Automorphic forms on higher rank groups: Fourier coefficients, L-functions, and arithmetic

The proposed research lies at the interface of number theory with algebra, geometry, analysis and mathematical physics. Motivated by fundamental conjectures, we propose to develop powerful new tools to investigate automorphic forms on higher rank groups in order to approach some of the deepest open problems in the field.

Automorphic forms are highly symmetric functions on Lie groups that constitute one of the most important concepts in modern mathematics. They are key to number theory, e.g., understanding polynomial equations with integer coefficients, and lie at the centre of many of the most important problems in the subject. For instance, Sir Andrew Wiles' proof of Fermat's Last Theorem in 1995 relied on a deep connection between modular forms (an example of automorphic forms) and elliptic curves. Together with their associated L-functions, automorphic forms are also central objects in the Langlands programme - a vast web of theorems and conjectures connecting algebra, geometry, number theory, and analysis - which is one of the most active areas of mathematical research today. Additionally, two of the seven Clay "million dollar" Millennium Prize Problems lie in the area of automorphic L-functions. Automorphic forms also have connections to several areas of mathematical physics, such as quantum chaos, string theory, and quantum field theory.

Over the last century, there has been considerable progress in our detailed understanding of modular forms and Maass forms, which are the two types of automorphic forms on the (rank 1) group GL(2). However, progress in the higher rank cases has been much more limited. Indeed, the analytic aspects of automorphic forms on higher-rank groups has come into focus only in the last few years, with progress largely limited to special cases such as GL(3). In higher rank settings, existing methods and paradigms break down requiring the development of new ideas and innovations.

This project sets out to make far-reaching breakthroughs relating to the circle of ideas around Fourier coefficients of automorphic forms, period formulas, L-functions, and arithmetic to resolve some of the most important and substantial problems in the field. In a significant departure from existing work in this field, we will approach these problems simultaneously from the analytic, algebraic, and arithmetic directions. This project unifies these research areas at the level of results (new "master theorems" that bring several previous results under one umbrella), methods (by combining distinct methodological frameworks), and fields (we will bring together different fields of mathematics which have seen relatively little interaction). This will allow us to make advances that were previously inaccessible.

Ultimately, this research will provide a new bridge between the Langlands programme and several topics in number theory, geometry, algebra, analysis, and mathematical physics. Moreover, it will resolve some of the most substantial problems in the field in higher rank settings such as the distribution of generalised Fourier coefficients, Quantum Unique Ergodicity, the sup-norm problem, subconvexity, moments of families of L-functions, and Deligne's conjecture on critical values of L-functions, as well as open up numerous avenues for further exploration.

For the UK to remain competitive internationally, the importance of funding research in pure sciences cannot be over-emphasized. Number theory constitutes one of the most important branches of pure mathematics. Through unifying different methodologies based on analytic, algebraic and arithmetic approaches our proposal will go beyond well-established barriers in number theory and raise the international profile of the UK in this area of research. This will create academic impact by opening up new directions of research and strengthening connections between research areas. In turn, this will benefit researchers working in these areas both in the UK and internationally.

To maximise the academic impact of the proposal we will present our findings in high-level workshops and conferences in the UK and internationally. We will also develop and maintain a broad network of academic collaboration with world-leading researchers in the UK, USA and continental Europe. This will be achieved through regular research visits and seminar talks and will lead to long-lasting impact in terms of strengthening collaborations and recruitment of outstanding researchers at all career stages. Moreover, we will organise an EPSRC funded workshop in 2021 that will bring together a diverse group of top experts in number theory, representation theory and mathematical physics. The findings from the project will be published in high-impact academic journals, with early access to all our results available on the arXiv e-print server.

Number theory naturally captures the interest of wider audiences beyond mathematicians since many of the most substantial problems in the field can be expressed in an elementary way making it possible to convey the beauty and complexity lying in the background to wider audiences. Queen Mary University of London provides opportunities to engage the wider community through public lecture series as well as through its Centre for Public Engagement. Taking advantage of these opportunities, we will inspire young general audiences by lecturing on number theory at the School of Mathematical Sciences "Taster Days" lecture series, which introduces students from nearby schools in London to advanced, yet accessible mathematics.

Like all foundational research in pure mathematics, the non-academic impacts of this research will likely be in the longer term, and are somewhat harder to exactly predict. However, developments in number theory have frequently led to unexpected real-world applications, especially in the field of applied computer science (e.g., in cryptography). We will aim for the widest possible dissemination of our work, including to researchers at the Heilbronn Institute of Mathematical Research and the Institute of Applied Data Science, to identify any real-world applications of our methods and results.

Finally, another way in which the proposal will create impact is through training the PDRA in advanced mathematics that requires a unique and valuable skill-set. For example, a PDRA on a previous EPSRC grant of the PI was hired by a technology company that produces virtual reality simulations. The skills he developed in problem solving and quantitative analysis are crucial for his current position in industry.