Transcendence and Linear Relations of 1-Periods
Transcendence and Linear Relations of 1-Periods
Gisbert Wüstholz, Eidgenössische Technische Hochschule Zürich
AUD$183.95
inc GSTThis 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.
- Builds a bridge between transcendence, algebraic geometry and cohomology theory, appealing to a wide readership
- Presents the basic concepts and background material on which the theory relies
- Written by leading experts in the area
Reviews & endorsements
'… the book under review is surely a foundational work, which finally settles many open conjectures involving periods of curves. It has also the merit of providing references and proofs for a vast amount of foundational material, including many variants of the theory of motives. As such, it will surely become a standard reference for many works to come.' Riccardo Pengo, zbMATH Open
Product details
May 2022Hardback
9781316519936
263 pages
235 × 157 × 20 mm
0.5kg
Available
Table of Contents
- Prologue
- Acknowledgments
- 1. Introduction
- Part I. 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 II. 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 III. Periods of Algebraic Varieties:
- 12. Periods of algebraic varieties
- 13. Relations between periods
- 14. Vanishing of periods of curves
- Part IV. 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 V. Appendices: A. Nori motives
- B. Voevodsky motives
- C. Comparison of realisations
- List of Notations
- References
- Index.
