EPSRC Reference: 
EP/S032460/1 
Title: 
Modular symbols and applications 
Principal Investigator: 
Diamantis, Professor N 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

Department: 
Sch of Mathematical Sciences 
Organisation: 
University of Nottingham 
Scheme: 
Standard Research 
Starts: 
01 April 2020 
Ends: 
31 March 2023 
Value (£): 
331,558

EPSRC Research Topic Classifications: 

EPSRC Industrial Sector Classifications: 
No relevance to Underpinning Sectors 


Related Grants: 

Panel History: 

Summary on Grant Application Form 
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 Lfunctions (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 Lfunction 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 Lfunction 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 socalled 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 nonvanishing 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 Lfunctions.
The significance of these aims goes beyond the geometric setting they originated in, because questions about vanishing of Lfunctions 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.

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Organisation Website: 
http://www.nottingham.ac.uk 