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

Lead Research Organisation:
Queen Mary University of London

Department Name: Sch of Mathematical Sciences

### Abstract

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.

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.

### Planned Impact

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.

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.

### Publications

Chatzakos D
(2021)

*On the distribution of lattice points on hyperbolic circles*in Algebra & Number Theory
Kurlberg P
(2023)

*Superscars for arithmetic point scatters II*in Forum of Mathematics, Sigma
Bui H
(2024)

*Weighted central limit theorems for central values of $L$-functions*in Journal of the European Mathematical Society
Jääsaari J
(2021)

*ON FUNDAMENTAL FOURIER COEFFICIENTS OF SIEGEL CUSP FORMS OF DEGREE 2*in Journal of the Institute of Mathematics of Jussieu
Jääsaari J
(2023)

*On signs of Fourier coefficients of Hecke-Maass cusp forms on 3*in Transactions of the American Mathematical Society
Huang B
(2022)

*Quantum variance for dihedral Maass forms*in Transactions of the American Mathematical SocietyDescription | Automorphic forms are highly symmetric functions on Lie groups that constitute one of the most important concepts in modern mathematics. We have undertaken a detailed study of the Fourier coefficients and moments of L-functions associated to certain automorphic forms, known as Siegel modular forms. We proved results on sign changes and upper and lower bounds on the Fourier coefficients, which was one of the goals of this project. We have also studied the fine distribution of lattice points lying on expanding circles in the hyperbolic plane, and shown that the angles are equidistributed in general. We established an asymptotic formula for the weighted quantum variance of dihedral Maass forms on in the large eigenvalue limit, for certain fixed discriminants. We have also proved the mass equidistribution of Saito-Kurokawa lifts, which was another main goal of the project. |

Exploitation Route | Our results and new method introduced should be very useful to other researchers working on analytic number theory and automorphic forms. |

Sectors | Creative Economy Education Other |

Description | Collaboration with Pitale and Schmidt |

Organisation | University of Oklahoma |

Department | Mathematics Department |

Country | United States |

Sector | Academic/University |

PI Contribution | I have partnered with Professors Pitale and Schmidt at the University of Oklahoma to make a series of investigations into the arithmetic aspects of automorphic forms. This has resulted in several papers (published, accepted, or currently under consideration by journals) and made a big impact on the field. |

Collaborator Contribution | Profs. Pitale and Schmidt have provided their expertise and intellectual input into the project and written papers with me. They have also invited me to Oklahoma to carry out this research and paid for my accommodation and subsistence while there. |

Impact | 1) Representations of SL2(R) and nearly holomorphic modular forms (with A. Pitale and R. Schmidt) RIMS Kôkyûroku (2016), 1973: 141-153. 2) Explicit refinements of Böcherer's conjecture for Siegel modular forms of squarefree level (with M. Dickson, A. Pitale and R. Schmidt), J. Math. Soc. Japan. (2020), 72(1): 251-301. 3) Lowest weight modules of Sp_4(R) and nearly holomorphic Siegel modular forms (with A. Pitale and R. Schmidt), Kyoto J. Math. 4) A note on the growth of nearly holomorphic vector-valued Siegel modular forms (with A. Pitale and R. Schmidt), L-functions and automorphic forms, Contrib. Math. Comput. Sci. (2018),. 5) On the standard L-function for GSp2n × GL1 and algebraicity of symmetric fourth L-values for GL2, Ann. Math. Québec. (2020). |

Start Year | 2014 |

Description | Erlass VI |

Form Of Engagement Activity | Participation in an activity, workshop or similar |

Part Of Official Scheme? | No |

Geographic Reach | Regional |

Primary Audience | Professional Practitioners |

Results and Impact | This was a research workshop that brought together local researchers in the Egham-Reading-London area interested in arithmetic statistics and number theory. |

Year(s) Of Engagement Activity | 2022 |

URL | https://sites.google.com/view/erlass-webpage/erlass-vi?authuser=0 |

Description | Workshop on Automorphic Forms and L-functions of higher rank |

Form Of Engagement Activity | Participation in an activity, workshop or similar |

Part Of Official Scheme? | No |

Geographic Reach | International |

Primary Audience | Professional Practitioners |

Results and Impact | This event brought together around 50 participants, including professional mathematicians and postgraduate students from around the world, working in higher rank automorphic forms, analytic aspects of L-functions, and representation theory. |

Year(s) Of Engagement Activity | 2023 |

URL | https://sites.google.com/view/automorphic-conference |