On algebraic solutions of Lamé's differential equation

Abstract

In two previous papers [ 1, 21 we reconsidered and improved Klein’s method for establishing whether a given second order linear differential equation with rational function coefficients (over an algebraic curve) has a full set of algebraic solutions. That procedure being rather complicated, we were interested in checking whether, when applied to classical examples, it would lead to nontrivial new results. The hypergeometric equation being in that respect well understood, Lame’s equation (see (0.1)) was the simplest type of differential equation for which we could hope to get interesting information by our method. Before explaining in detail the content of this paper, it may be useful to the reader if we recall a few facts, proved in [ 1, 21, of which constant use will be made throughout this paper. Let C be a nonsingular algebraic curve over the complex field C with function field K, and let L be a second order linear differential operator on C; that is,