Algebraic A-hypergeometric functions and their monodromy

Abstract

The study of hypergeometric functions started in 1813 with a paper by Gauss. Hypergeometric functions are generalizations of classical elementary functions such as arcsin and log. Around 1900, Appell, Lauricella and Horn studied hypergeometric functions in several variables. At the end of the 1980's these functions were further generalized to A-hypergeometric functions by Gelfand, Graev, Kapranov and Zelevinsky. They defined an A-hypergeometric function to be a function satisfying a system of partial differential equations. This system depends on a set of lattice points A in Zn (hence the name A-hypergeometric function) and a parameter vector β. For some choices of the set A and the parameter vector β, the solutions of the A-hypergeometric system are algebraic functions, i.e. solutions of an algebraic equation. It is a natural question for which A and β the solutions are algebraic. A first answer to this question was given by Schwarz in 1873. He gave a list of all irreducible algebraic Gauss functions. This list has been extended to several other hypergeometric functions. These results were obtained before A-hypergeometric functions were defined in general, and hence do not use A-hypergeometric theory. Some years ago Beukers found a combinatorial criterion by which it can easily be checked whether an A-hypergeometric function is algebraic. In this PhD-thesis, Esther Bod uses this criterion to extend Schwarz' list to all irreducible algebraic Appell, Lauricella and Horn functions. Furthermore, she gives a classification of all sets A and parameter vectors β such that the convex hull of the points in A is at most 2-dimensional and the solutions of the A-hypergeometric system are algebraic. An important aspect of hypergeometric functions is their monodromy. The monodromy group is a representation of the fundamental group on the solution space of the system of differential equations. Hence it contains information about the analytic continuation of the solutions. It can be shown that the monodromy group is finite if and only if the solutions are algebraic. Up to now, algebraic functions were usually found by computing the monodromy group and determining the parameters for which it is finite. However, in this thesis Esther Bod first computes the algebraic functions by the combinatorial criterion mentioned above and then computes the monodromy groups of the algebraic Appell and Horn functions, using the fact that these groups are finite. For some of the Appell, Lauricella and Horn functions, there are 1-parameter families of algebraic functions. In this thesis explicit formulas, depending on the parameter, are given for these functions. The minimal polynomials and the Galois groups of these functions can easily computed from the formulas