Transcendence and Linear Relations of 1-Periods (Cambridge Tracts in Mathematics)

"This exploration of the relation between periods and transcendental numbers brings Baker's theory of linear forms in logarithms into its most general framework, the theory of 1-motives. Written by leading experts in the field, it contains original results and finalises the theory of linear relations of 1-periods, answering long-standing questions in transcendence theory. It provides a complete exposition of the new theory for researchers, but also serves as an introduction to transcendence for graduate students and newcomers. It begins with foundational material, including a review of the theory of commutative algebraic groups and the analytic subgroup theorem as well as the basics of singular homology and de Rham cohomology. Part II addresses periods of 1-motives, linking back to classical examples like the transcendence of π, before the authors turn to periods of algebraic varieties in Part III. Finally, Part IV aims at a dimension formula for the space of periods of a 1-motive in terms of its data."-- Provided by publisher Frontmatter Dedication Contents Prologue Acknowledgements 1 Introduction PART ONE FOUNDATIONS 2 Basics on Categories 3 Homology and Cohomology 4 Commutative Algebraic Groups 5 Lie Groups 6 The Analytic Subgroup Theorem 7 The Formalism of the Period Conjecture PART TWO PERIODS OF DELIGNE 1-MOTIVES 8 Deligne’s 1-Motives 9 Periods of 1-Motives 10 First Examples 11 On Non-closed Elliptic Periods PART THREE PERIODS OF ALGEBRAIC VARIETIES 12 Periods of Algebraic Varieties 13 Relations Between Periods 14 Vanishing of Periods of Curves PART FOUR DIMENSIONS OF PERIOD SPACES 15 Dimension Computations: An Estimate 16 Structure of the Period Space 17 Incomplete Periods of the Third Kind 18 Elliptic Curves 19 Values of Hypergeometric Functions PART FIVE APPENDICES Appendix A Nori Motives Appendix B Voevodsky Motives Appendix C Comparison of Realisations List of Notation References Index This work answers long-standing open questions in transcendence theory and finalises the theory of linear relations of 1-periods. It serves as a detailed and modern introduction for graduate students and young researchers to the beautiful world of transcendence. The authors include foundational material and link examples back to classical results.