LMF: L-Functions and Modular Forms

L-functions and modular forms are fundamental mathematical objects that encode much of our knowledge of contemporary number theory. They form part of a web of interconnected objects, the understanding of which in the most basic cases lies at the foundations of much of modern mathematics. A spectacular example is Wiles' proof of Fermat's Last Theorem, which was an application of a fundamental "modularity" link between L-functions, modular forms and elliptic curves. This project will greatly extend and generalize such connections, both theoretically and computationally.

The research vision inspiring our programme can be summarised as: "Breaking the boundaries of classical L-functions and modular forms, and exploring their applications to 21st-century mathematics, physics, and computer science". Our guiding goal is to push forward both theoretical and algorithmic developments, in order to develop L-functions and modular forms far beyond current capabilities. This programme will systematically develop an extensive catalogue of number theoretic objects, and will make this information available through an integrated online resource that will become an indispensable tool for the world's research community.

L-functions are to pure mathematics what fundamental particles are to physics: their interaction reveal fundamental truths. To continue the analogy, computers are to number theorists what colliders are to particle physicists. Aside from their established role as empirical "testers" for conjectures and theories, experiments can often throw up quite unexpected phenomena which go on to reshape modern theory. Our programme will establish a major database and encyclopedia of knowledge about L-functions and related objects, which will play a role analogous to that of the LHC for the scientists at CERN. Both are at the threshold of tantalising glimpses into completely uncharted territories: higher degree L-functions for us and the Higgs boson for them.

Theoretical and computational work on higher degree L-functions has only started to make substantial progress in the past few years. There do not currently exist efficient methods to work with these, and rigorous computations with them are not yet possible. Neither is there yet an explicit description of all ways in which degree 3 L-functions can arise. We will address these facets in our research programme: both algorithmic development and theoretical classification.

As well as having theoretical applications to modularity relationships as in Wiles' proof, detailed knowledge of L-functions has more far-reaching implications. Collections of L-functions have statistical properties which first arose in theoretical physics. This surprising connection, which has witnessed substantial developments led by researchers in Bristol, has fundamental predictive power in number theory; the synergy will be vastly extended in this programme. In another strand, number theory plays an increasingly vital role in computing and communications, as evidenced by its striking applications to both cryptography and coding theory.

The Riemann Hypothesis (one of the Clay Mathematics Million Dollar Millennium Problems) concerns the distribution of prime numbers, and the correctness of the best algorithms for testing large prime numbers depend on the truth of a generalised version of this 150-year-old unsolved problem. These are algorithms which are used by public-key cryptosystems that everyone who uses the Internet relies on daily, and that underpin our digital economy. Our programme involves the creation of a huge amount of data about a wide range of modular forms and L-functions, which will far surpass in range and depth anything computed before in this area. This in turn will be used to analyse some of the most famous outstanding problems in mathematics, including the Riemann Hypothesis and another Clay problem, the Birch and Swinnerton-Dyer conjecture.

Those to benefit from the planned research certainly include mathematicians (especially number theorists), physicists and computer scientists, as well as a wider range of scientists who need to manage large amounts of data.

UK universities will benefit through the raised international profile of its mathematics research, through greater engagement of the public in its research, and through the attraction of well-qualified international students to study in their thriving, world-class research institutes.

The UK economy will benefit from the increased number of skilled mathematicians who are not only experts in their specialised field, but who have spent three years on a project where the presentation of their results to potential users has been key. Many of these are likely to continue to academic careers, while others will be highly sought-after for posts in the digital and knowledge economy, having acquired transferable skills in database management, website design and the visualisation of complex sets of data.

Economic and societal benefits will follow (in due course) from a better understanding by physicists of complex quantum systems (all modern electronic systems depend on quantum physics), and this understanding will be assisted by the study of L-functions since they exhibit all of the characteristic features of the spectra of complex quantum systems, yet are substantially more amenable to numerical modelling.

Additional societal benefits will come from applications of our work in cryptography, which is vital for the security and reliability of the internet, digital commerce, and national security.

One of the most spectacular results of the last 25 years, which made front-page news worldwide, was the proof of Fermat's Last Theorem through greater structural understanding of L-functions, which are the objects of study in our proposal. Three of the ten Fields medals awarded so far in this century were for work closely connected to the our L-function database, and we expect that applications of higher degree L-functions will similarly produce some of the greatest results of 21st century mathematics.