Many Variations of Mahler Measures
A Lasting Symphony
Published on 6. May 2020
978-1-108-88919-3 (ISBN)
Description
The Mahler measure is a fascinating notion and an exciting topic in contemporary mathematics, interconnecting with subjects as diverse as number theory, analysis, arithmetic geometry, special functions and random walks. This friendly and concise introduction to the Mahler measure is a valuable resource for both graduate courses and self-study. It provides the reader with the necessary background material, before presenting the recent achievements and the remaining challenges in the field. The first part introduces the univariate Mahler measure and addresses Lehmer's question, and then discusses techniques of reducing multivariate measures to hypergeometric functions. The second part touches on the novelties of the subject, especially the relation with elliptic curves, modular forms and special values of L-functions. Finally, the Appendix presents the modern definition of motivic cohomology and regulator maps, as well as Deligne-Beilinson cohomology. The text includes many exercises to test comprehension and challenge readers of all abilities.
Reviews / Votes
'... the book will serve as a great introduction to the subject of Mahler's measure, in some of its manifold variations, with a special focus on its links with special values of L-functions. It is particularly suited for a student or research seminar, as well as for individual work, because of its concise nature, which emphasizes the most important points of the theory, while not leaving out crucial details when needed.' Riccardo Pengo, zbMATHMore details
Series
Language
English
Place of publication
Cambridge
United Kingdom
Target group
Professional and scholarly
Illustrations
Worked examples or Exercises; 1 Halftones, black and white; 6 Line drawings, black and white
File size
2,12 MB
ISBN-13
978-1-108-88919-3 (9781108889193)
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Schweitzer Classification
Other editions
Additional editions
Book
05/2020
Cambridge University Press
€52.50
Shipment within 15-20 days
Persons
Content
1. Some basics; 2. Lehmer's problem; 3. Multivariate setting; 4. The dilogarithm; 5. Differential equations for families of Mahler measures; 6. Random walk; 7. The regulator map for $K_2$ of curves; 8. Deninger's method for multivariate polynomials; 9. The Rogers-Zudilin method; 10. Modular regulators; Appendix. Motivic cohomology and regulator maps; References; Author Index; Subject index.
