2003/09/25 by Lei Yang, Yang, Lei · 2 citations
Mathematics · Physics and Astronomy · #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Appell series #Basic hypergeometric series #Bilateral hypergeometric series #Differential algebraic equation #Differential algebraic geometry #Differential equation #First-order partial differential equation #Generalized hypergeometric function #Hypergeometric distribution #Hypergeometric function #Hypergeometric function of a matrix argument #Mathematical analysis #Mathematics #Moduli #Monodromy #Nonlinear Waves and Solitons #Nonlinear system #Ordinary differential equation #Partial differential equation #Physics #Pure mathematics #math.AG #math.NT #msc:11F55 #msc:11G15 #msc:14K20 #msc:22E40 #msc:33Cxx
paper · pdf · doi:10.48550/arxiv.math/0309415
published in arXiv (Cornell University) (Cornell University) · 119 pages
openalex publication_date 2003/09/25 · arxiv created 2004/12/16 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
In this paper, we study the monodromy of Appell hypergeometric partial differential equations, which lead us to find four derivatives which are associated to the group GL(3). Our four derivatives have the remarkable properties. We find that Appell hypergeometric partial differential equations can be reduced to four nonlinear partial differential equations by the use of our four derivatives. We investigate these four nonlinear partial differential equations. We are interested in some particular cases. In these cases, the system of linear partial differential equations has the remarkable properties. We find the η-functions associated to the unitary group U(2, 1), which has the interesting properties. Our η-functions lead us to study the arithmetic of Picard curves, especially the transform problems (moduli problems) and the corresponding modular equations. We give a rational transformation between two irrational integrals which are associated to some algebraic surfaces of degree seven. We find a modular equation which give the relation between the moduli of two Picard curves.