Transcendence and Linear Relations of 1-Periods

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.

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.