Transcendence and Linear Relations of 1-Periods

Annette Huber; Gisbert Wüstholz

doi:10.1017/9781009019729

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Cresson, Jacky and Viu-Sos, Juan 2022. On the equality of periods of Kontsevich–Zagier. Journal de théorie des nombres de Bordeaux, Vol. 34, Issue. 2, p. 323.

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Kaiser, Tobias 2024. Periods, power series, and integrated algebraic numbers. Mathematische Annalen, Vol. 390, Issue. 2, p. 2043.

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Fresán, Javier 2024. Une introduction aux périodes. Journées mathématiques X-UPS, p. 1.

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Annette Huber, Albert-Ludwigs-Universität Freiburg, Germany, Gisbert Wüstholz, Eidgenössische Technische Hochschule Zürich

Publisher:: Cambridge University Press
Online publication date:: May 2022
Print publication year:: 2022
Online ISBN:: 9781009019729
DOI:: https://doi.org/10.1017/9781009019729

Subjects:: Mathematics (general), Mathematics, Number Theory, Algebra
Series:: Cambridge Tracts in Mathematics

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Transcendence and Linear Relations of 1-Periods
Annette Huber, Gisbert Wüstholz
Online ISBN: 9781009019729

Book DOI: https://doi.org/10.1017/9781009019729
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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.

‘… 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 Source: zbMATH Open

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