Overview

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.

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9781316519936

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Reviews

'... 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

Author Information

Annette Huber is Professor for Number Theory at Albert-Ludwigs-Universität Freiburg. She works in arithmetic geometry and is a leading specialist in the theory of motives. Together with Stefan Müller-Stach, she authored the book Periods and Nori motives (2017). She was a speaker at the 2002 ICM and is a member of the German National Academy of Sciences, the Leopoldina. Gisbert Wüstholz is Professor Emeritus at ETH Zurich. He is a leading researcher in transcendence theory and diophantine geometry. In 1986, he was an invited speaker at the ICM in Berkeley, in 1992 he gave the Mordell Lecture and in 2001 the Kuwait Foundation Lecture. He is Honorary Professor at Togji University Shanghai and at TU Graz. He is an elected member of four academies including the Leopoldina and has published six books.

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