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

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

2001-05-07

arxiv.org/abs/math/0105051v3

Trans.Am.Math.Soc. 356 (2004) 2989-3004

Mathematics

Algebraic Geometry

20 pages, LaTeX. Subsection added on the relation between special Riemann surfaces and Jacobians with complex multiplication,

Scientific paper

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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