Three counterexamples to a conjecture of Colin de Verdière on multiplicity

Abstract

We exhibit closed hyperbolic surfaces of genus $10$, $17$, and $37$ such that the multiplicity of the first nonzero eigenvalue of their Laplacian is larger than the maximum conjectured by Yves Colin de Verdière in 1986. In order to determine these multiplicities, we apply the twisted Selberg trace formula to the representations induced by the isometry groups of these surfaces on corresponding triangle groups.