Eigenfunctions of the Laplacian Acting on Degree Zero Bundles over Special Riemann Surfaces
arXiv:math/0105051 · doi:10.1090/S0002-9947-04-03587-1
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.
20 pages, LaTeX. Subsection added on the relation between special Riemann surfaces and Jacobians with complex multiplication, typos corrected, one reference added. v3. Final version to appear in Trans.Am.Math.Soc