevance to cryptography and could in the long-term improve on security measures against quantum computing for example. As we cannot predict this at the current time, this will be very much open for others to explore using the results of the project.