Explicit Methods for non-holomorphic Hilbert Modular Forms

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The theory of modular forms and L-functions 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 half-plane 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 so-called Laplace-Beltrami 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 non-holomorphic 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 non-holomorphic Hilbert modular forms, also called Hilbert - Maass cusp forms. These are functions defined on higher-dimensional 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 non-holomorphic Hilbert modular forms is towards the resolution of Hilbert's 11-th problem about quadratic forms in many variables. This, in turn, has far-reaching applications, for instance in quantum computing, where the so-called 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, so-called L-functions, which can be constructed from a given Hecke - Maass cusp form. In the case of classical modular forms, the theory of L-functions is very well understood and a large number of examples have been studied numerically, with all evidence supporting the so-called Grand Riemann Hypothesis (GRH). Recall that the Riemann Hypothesis (RH), one of the currently open Clay millennium problems, states that that all non-trivial 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 L-functions, including in particular L-functions associated with classical Modular forms and Maass cusp forms, as well as Hilbert modular forms and Hilbert - Maass forms. Although the L-functions associated with Hilbert - Maass forms have been studied extensively in certain cases from a theoretical perspective, no-one 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.