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The multiplicity of an eigenvalue $\lambda$ of the Laplace-Beltrami operator refers to the dimension of the eigenspace $$\mathcal{M}(\Gamma,\chi,\lambda)$$ of Maass newforms on the group $$\Gamma$$ of character $$\chi$$ with the same Laplace eigenvalue $$\lambda$$.

It is generally believed that these eigenspaces are one-dimensional unless there is a symmetry present. There are essentially three known cases where the dimension is greater than 1:

1. If the eigenvalue corresponds to an oldspace in the sense of Atkin and Lehner.

2. If the character $$\chi$$ has quadratic characters as factors.

3. If $$\Gamma=\Gamma_{0}(N)$$ with $9|N$

The above three cases are proven and the generalization of 3. to other square factors greater than 9 is conjectured.

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• Review status: reviewed
• Last edited by Andrew Sutherland on 2020-07-23 16:03:15
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