# Arithmetic aspects of automorphic forms: Petersson norms and special values of L-functions

Lead Research Organisation:
University of Bristol

Department Name: Mathematics

### Abstract

The Langlands program has been an area of active research in mathematics for the last forty years and consists of a vast web of theorems and conjectures connecting objects in number theory, representation theory, analysis and geometry. Central to the Langlands program are automorphic forms and their associated L-functions. These objects arise from analysis and representation theory. Despite slow and steady progress, many fundamental questions about the relationship between automorphic forms and number theory/arithmetic geometry remain unanswered. The goal of this project is gain new insights into some of these questions by making a deep investigation of three key topics: a) Nearly holomorphic Siegel modular forms, b) Deligne's conjecture on algebraicity of critical L-values, c) Ratios of Petersson norms for functorially related automorphic forms.

Nearly holomorphic Siegel modular forms were first introduced by Shimura and have been indispensable for studying special values of automorphic L-functions. However, despite their ubiquity, their arithmetic properties and place in the Langlands framework have not yet been fully understood. This project will study their representation-theoretic and arithmetic properties and prove a close link between them and "vector valued Siegel modular forms".

The insights gained from the study of nearly holomorphic modular forms will be used to tackle some special cases of a famous conjecture made by the Fields medal winning mathematician Pierre Deligne. The simplest example of Deligne's conjecture is the classical fact that the value of the Riemann zeta function at all positive even integers is a power of pi times a rational number. This research will prove the expected rationality results for much more complicated L-functions that are of great importance in several fields of mathematics.

This research will also aim to prove a well-known conjecture made by Ibukiyama and Katsurada concerning ratios of Petersson norms for certain Siegel modular forms. This problem has deep significance as it concerns the behavior of arithmetic properties of automorphic forms under Langlands lifting. Solving this will require innovative adaptation to existing methods.

The methods used in this project will be a powerful mix of classical techniques, modern representation theoretic methods, and group cohomology. This research will open several new avenues for further exploration.

Nearly holomorphic Siegel modular forms were first introduced by Shimura and have been indispensable for studying special values of automorphic L-functions. However, despite their ubiquity, their arithmetic properties and place in the Langlands framework have not yet been fully understood. This project will study their representation-theoretic and arithmetic properties and prove a close link between them and "vector valued Siegel modular forms".

The insights gained from the study of nearly holomorphic modular forms will be used to tackle some special cases of a famous conjecture made by the Fields medal winning mathematician Pierre Deligne. The simplest example of Deligne's conjecture is the classical fact that the value of the Riemann zeta function at all positive even integers is a power of pi times a rational number. This research will prove the expected rationality results for much more complicated L-functions that are of great importance in several fields of mathematics.

This research will also aim to prove a well-known conjecture made by Ibukiyama and Katsurada concerning ratios of Petersson norms for certain Siegel modular forms. This problem has deep significance as it concerns the behavior of arithmetic properties of automorphic forms under Langlands lifting. Solving this will require innovative adaptation to existing methods.

The methods used in this project will be a powerful mix of classical techniques, modern representation theoretic methods, and group cohomology. This research will open several new avenues for further exploration.

### Planned Impact

This research will have significant impact on the academic areas of Siegel modular forms, automorphic representations and special values of L-functions. In particular, the successful synthesis of various different mathematical tools for the purpose of this project, together with the key results obtained, will influence future work in this area and immensely benefit other researchers.

Furthermore, holomorphic and nearly holomorphic Siegel modular forms arise not just in pure mathematics but also in the physics of black holes. So the detailed study of nearly holomorphic and vector valued Siegel modular forms of degree 2, which is one of the goals of this project, has potential applications to physics.

Regarding non-academic impacts, it is worth noting that number theory has many important applications in everyday life. As an example, the study of elliptic curves (which is related to the subject matter of this proposal) led to elliptic curve cryptography which is widely used today to protect sensitive data. Another example is the role played by modular forms in coding theory which is of key importance in data compression. Indeed, Siegel modular forms -- central to this research -- have been used by Choie and Duke to study multiple weight enumerators of certain binary linear codes. The PI will keep in close touch with the latest developments in the area of coding theory, and will discuss possible applications of this project with information technology researchers.

Finally, the proposed research will raise the UK's profile internationally. In order for the UK to remain competitive, the importance of ``basic research" cannot be over-emphasized. Developments in basic research can lead to unexpected applications in seemingly unrelated areas. A successful completion of this project would establish the UK as a new center for excellence of research in this field and will forge new links between the University of Bristol and two other leading international research institutes in the field: Oklahoma University, USA, and IISER Pune, India.

Furthermore, holomorphic and nearly holomorphic Siegel modular forms arise not just in pure mathematics but also in the physics of black holes. So the detailed study of nearly holomorphic and vector valued Siegel modular forms of degree 2, which is one of the goals of this project, has potential applications to physics.

Regarding non-academic impacts, it is worth noting that number theory has many important applications in everyday life. As an example, the study of elliptic curves (which is related to the subject matter of this proposal) led to elliptic curve cryptography which is widely used today to protect sensitive data. Another example is the role played by modular forms in coding theory which is of key importance in data compression. Indeed, Siegel modular forms -- central to this research -- have been used by Choie and Duke to study multiple weight enumerators of certain binary linear codes. The PI will keep in close touch with the latest developments in the area of coding theory, and will discuss possible applications of this project with information technology researchers.

Finally, the proposed research will raise the UK's profile internationally. In order for the UK to remain competitive, the importance of ``basic research" cannot be over-emphasized. Developments in basic research can lead to unexpected applications in seemingly unrelated areas. A successful completion of this project would establish the UK as a new center for excellence of research in this field and will forge new links between the University of Bristol and two other leading international research institutes in the field: Oklahoma University, USA, and IISER Pune, India.

### Publications

Pitale A
(2016)

*Representations of SL_2(R) and nearly holomorphic modular forms*
Pitale A
(2020)

*On the standard L-function for $$\mathrm{GSp}_{2n} \times \mathrm{GL}_1$$ and algebraicity of symmetric fourth L-values for $$\mathrm{GL}_2$$*in Annales mathématiques du Québec
Pitale A
(2020)

*Lowest weight modules of Sp_4(R) and nearly holomorphic Siegel modular forms*in Kyoto Journal of Mathematics
Saha A
(2017)

*Hybrid sup-norm bounds for Maass newforms of powerful level*in Algebra & Number Theory
Saha A
(2017)

*On sup-norms of cusp forms of powerful level*in Journal of the European Mathematical Society
Saha A
(2016)

*Large Values of Newforms on GL(2) with Highly Ramified Central Character*in International Mathematics Research NoticesDescription | A major achievement of the research funded by this grant has been the discovery of new results on the arithmetic properties of automorphic forms and the special values of their L-functions, especially in the context of Siegel modular forms. The critical values of L-functions attached to cohomological cuspidal automorphic representations are objects of deep arithmetic significance. In particular, a famous conjecture of Deligne predicts that these values are algebraic numbers up to multiplication by suitable periods. In joint work with Pitale and Schmidt, I have proved new results in this direction for Siegel modular forms (in review at Annales de Mathematics at Quebec). Moreover, by explicating the algebraic structure of the relevant space of n-finite automorphic forms, we were able to prove a structure theorem for the space of nearly holomorphic vector-valued Siegel modular forms of (arbitrary) weight with respect to an arbitrary congruence subgroup of Sp_4 (Q) (to appear in Kyoto J. Math). In another significant work carried out jointly with Dickson, Pitale and Schmidt, we explicitly computed certain local factors which together with recent work of Furusawa and Morimoto has yielded a proof of the famous Böcherer conjecture. This has been accepted in Journal of the Mathematical Society of Japan. Another major achievement of the research funded by this grant in a very different direction has been in the field of supremum norms of eigenfunctions. Since the seminal work of Iwaniec and Sarnak [Ann. of Math. (2), 141(2):301-320, 1995] it has been understood that for Hecke eigenfunctions on negatively curved arithmetic surfaces, one can exploit certain symmetries to prove strong bounds on the sup-norms that go beyond what can be proved by general tools from analysis and geometry. In this case, the eigenfunctions are example of automorphic forms, which play a fundamental role in number theory and in the deep web of conjectures known as the Langlands program. In the last few years, there has been a lot of work on proving strong sup-norm bounds for higher dimensional arithmetic manifolds, with outstanding results by many top researchers, including Blomer, Harcos, Marshall, Michel, and Templier. However, virtually all results proved so far in the higher dimensional situation assume that the manifold is fixed. Thus, a major open problem here is to understand the dependence of the sup-norms on the manifold itself; i.e., on the level of the associated automorphic form. Of particular interest is the case where high powers of prime numbers divide the level; in this so called powerful level situation, existing methods are insufficient and new tools are needed. At the same time, the case of powerful levels is full of unexpected phenomena (such as stronger bounds!) and rich connections with the Langlands program. My two single-author papers on this topic [published in JEMS and Algebra and Number Theory] have introduced completely new techniques to solve the longstanding and wide open problem for sup-norms of powerful levels on the group GL(2). These latter papers, which were a direct result of the research funded by this grant, were the first time anyone had proved an upper bound for the sup-norm for powerful levels, for any group. Another key finding from the research funded by this grant has been a recent work with Corbett that gives an exact formula for the ramification index of a modular parametrization at any cusp (published in Math. Res. Lett.). |

Exploitation Route | Our results on the arithmetic of nearly holomorphic forms and L-values should be very useful to other researchers working on arithmetic aspects of automorphic forms. My results on sup-norms have also opened up a whole new area related to analytic properties of automorphic forms of powerful level with lots of important basic questions which are currently being pursued by myself as well as several mathematicians around the world (Blomer, Marshall, Milicevic, Hu, Templier and others). |

Sectors | Creative Economy,Digital/Communication/Information Technologies (including Software),Education,Other |

URL | http://www.maths.qmul.ac.uk/~asaha/research/ |

Description | This research has raised the UK's profile internationally. The area of Siegel modular forms and special values of automorphic L-functions has so far been under- represented in the UK, with much of the global activity occurring in the USA, France, Germany and Japan. This project has established the UK as a new center for excellence of research in this field. The collaborations between the PI, Prof. Schmidt, and Prof. Pitale has created an international research network which has had wide academic and non-academic benefits and led to a stream of visits and communications between researchers based in UK, USA and India. Further, this research has had significant impact on people via generating new academic positions (e.g., Martin Dickson). One of the team members (M. DIckson) has used the expertise gained as part of this project to make impact in the area of data science and in the creation of new datasets (related to the LMFDB project) which has had a significant impact on information technology. |

First Year Of Impact | 2016 |

Sector | Digital/Communication/Information Technologies (including Software),Education |

Impact Types | Societal,Economic |

Description | New investigations in automorphic forms: Arithmetic and analytic interfaces |

Amount | £293,685 (GBP) |

Funding ID | RPG-2018-401 |

Organisation | The Leverhulme Trust |

Sector | Charity/Non Profit |

Country | United Kingdom |

Start | 03/2019 |

End | 09/2023 |

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 |