Moduli spaces for Lamé functions and Abelian differentials of the second kind

The topology of the moduli space for Lamé functions of degree m is determined: this is a Riemann surface which consists of two connected components when m ≥ 2; we find the Euler characteristics and genera of these components. As a corollary we prove a conjecture of Maier on degrees of Cohn's polynomials. These results are obtained with the help of a geometric description of these Riemann surfaces, as quotients of the moduli spaces for certain singular flat triangles. An application is given to the study of metrics of constant positive curvature with one conic singularity with the angle 2π(2m + 1) on a torus. We show that the degeneration locus of such metrics is a union of smooth analytic curves and we enumerate these curves.