The accessory parameter problem in positive characteristic

We study the existence of Fuchsian differential equations in positive characteristic with nilpotent p-curvature, and given local invariants. In the case of differential equations with logarithmic local mononodromy, we determine the minimal possible degree of a polynomial solution. 2000 Mathematical Subject Classification: Primary 14D10, 12H20 This paper deals with second order differential equations with regular singularities in characteristic p > 0. Our main interest is to characterize those differential equations with nilpotent (resp. nilpotent but nonzero) p-curvature. This problem is known as Dwork’s accessory parameter problem. Differential equations with nilpotent p-curvature arise naturally in algebraic geometry. For example, Katz ([13]) showed that differential equations “coming from geometry”, such as Picard–Fuchs differential equations, have nilpotent p-curvature. The nilpotence (resp. nonvanishing) of the p-curvature may be characterized in terms of the existence of polynomial solutions. The study of polynomial solution of differential equations in positive characteristic goes back to Dwork ([7], [8]) and Honda ([11]). More recently, differential equations with nilpotent but nonzero p-curvature came up in Mochizuki’s work on p-adic uniformization ([14], [15]). Mochizuki develops the theory of indigenous bundles. On a curve of genus zero, these may be identified with differential equations with nilpotent but nonzero p-curvature ([6, §5]). To prove concrete existence results, the description of indigenous bundles as differential equations turns out to be more convenient. Differential equation with nilpotent but nonzero p-curvature also arise in the theory of reduction to characteristic p of Galois covers of curves. Solutions of differential equations arise in this context in the form of deformation data. In [6] one finds a correspondence between indigenous bundles and deformation data. (See also §2.2.) This paper combines results from the work of Mochizuki on indigenous bundles with results on deformation data and techniques from the work of Honda and Dwork. It turns out that combing these techniques is very fruitful and allows to answer questions which are interesting from all three points of view. We now give a more detailed description of the content of the paper.