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Eigenfunctions of the Laplacian Acting on Degree Zero Bundles over Special Riemann Surfaces

Marco Matone
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Abstract

We find an infinite set of eigenfunctions for the Laplacian with respect to a flat metric with conical singularities and acting on degree zero bundles over special Riemann surfaces of genus greater than one. These special surfaces correspond to Riemann period matrices satisfying a set of equations which lead to a number theoretical problem. It turns out that these surfaces precisely correspond to branched covering of the torus. This reflects in a Jacobian with a particular kind of complex multiplication.

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Marco Matone. 2001-05-07. Eigenfunctions of the Laplacian Acting on Degree Zero Bundles over Special Riemann Surfaces. https://doi.org/10.1090/s0002-9947-04-03587-1

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