Solving Linear Differential Equations with Parameters: An algorithmic approach with applications to arithmetic geometry

Towards the end of the last century J. Kovacic constructed an algorithm which: (i) determines the differential Galois group of a secondorder linear homogeneous ordinary differential equation with coefficients in a field of rational functions over an algebraically closed field of characteristic 0; and (ii) determines from that group if the equation admits “elementary” solutions. We have been successful in extending Kovacic’s methods to cover the case of one-parameter families of second-order linear homogeneous equations, and the talk will outline the basic ideas.

The Galois theory of differential equations with parameters is originally due to Peter Landesman, and the special case of linear differential equations with parameters (Parametrized Picard-Vessiot Theory) has been recast quite recently in close analogy with the classical Picard-Vessiot theory by P. Cassidy and M. Singer. In recent work of Gorchinskiy and Ovchinnikov the work of Cassidy-Singer has been related to Gauss-Manin connections. We will derive the Picard-Fuchs equation as a simple application of our algorithm in order to illustrate our methods, and discuss other potential applications to arithmetic and algebraic geometry if time permits.