Transcendence and Linear Relations of 1-Periods

Leading experts explore the relation between periods and transcendental numbers, using a modern approach derived from the theory of motives.

Annette Huber (Author), Gisbert Wüstholz (Author)

9781316519936, Cambridge University Press

Hardback, published 26 May 2022

263 pages
23.5 x 15.7 x 2 cm, 0.5 kg

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 ?, 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.

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.

Subject Areas: Number theory [PBH], Algebra [PBF]