EPSRC Reference: 
EP/V026321/1 
Title: 
Explicit Methods for nonholomorphic Hilbert Modular Forms 
Principal Investigator: 
Stromberg, Dr F K 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

Department: 
Sch of Mathematical Sciences 
Organisation: 
University of Nottingham 
Scheme: 
Standard Research 
Starts: 
01 January 2022 
Ends: 
30 September 2025 
Value (£): 
356,966

EPSRC Research Topic Classifications: 

EPSRC Industrial Sector Classifications: 
No relevance to Underpinning Sectors 


Related Grants: 

Panel History: 

Summary on Grant Application Form 
The theory of modular forms and Lfunctions is an important part of modern number theory. For instance, it is one of the fundamental building blocks used in the celebrated proof of Fermat's last theorem by Wiles and Wiles  Taylor. Classical modular forms are holomorphic functions defined on the complex upper halfplane transforming in a certain way with respect to congruence subgroups of the modular group.
One particularly interesting generalisation is to functions that are no longer holomorphic. Instead, one considers eigenfunctions of the socalled LaplaceBeltrami operator. This leads to the theory of Maass waveforms, and, especially, Maass cusp forms, which can be interpreted in terms of quantum mechanical particles moving on a hyperbolic surface. The study of these nonholomorphic modular forms is therefore intricately linked to the spectral theory of the surface they are defined on.
In this project we extend this even further and study nonholomorphic Hilbert modular forms, also called Hilbert  Maass cusp forms. These are functions defined on higherdimensional spaces (the 'Hilbert modular varieties') which in some cases are like surfaces but, generally, they are quite complex and difficult to understand geometrically. As a result, apart from some special cases, there are no explicit examples of general Hilbert  Maass cusp forms. One of the primary goal of this project is to develop algorithms to find explicit numerical examples. These will help us to obtain a better understanding of both individual Hilbert  Maass cusp forms as well as the tools required to study them (notably, spectral theory). One of the most striking applications of nonholomorphic Hilbert modular forms is towards the resolution of Hilbert's 11th problem about quadratic forms in many variables. This, in turn, has farreaching applications, for instance in quantum computing, where the socalled strong approximation properties of certain quadratic forms can be used to design universal quantum gates.
The second aim of this project is to study a class of associated functions, socalled Lfunctions, which can be constructed from a given Hecke  Maass cusp form. In the case of classical modular forms, the theory of Lfunctions is very well understood and a large number of examples have been studied numerically, with all evidence supporting the socalled Grand Riemann Hypothesis (GRH). Recall that the Riemann Hypothesis (RH), one of the currently open Clay millennium problems, states that that all nontrivial zeros of the Riemann zeta function lie on a certain vertical line. The more general GRH asserts that the same should hold true for a much larger class of Lfunctions, including in particular Lfunctions associated with classical Modular forms and Maass cusp forms, as well as Hilbert modular forms and Hilbert  Maass forms.
Although the Lfunctions associated with Hilbert  Maass forms have been studied extensively in certain cases from a theoretical perspective, noone has made an attempt at their computation so far. One of the aims of this project is to open up this field for further exploration by constructing algorithms and generate large classes of examples in a number of interesting cases. Our algorithms and results will all be made available to other researchers and we hope that these will inspire new theoretical conjectures and theorems.

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