ARITHMETIC INSIGHTS
CONTENTS INTRODUCTION 0 1 OUTLINE OF THE THEORY 0 2 COMPARISON WITH OTHER THEORIES CHAPTER 1 ALGEBRAIC BACKGROUND 1 1 ALGEBRA 1 2 ALGEBRAIC GEOMETRY 1 3 SUPERALGEBRA CHAPTER 2 CLASSICAL DI?ERENTIAL GEOMETRY REVISITED 2 1 CONNECTIONS IN PRINCIPAL BUNDLES AND CURVATURE 2 2 LIE ALGEBRA AND CLASSICAL GROUPS 2 3 INVOLUTIONS AND SYMMETRIC SPACES 2 4 LOGARITHMIC DERIVATIVE AND DI?ERENTIAL GALOIS GROUPS 2 5 CHERN CONNECTIONS: THE SYMMETRIC/ANTI-SYMMETRIC CASE 2 6 CHERN CONNECTIONS: THE HERMITIAN CASE 2 7 LEVI-CIVIT`A CONNECTION AND FEDOSOV CONNECTION 2 8 LOCALLY SYMMETRIC CONNECTIONS 2 9 EHRESMANN CONNECTIONS ATTACHED TO INNER INVOLUTIONS 2 10 CONNECTIONS IN VECTOR BUNDLES 2 11 LAX CONNECTIONS 2 12 HAMILTONIAN CONNECTIONS 2 13 CARTAN CONNECTION 2 14 WEIERSTRASS AND RICCATI CONNECTIONS 2 15 DI?ERENTIAL GROUPS: CASSIDY AND PAINLEV´E CHAPTER 3 ARITHMETIC DI?ERENTIAL GEOMETRY: GENERALITIES 3 1 GLOBAL CONNECTIONS AND THEIR CURVATURE 3 2 ADELIC CONNECTIONS 3 3 SEMIGLOBAL CONNECTIONS AND THEIR CURVATURE; GALOIS CONNECTIONS 3 4 CURVATURE VIA ANALYTIC CONTINUATION BETWEEN PRIMES 3 5 CURVATURE VIA ALGEBRAIZATION BY CORRESPONDENCES 3 6 ARITHMETIC JET SPACES AND THE CARTAN CONNECTION 3 7 ARITHMETIC LIE ALGEBRAS AND ARITHMETIC LOGARITHMIC DERIVATIVE 3 8 COMPATIBILITY WITH TRANSLATIONS AND INVOLUTIONS 3 9 ARITHMETIC LIE BRACKETS AND EXPONENTIAL 3 10 HAMILTONIAN FORMALISM AND PAINLEV´E CONTENTS 3 11 P-ADIC CONNECTIONS ON CURVES: WEIERSTRASS AND RICCATI CHAPTER 4 ARITHMETIC DI?ERENTIAL GEOMETRY: THE CASE OF GLN 4 1 ARITHMETIC LOGARITHMIC DERIVATIVE AND EHRESMANN CONNECTIONS 4 2 EXISTENCE OF CHERN CONNECTIONS 4 3 EXISTENCE OF LEVI-CIVIT`A CONNECTIONS 4 4 EXISTENCE/NON-EXISTENCE OF FEDOSOV CONNECTIONS 4 5 EXISTENCE/NON-EXISTENCE OF LAX-TYPE CONNECTIONS 4 6 EXISTENCE OF SPECIAL LINEAR CONNECTIONS 4 7 EXISTENCE OF EULER CONNECTIONS 4 8 CURVATURE FORMALISM AND GAUGE ACTION ON GLN 4 9 NON-EXISTENCE OF CLASSICAL ?-COCYCLES ON GLN 4 10 NON-EXISTENCE OF ?-SUBGROUPS OF SIMPLE GROUPS 4 11 NON-EXISTENCE OF INVARIANT ADELIC CONNECTIONS ON GLN CHAPTER 5 CURVATURE AND GALOIS GROUPS OF EHRESMANN CONNECTIONS 5 1 GAUGE AND CURVATURE FORMULAS 5 2 EXISTENCE, UNIQUENESS, AND RATIONALITY OF SOLUTIONS 5 3 GALOIS GROUPS: GENERALITIES 5 4 GALOIS GROUPS: THE GENERIC CASE CHAPTER 6 CURVATURE OF CHERN CONNECTIONS 6 1 ANALYTIC CONTINUATION ALONG TORI 6 2 NON-VANISHING/VANISHING OF CURVATURE VIA ANALYTIC CONTINUATION 6 3 CONVERGENCE ESTIMATES 6 4 THE CASES N = 1 AND M = 6 5 NON-VANISHING/VANISHING OF CURVATURE VIA CORRESPONDENCES CHAPTER 7 CURVATURE OF LEVI-CIVIT`A CONNECTIONS 7 1 THE CASE M = 1: NON-VANISHING OF CURVATURE MOD P 7 2 ANALYTIC CONTINUATION ALONG THE IDENTITY 7 3 MIXED CURVATURE CHAPTER 8 CURVATURE OF LAX CONNECTIONS 8 1 ANALYTIC CONTINUATION ALONG TORSION POINTS 8 2 NON-VANISHING/VANISHING OF CURVATURE CHAPTER 9 OPEN PROBLEMS 9 1 UNIFYING HOLQ AND ?Q 9 2 UNIFYING “?/?P” AND “?/??P?” 9 3 UNIFYING SH AND GLN 9 4 FURTHER CONCEPTS AND COMPUTATIONS 9 5 WHAT LIES AT IN?NITY?