Modular symbols and applications

One of the most effective ways to gain insight into difficult arithmetic questions in modern Number Theory is by visualising them through a certain type of curves, called elliptic curves. These are curves with a natural form and a simple defining equation but their structure has far reaching consequences, famously including the proof of Fermat's Last Theorem by Wiles et al. Some of the deepest properties of elliptic curves, in turn, can be formulated in terms of functions on the complex plane called L-functions (a fact at the heart of Wiles' proof). For example, there are infinitely many rational solutions of the equation defining an elliptic curve if the corresponding L-function vanishes upon evaluation at 1. This is a special case of one of the 'Millennium Problems' for whose solution a 1 million USD prize is available.

Understanding better the vanishing of a certain L-function at 1 should also answer a question recently asked by Mazur and Rubin, two of the most eminent number theorists in the last hundred years. Instead of the set of rational numbers, consider a so-called number field, a larger set of numbers with properties mirroring those of the rationals. The question they asked is: How do the solutions of the equation defining an elliptic curve change if we ask for those solutions to belong to varying number fields?

They noticed that the non-vanishing problem behind this question can be expressed in terms of a fundamental invariant called modular symbol. This motivated them to study the statistics of modular symbols and especially their "mean" and "variance". Upon analysing numerical data, they, together with W. Stein, formulated conjectures describing precisely the large scale behaviour of those means and variances.

In a recent paper, we fully proved one of those two conjectures. With that breakthrough achieved, the three main aims of the proposed project then are:

1. To establish the full proof of the conjecture by Mazur et al. that deals with the "variance" of modular symbols.

2. To formulate and prove analogous conjectures for Theta coefficients and Theta elements, which are important algebraic counterparts of the objects featuring in the above conjectures.

3. To use the outcomes of Aims 1 and 2 to derive information about vanishing of L-functions.

The significance of these aims goes beyond the geometric setting they originated in, because questions about vanishing of L-functions is an area of huge significance for analytic number theory. The project will not deal directly with applications to arithmetic and geometric problems but the methods introduced will be pivotal for progress in arithmetic and geometric as well as analytic aspects. Likewise, although applications outside Mathematics is not part of the proposal, results from the project will indirectly have potential impact on Elliptic Curve Cryptography.

Who will benefit from this research?

1. Postgraduate students and young mathematicians.
2. Wider public.
3. Security systems companies (indirectly).

How will they benefit from this research?

1. The project will have a strong training element for the PDRA and other young mathematicians connected with the project. It will provide sophisticated experience into analytic techniques motivated by an important conjecture and, especially in Aims 2 and 3, it will involve a mix of conceptually and technically challenging work. In addition, the project includes significant input from algebraic and geometric subjects, further increasing its training value for younger mathematicians.

2. I will use the proposed research to inform the general public about fundamental research. The motivating questions of the project involve elliptic curves which can be visualised through the use of computers and visual technology. The actual questions studied in the proposal (e.g. statistics about the growth of important quantities) can likewise be visualised in a way which is meaningful to the general public. This has the potential to more widely engage the public, encouraging further interest in studying Pure Mathematics and appreciation of the value of the subject.

3. A potential indirect impact of the proposed research is applications to security systems. Specifically, some of the motivating problems of the project are formulated in terms of elliptic curves. Although it is not an aim of the proposal to investigate direct applications to elliptic curves, our results will provide key insight which, as discussed in "Case for Support" and "Pathway to Impact", will be valuable for mathematicians working on elliptic curves. On the other hand, advances in the theory of elliptic curves are used by Elliptic Curve Cryptography, an important modern cryptosystem. Therefore, achieving the aims of the proposal, may, indirectly, have an essential impact on Elliptic Curve Cryptography.