GLOBALLY ANALYZABLE FUCHSIAN DIFFERENTIAL EQUATIONS

摘要

1. Introduction In various fields in mathematics and physics, such as the theory of automorphic functions, transcendental number theory, algebraic geometry, differential geometry, representation theory of Lie groups and Lie algebras, mathematical physics, and of course analysis, Fuchsian ordinary differential equations play a basic role. About 150 years ago, Fuchs, Frobenius et al. constructed a general theory of Fuchsian ordinary differential equations. The theory consists of a local theory at a regular singular point and a global relation for the sum of characteristic exponents, which we call the Fuchs relation. However, after that it seems there has been no essential progress in the general theory of Fuchsian ordinary differential equations. In particular, we do not yet have a method of calculating global quantities such as monodromies and connection coefficients, unless some special conditions hold. Here by special conditions we mean that the Fuchsian ordinary differential equation is free of accessory parameters or has an integral representation of solutions, etc. For Fuchsian ordinary differential equations free of accessory parameters, Okubo [55], [56] gave a perspective of the study by supplying many excellent ideas. Along Okubo's line, Yokoyama [74], [29] found a way to obtain all Fuchsian ordinary differential equations free of accessory parameters and clarified the structure of the set of such equations. Yokoyama's result looks like a goal in this direction. At almost the same time, Katz [42] also obtained a similar result, which covers the exceptional cases in Yokoyama's result. Dettweiler and Reiter [13], [14] noticed the relation between Katz's theory and Okubo's theory, and recently Oshima [58] clarified the relation between Yokoyama's theory and Katz's theory.