On a property of Fermi curves of 2-dimensional periodic Schr\"odinger operators

Abstract

We consider a compact Riemann surface with a holomorphic involution, two marked fixed points of the involution and a divisor obeying an equation up to linear equivalence of divisors involving all this data. Examples of such data are Fermi curves of 2-dimensional periodic Schr\"odinger operators. We show that the equation has a solution if and only if the two marked points are the only fixed points of the involution.