Abstract
We describe three new algorithms related to the rigorous computation of Maass cusp forms.Firstly, we describe a novel algorithm to compute and rigorously verify the Laplace eigenvalue and Hecke eigenvalues of Maass cusp forms of squarefree level and trivial character. The main tool we use is an explicit version of the Selberg trace formula.
We then describe a new algorithm to unconditionally compute the class numbers of real quadratic fields. Again, the main tool used here is an explicit trace formula for Maass forms of level 1 and a dataset of rigorously verified Maass forms.
Finally, we describe a method to extend Hejhal's algorithm to rigorously zoom into a Laplace eigenvalue of a Maass form, once we know it exists in a small interval. With this, we derive a test to show whether or not the main matrix appearing in this algorithm for level 1 Maass forms is well-conditioned as the matrix size increases.
| Date of Award | 9 May 2023 |
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| Original language | English |
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| Supervisor | Andrew R Booker (Supervisor) |