HESSIAN POLYHEDRA, INVARIANT THEO & APPELL HYPERGEOME FUNCT

by Lei Yang

Overview

Our book gives the complex counterpart of Klein's classic book on the icosahedron. We show that the following four apparently disjoint theories: the symmetries of the Hessian polyhedra (geometry), the resolution of some system of algebraic equations (algebra), the system of partial differential equations of Appell hypergeometric functions (analysis) and the modular equation of Picard modular functions (arithmetic) are in fact dominated by the structure of a single object, the Hessian group $mathfrak{G}’_{216}$. It provides another beautiful example on the fundamental unity of mathematics.

Product Details

ISBN-13:	9789813209497
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	03/13/2018
Sold by:	Barnes & Noble
Format:	eBook
Pages:	316
File size:	70 MB
Note:	This product may take a few minutes to download.

Table of Contents

Part I Geometry and arithmetic associated with Appell hypergeometric functions

1 Introduction 1

2 Four derivatives and their properties 32

3 Nonlinear partial differential equations and modular functions associated to U(2, 1) 53

4 η-functions associated to U(2, 1) and Picard curves 72

5 Transform problems and modular equations associated to algebraic surfaces 86

6 Triangular s-functions associated to U(2, 1) and Picard curves 95

7 Four derivatives and nonlinear evolution equations 108

References for Part I 112

Part II Hessian polyhedra, invariant theory and Appell hypergeometric functions

1 Introduction 117

2 The reciprocity law about the Appell hypergeometric functions 149

3 The algebraic solutions of Appell hypergeometric partial differential equations 155

4 The Hessian polyhedral equations 160

5 Invariant theory for the system of algebraic equations 188

6 Ternary cubic forms associated to Hessian polyhedra 201

7 Some rational invariants on CP² 211

References for Part II 219

Part III Galois representations arising from twenty-seven lines on a cubic surface and the arithmetic associated with Hessian polyhedra

1 Introduction 225

2 Hessian polyhedra and cubic forms associated to G_{25, 920} 239

3 Hessian polyhedra and Picard modular forms 247

4 Hessian polyhedra and Galois representations associated with cubic surfaces 274

5 Hessian polyhedra and the arithmetic of rigid Calabi-Yau threefolds 291

References for Part III 304