Abstract

In the first three chapters of this dissertation we give an introduction to the theory of ordinary linear differential equations with coeffcients in C(z). In particular we consider the case of Fuchsian differential equations. Classically one is interested in describing all Fuchsian equations whose solution spaces only consist of algebraic
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functions. Such equations are called algebraic. To each Fuchsian differential equation a group can be defined, the monodromy group of the equation. This monodromy group determines whether or not a given Fuchsian equation is algebraic. If the monodromy group is finite then the equation is algebraic, otherwise it is not.
It is known when a given Fuchsian equation of order 1 only has algebraic solutions. In this case the monodromy group is cyclic. It is a lot more complicated to give a classification of all algebraic Fuchsian equations of order at least 2. A classical example of a second order Fuchsian equation with 3 singular points is the hypergeometric equation. A complete classification of all algebraic hypergeometric equations was given by H.A. Schwarz in 1873. It induced a description of all algebraic Fuchsian equations of order 2 with 3 singular points.
The next category of Fuchsian equations of order 2 consists of second order equations with 4 singular points. Little is known about their algebraicity. One reason for this is the appearance of an accessory parameter in each of the equations. In this thesis we consider the specific second order Fuchsian equation with 4 singular points known as the Lamé (differential) equation Ln(y) = 0, see Definition 4.1.1. The general question we try to answer in Chapters 4, 5 and 6 is when we have an algebraic Lamé equation. In the work of F. Baldassarri and B. Chiarellotto a systematic approach for solving this question was given for the first time. In Chapters 4 en 5 we reprove some of their results in a simplified way. Moreover, we give a complete list of the finite groups that are the most likely to occur as the monodromy groups of the algebraic Lamé equations.
In Chapter 6 we give an algorithm to carry out the procedure of determining all algebraic Lamé equations with a given monodromy group and index n. The underlying ideas in the algorithm are based on the invariant action of the monodromy group on polynomials in two variables and polynomial solutions of symmetric powers of Ln. This kind of technique has been used on several previous occasions. To conclude this thesis, we present the beginning of the enumeration of the algebraic Lamé equations as tables in Appendix A.
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